X-R Solutionis in the Virtual Reality Industry

X-R Solution

xrsolution

Mukul Garg, an entrepreneur and engineering enthusiast, boasts a career spanning over 16 years, marked by a diverse array of accomplishments. He’s been invited to judge Stanford hackathons, the Alameda County Science and Engineering Fair, the Stevie® Awards for Sales & Customer Service, joined the Forbes Tech Council, and even written a feature for the publication. Mukul currently serves as the Head of Support Engineering for PubNub, orchestrating innovative solutions by amalgamating his technical expertise with a staunch dedication to customer satisfaction. He’s a forward-thinking technical leader looking to teach others how to harness the power of AI for the benefit of customers and the support staff who serve them. Today, Mukul Garg has been generous enough to answer some questions to teach us more about how he’s using AI to raise industry standards in customer support. Q: To start with, why don’t you explain what inspires you in your day-to-day? You’ve taken on a very high-tech project that requires not only learning a whole new frontier of software but, in many ways, creating it. A: Throughout my career, I've witnessed a significant disparity between the technical capabilities of products and the actual experience customers receive. This realization sparked a desire to infuse empathy and understanding into the realm of customer support, leveraging AI as a tool to enhance rather than replace human interaction. By harnessing the power of technology, we can streamline processes, anticipate customer needs, and ultimately deliver a more personalized and empathetic experience that truly resonates with users. Q: Can you share a specific example of how you've utilized AI to enhance customer support and experiences? A: In my role, I've spearheaded initiatives aimed at integrating AI-driven solutions into our customer support systems with remarkable results. By automating routine inquiries and tasks, we've been able to free up our support team to focus on more complex issues, resulting in faster response times and higher overall customer satisfaction levels. Q: That feels like one great data point, but there’s a lot more happening on a much larger scale. What can you say about how AI might affect the future of sales and the customer experience? A: The potential impact of AI on the future of customer support is immense, extending far beyond simple automation to encompass a fundamental shift in the way organizations interact with their customers. From predictive support and proactive issue resolution to hyper-personalized recommendations, the possibilities are endless. Q: What role do you see AI playing in setting new industry standards for customer support and go-to-market strategies? A: AI has the potential to redefine industry standards for customer support by raising the bar for efficiency, effectiveness, and personalization. Organizations that leverage AI-driven solutions are able to deliver faster response times, higher accuracy rates, and more personalized interactions than ever before, setting a new benchmark for what constitutes excellent customer support. Q: What advice would you offer to organizations looking to leverage AI to enhance their customer support strategies? A: Over 16 years leading customer-focused teams, I've learned that building strong, capable teams is key to success. This requires fostering collaboration between diverse skill sets like customer success, engineering, and sales while ensuring everyone works towards a shared vision. My advice? Focus on building trust, invest in your team's growth, and celebrate their achievements. Together, you can achieve remarkable things. Mukul Garg's insights into the integration of AI and engineering in customer support highlight evolving strategies for efficiency and personalization. His experiences provide valuable perspectives for organizations adapting to meet evolving user needs with innovative solutions. Mukul Garg, an entrepreneur and engineering enthusiast, boasts a career spanning over 16 years, marked by a diverse array of accomplishments. He’s been invited to judge Stanford hackathons, the Alameda County Science and Engineering Fair, the Stevie® Awards for Sales & Customer Service, joined the Forbes Tech Council, and even written a feature for the publication . Mukul currently serves as the Head of Support Engineering for PubNub, orchestrating innovative solutions by amalgamating his technical expertise with a staunch dedication to customer satisfaction. He’s a forward-thinking technical leader looking to teach others how to harness the power of AI for the benefit of customers and the support staff who serve them. publication publication Today, Mukul Garg has been generous enough to answer some questions to teach us more about how he’s using AI to raise industry standards in customer support. Mukul Garg Mukul Garg Q: To start with, why don’t you explain what inspires you in your day-to-day? You’ve taken on a very high-tech project that requires not only learning a whole new frontier of software but, in many ways, creating it. Q: To start with, why don’t you explain what inspires you in your day-to-day? You’ve taken on a very high-tech project that requires not only learning a whole new frontier of software but, in many ways, creating it. A: Throughout my career, I've witnessed a significant disparity between the technical capabilities of products and the actual experience customers receive. This realization sparked a desire to infuse empathy and understanding into the realm of customer support, leveraging AI as a tool to enhance rather than replace human interaction. By harnessing the power of technology, we can streamline processes, anticipate customer needs, and ultimately deliver a more personalized and empathetic experience that truly resonates with users. A: Throughout my career, I've witnessed a significant disparity between the technical capabilities of products and the actual experience customers receive. This realization sparked a desire to infuse empathy and understanding into the realm of customer support, leveraging AI as a tool to enhance rather than replace human interaction. By harnessing the power of technology, we can streamline processes, anticipate customer needs, and ultimately deliver a more personalized and empathetic experience that truly resonates with users. Q: Can you share a specific example of how you've utilized AI to enhance customer support and experiences? Q: Can you share a specific example of how you've utilized AI to enhance customer support and experiences? A: In my role, I've spearheaded initiatives aimed at integrating AI-driven solutions into our customer support systems with remarkable results. By automating routine inquiries and tasks, we've been able to free up our support team to focus on more complex issues, resulting in faster response times and higher overall customer satisfaction levels. A: In my role, I've spearheaded initiatives aimed at integrating AI-driven solutions into our customer support systems with remarkable results. By automating routine inquiries and tasks, we've been able to free up our support team to focus on more complex issues, resulting in faster response times and higher overall customer satisfaction levels. Q: That feels like one great data point, but there’s a lot more happening on a much larger scale. What can you say about how AI might affect the future of sales and the customer experience? Q: That feels like one great data point, but there’s a lot more happening on a much larger scale. What can you say about how AI might affect the future of sales and the customer experience? A: The potential impact of AI on the future of customer support is immense, extending far beyond simple automation to encompass a fundamental shift in the way organizations interact with their customers. From predictive support and proactive issue resolution to hyper-personalized recommendations, the possibilities are endless. A: The potential impact of AI on the future of customer support is immense, extending far beyond simple automation to encompass a fundamental shift in the way organizations interact with their customers. From predictive support and proactive issue resolution to hyper-personalized recommendations, the possibilities are endless. Q: What role do you see AI playing in setting new industry standards for customer support and go-to-market strategies? Q: What role do you see AI playing in setting new industry standards for customer support and go-to-market strategies? A: AI has the potential to redefine industry standards for customer support by raising the bar for efficiency, effectiveness, and personalization. Organizations that leverage AI-driven solutions are able to deliver faster response times, higher accuracy rates, and more personalized interactions than ever before, setting a new benchmark for what constitutes excellent customer support. A: AI has the potential to redefine industry standards for customer support by raising the bar for efficiency, effectiveness, and personalization. Organizations that leverage AI-driven solutions are able to deliver faster response times, higher accuracy rates, and more personalized interactions than ever before, setting a new benchmark for what constitutes excellent customer support. Q: What advice would you offer to organizations looking to leverage AI to enhance their customer support strategies? Q: What advice would you offer to organizations looking to leverage AI to enhance their customer support strategies? A: Over 16 years leading customer-focused teams, I've learned that building strong, capable teams is key to success. This requires fostering collaboration between diverse skill sets like customer success, engineering, and sales while ensuring everyone works towards a shared vision. My advice? Focus on building trust, invest in your team's growth, and celebrate their achievements. Together, you can achieve remarkable things. A: Over 16 years leading customer-focused teams, I've learned that building strong, capable teams is key to success. This requires fostering collaboration between diverse skill sets like customer success, engineering, and sales while ensuring everyone works towards a shared vision. My advice? Focus on building trust, invest in your team's growth, and celebrate their achievements. Together, you can achieve remarkable things. Mukul Garg's insights into the integration of AI and engineering in customer support highlight evolving strategies for efficiency and personalization. His experiences provide valuable perspectives for organizations adapting to meet evolving user needs with innovative solutions.

Innovating with AI: Mukul Garg's Vision for Future Tech Solutions

AI এর সাথে উদ্ভাবন: ভবিষ্যতের প্রযুক্তি সমাধানের জন্য মুকুল গার্গের দৃষ্টি

Innovation mit KI: Mukul Gargs Vision für zukünftige technische Lösungen

Innovando con IA: la visión de Mukul Garg para las soluciones tecnológicas futuras

Innover avec l'IA : la vision de Mukul Garg pour les futures solutions technologiques

एआई के साथ नवाचार: भविष्य के तकनीकी समाधानों के लिए मुकुल गर्ग का दृष्टिकोण

AI によるイノベーション: ムクル・ガーグの未来のテクノロジーソリューションのビジョン

AI를 통한 혁신: 미래 기술 솔루션에 대한 Mukul Garg의 비전

Inovando com IA: a visão de Mukul Garg para soluções tecnológicas futuras

Инновации с использованием искусственного интеллекта: видение Мукула Гарга будущих технологических решений

Yapay Zeka ile İnovasyon: Mukul Garg'ın Geleceğin Teknoloji Çözümlerine Yönelik Vizyonu

Đổi mới bằng AI: Tầm nhìn của Mukul Garg về các giải pháp công nghệ tương lai

利用人工智能进行创新：Mukul Garg 对未来技术解决方案的愿景

Scaling Ethereum: Data Bloat, Data Availability, and the Cloudless Solution

স্কেলিং ইথেরিয়াম: ডেটা ব্লোট, ডেটা উপলব্ধতা এবং ক্লাউডহীন সমাধান

Skalierung von Ethereum: Datenaufblähung, Datenverfügbarkeit und die Cloudless-Lösung

Escalando Ethereum: exceso de datos, disponibilidad de datos y la solución sin nubes

Mise à l'échelle d'Ethereum : gonflement des données, disponibilité des données et solution sans cloud

एथेरियम का विस्तार: डेटा ब्लोट, डेटा उपलब्धता और क्लाउडलेस समाधान

Ethereum のスケーリング: データの肥大化、データの可用性、クラウドレス ソリューション

Ethereum 확장: 데이터 팽창, 데이터 가용성 및 클라우드리스 솔루션

Dimensionando o Ethereum: inchaço de dados, disponibilidade de dados e a solução sem nuvem

Масштабирование Ethereum: раздувание данных, доступность данных и безоблачное решение

Ethereum'u Ölçeklendirmek: Veri Şişmesi, Veri Kullanılabilirliği ve Bulutsuz Çözüm

Mở rộng quy mô Ethereum: Dữ liệu phình to, Tính sẵn có của Dữ liệu và Giải pháp Không có Đám mây

扩展以太坊：数据膨胀、数据可用性和无云解决方案

Advances in technology are necessary to move us forward, keeping us connected, productive, and efficient in the process. But this also drives innovation in crime, revealing new, enhanced ways for bad actors to steal data, violate intellectual property rights, distribute fake products, and find more methods to profit at the expense of businesses and consumers. In recent years, counterfeiting has become a formidable challenge for brands across various industries—from pharmaceuticals and automotive to luxury goods and electronics. Individual and small-scale retailers without a system of checks and balances in place to swiftly identify, disrupt, and prevent counterfeiters from completing sales are at risk. Plus, the surge in DIY e-commerce businesses (like Amazon Storefront) further complicates the landscape, giving counterfeiters more avenues to infiltrate markets with fake products. But when sales of counterfeit goods are booming in an industry worth $2.3 trillion, it’s hard to ignore the appeal for criminals. In fact, a desire for branded goods at lower costs fuels the drive for fake products: 52% of consumers knowingly purchase counterfeit items and 21% frequently make counterfeit purchases. As criminals continue to push the limits and find ways to infringe on brands, they erode the trust and reputation built by companies and online marketplaces, forcing consumers to navigate authenticity at a much greater scale. One way of addressing these growing concerns is to fight fire with technological fire. Ennoventure is a disruptor in this space, leveraging AI-driven technologies to combat counterfeiting through protective, proactive brand strategies. Driven by anti-counterfeit solutions, these innovations are setting new standards in brand protection, proving to be game-changers in ensuring product authenticity and consumer safety. Rethinking Anti-Counterfeiting Through AI Traditional anti-counterfeit measures often rely on physical markers or tags, like labels, logos, and patterns. These methods are not only vulnerable to duplication, but they also add to production costs and environmental waste. Ennoventure's approach breaks away from these outdated practices by integrating cutting-edge AI technologies that are invisible and impossible to replicate. How is this possible? By embedding these solutions directly into the product packaging or the products themselves, Ennoventure ensures that authenticity verification becomes a seamless part of the consumer experience without the added layers of complexity and cost. The core of Ennoventure's technology is its sophisticated use of invisible encryption. This encryption can be incorporated into any part of a product without altering its design or aesthetics, leaving behind a secure, tamper-proof, and unalterable invisible signature to protect brands and their products. This encryption is only detectable through a specialized mobile application, making it an effective deterrent against counterfeit attempts. The AI component comes into play by continuously learning and adapting to new counterfeiting methods, thus staying several steps ahead of counterfeiters. Chad Kite, Chief Commercial Officer of Ennoventure, explains that using emerging technologies like AI and cryptography empowers brands to develop flexible, strategic digital solutions. "At Ennoventure, we believe that the battle against counterfeiting is not just about keeping counterfeiters at bay, but staying ahead of them," said Chad. "Our Al-driven technology does exactly that – providing an invisible yet invincible shield that integrates seamlessly into any brand's anti-counterfeit effort without any process change or capital expenditure. This is the future of anti-counterfeiting, where advanced protection is covert and integrated naturally into the consumer experience" By equipping brands with comprehensive security features, solutions like Ennoventure protect brand integrity across the supply chain, seamlessly integrating anti-counterfeiting technology into existing brand packaging. Cost-Effective Implementation The list of innovations in the anti-counterfeiting space is long and includes embedding QR codes, using digital printing technologies, implementing anti-piracy algorithms, leveraging GPS technology, and creating unique coding systems to stop counterfeiters. Even with so many options available to brands, one of the most significant barriers to adopting new anti-counterfeiting technologies is the associated cost, particularly the capital expenditures required for new equipment or materials. Ennoventure tackles this challenge head-on by offering solutions that require no capital investment to implement. This approach makes it easier and faster for brands to adopt and removes the financial risk involved in transitioning to new anti-counterfeit technologies. Enhancing Brand Integrity and Consumer Trust At the heart of Ennoventure’s mission is the safety and satisfaction of the consumer. Counterfeit products not only undermine brand reputation but also pose serious risks to consumers—risks that can translate into substantial legal and financial repercussions for brands or physical harm to consumers. In fact, faulty, counterfeit electronics are responsible for over 70 deaths and 350,000 serious injuries each year in the United States alone. Consumers today are more savvy than ever, taking time to research products before making a purchase. In fact, a whopping 89-96% of consumers reportedly spend more than 10 minutes researching online purchases, fielding a flurry of options in the process. By ensuring product authenticity, Ennoventure's solutions help brands protect their consumers and, in turn, solidify their market position and brand value. This trust is crucial in today's market, where consumer loyalty is increasingly dependent on brand transparency and reliability. Global Scalability and Impact Ennoventure's anti-counterfeit solutions are designed for global scalability, allowing brands to protect their products across diverse markets with varying challenges and regulatory environments. Whether navigating complex supply chains or meeting regional compliance requirements, Ennoventure’s platform adapts to meet these needs effectively. Kite highlights the global scope of Ennoventure’s mission, stating, “Our vision extends beyond individual markets. We’re here to revolutionize brand protection on a global scale, making it easier and more cost-effective for brands everywhere to adopt and adapt our solutions. This is how we create a safer marketplace—not just locally, but across borders, ensuring that every product that reaches a consumer is genuine and safe.” For brands looking to expand their reach without compromising on safety, Ennoventure provides the perfect blend of global vision on a personal level. The Road Ahead: AI and the Future of Brand Protection As counterfeiters become more sophisticated, the battle to protect authenticity will only intensify, putting the onus on brands to inform, protect, and secure consumers throughout the buyer journey. Ennoventure continues to innovate in this space, ensuring its technologies are not just relevant for today’s threats but well-positioned to anticipate future threats and advances in counterfeit technology. With ongoing developments in AI and machine learning, and the flexibility to adapt technologies to create modern solutions, Ennoventure is at the forefront, leading the charge for protecting brands and consumers with anti-counterfeit measures. For brands seeking to safeguard their reputation and ensure the integrity of their products, turning to Ennoventure’s advanced technologies isn’t just an option—it’s a necessity. Advances in technology are necessary to move us forward, keeping us connected, productive, and efficient in the process. But this also drives innovation in crime, revealing new, enhanced ways for bad actors to steal data, violate intellectual property rights, distribute fake products, and find more methods to profit at the expense of businesses and consumers. In recent years, counterfeiting has become a formidable challenge for brands across various industries—from pharmaceuticals and automotive to luxury goods and electronics. Individual and small-scale retailers without a system of checks and balances in place to swiftly identify, disrupt, and prevent counterfeiters from completing sales are at risk. Plus, the surge in DIY e-commerce businesses (like Amazon Storefront) further complicates the landscape, giving counterfeiters more avenues to infiltrate markets with fake products. But when sales of counterfeit goods are booming in an industry worth $2.3 trillion , it’s hard to ignore the appeal for criminals. worth $2.3 trillion worth $2.3 trillion In fact, a desire for branded goods at lower costs fuels the drive for fake products: 52% of consumers knowingly purchase counterfeit items and 21% frequently make counterfeit purchases. As criminals continue to push the limits and find ways to infringe on brands, they erode the trust and reputation built by companies and online marketplaces, forcing consumers to navigate authenticity at a much greater scale. 52% of consumers 52% of consumers One way of addressing these growing concerns is to fight fire with technological fire. Ennoventure is a disruptor in this space, leveraging AI-driven technologies to combat counterfeiting through protective, proactive brand strategies. Driven by anti-counterfeit solutions, these innovations are setting new standards in brand protection, proving to be game-changers in ensuring product authenticity and consumer safety. Ennoventure Ennoventure Rethinking Anti-Counterfeiting Through AI Rethinking Anti-Counterfeiting Through AI Traditional anti-counterfeit measures often rely on physical markers or tags, like labels, logos, and patterns. These methods are not only vulnerable to duplication, but they also add to production costs and environmental waste. Ennoventure's approach breaks away from these outdated practices by integrating cutting-edge AI technologies that are invisible and impossible to replicate. How is this possible? By embedding these solutions directly into the product packaging or the products themselves, Ennoventure ensures that authenticity verification becomes a seamless part of the consumer experience without the added layers of complexity and cost. The core of Ennoventure's technology is its sophisticated use of invisible encryption. This encryption can be incorporated into any part of a product without altering its design or aesthetics, leaving behind a secure, tamper-proof, and unalterable invisible signature to protect brands and their products. This encryption is only detectable through a specialized mobile application, making it an effective deterrent against counterfeit attempts. The AI component comes into play by continuously learning and adapting to new counterfeiting methods, thus staying several steps ahead of counterfeiters. Chad Kite, Chief Commercial Officer of Ennoventure, explains that using emerging technologies like AI and cryptography empowers brands to develop flexible, strategic digital solutions. "At Ennoventure, we believe that the battle against counterfeiting is not just about keeping counterfeiters at bay, but staying ahead of them," said Chad. "Our Al-driven technology does exactly that – providing an invisible yet invincible shield that integrates seamlessly into any brand's anti-counterfeit effort without any process change or capital expenditure. This is the future of anti-counterfeiting, where advanced protection is covert and integrated naturally into the consumer experience" "At Ennoventure, we believe that the battle against counterfeiting is not just about keeping counterfeiters at bay, but staying ahead of them," said Chad. "Our Al-driven technology does exactly that – providing an invisible yet invincible shield that integrates seamlessly into any brand's anti-counterfeit effort without any process change or capital expenditure. This is the future of anti-counterfeiting, where advanced protection is covert and integrated naturally into the consumer experience" By equipping brands with comprehensive security features, solutions like Ennoventure protect brand integrity across the supply chain, seamlessly integrating anti-counterfeiting technology into existing brand packaging. Cost-Effective Implementation The list of innovations in the anti-counterfeiting space is long and includes embedding QR codes, using digital printing technologies, implementing anti-piracy algorithms, leveraging GPS technology, and creating unique coding systems to stop counterfeiters. Even with so many options available to brands, one of the most significant barriers to adopting new anti-counterfeiting technologies is the associated cost, particularly the capital expenditures required for new equipment or materials. Ennoventure tackles this challenge head-on by offering solutions that require no capital investment to implement. This approach makes it easier and faster for brands to adopt and removes the financial risk involved in transitioning to new anti-counterfeit technologies. Enhancing Brand Integrity and Consumer Trust At the heart of Ennoventure’s mission is the safety and satisfaction of the consumer. Counterfeit products not only undermine brand reputation but also pose serious risks to consumers—risks that can translate into substantial legal and financial repercussions for brands or physical harm to consumers. In fact, faulty, counterfeit electronics are responsible for over 70 deaths and 350,000 serious injuries each year in the United States alone. counterfeit electronics counterfeit electronics Consumers today are more savvy than ever, taking time to research products before making a purchase. In fact, a whopping 89-96% of consumers reportedly spend more than 10 minutes researching online purchases, fielding a flurry of options in the process. By ensuring product authenticity, Ennoventure's solutions help brands protect their consumers and, in turn, solidify their market position and brand value. This trust is crucial in today's market, where consumer loyalty is increasingly dependent on brand transparency and reliability. 89-96% of consumers 89-96% of consumers brand transparency and reliability. brand transparency and reliability. Global Scalability and Impact Ennoventure's anti-counterfeit solutions are designed for global scalability, allowing brands to protect their products across diverse markets with varying challenges and regulatory environments. Whether navigating complex supply chains or meeting regional compliance requirements, Ennoventure’s platform adapts to meet these needs effectively. Kite highlights the global scope of Ennoventure’s mission, stating, “Our vision extends beyond individual markets. We’re here to revolutionize brand protection on a global scale, making it easier and more cost-effective for brands everywhere to adopt and adapt our solutions. This is how we create a safer marketplace—not just locally, but across borders, ensuring that every product that reaches a consumer is genuine and safe.” Kite highlights the global scope of Ennoventure’s mission, stating, “Our vision extends beyond individual markets. We’re here to revolutionize brand protection on a global scale, making it easier and more cost-effective for brands everywhere to adopt and adapt our solutions. This is how we create a safer marketplace—not just locally, but across borders, ensuring that every product that reaches a consumer is genuine and safe.” For brands looking to expand their reach without compromising on safety, Ennoventure provides the perfect blend of global vision on a personal level. The Road Ahead: AI and the Future of Brand Protection As counterfeiters become more sophisticated, the battle to protect authenticity will only intensify, putting the onus on brands to inform, protect, and secure consumers throughout the buyer journey. Ennoventure continues to innovate in this space, ensuring its technologies are not just relevant for today’s threats but well-positioned to anticipate future threats and advances in counterfeit technology. With ongoing developments in AI and machine learning, and the flexibility to adapt technologies to create modern solutions, Ennoventure is at the forefront, leading the charge for protecting brands and consumers with anti-counterfeit measures. For brands seeking to safeguard their reputation and ensure the integrity of their products, turning to Ennoventure’s advanced technologies isn’t just an option—it’s a necessity.

Ennoventure's AI Advantage: The Future of Anti-Counterfeit Solutions for Enhanced Brand Safety

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References 5. Applications to Calabi-Yau complete intersections Therefore, (5.A) and (5.B) gives an equivalence Proposition 5.1. The restriction of (5.F to a magic window and the functor (5.G) are equivalences. Since the bottom functor is an equivalence by Theorem A.5, so is (5.G). for derived factorization categories is an equivalence by Theorem A.5. The following shows that the equivalences of magic windows generating the group action (5.D) correspond to mutation functors between noncommutative matrix factorizations. Proof. We only show that the left square commutes, since the commutativity of the right one follows from a similar argument. Consider the following diagram commutes, where the vertical equivalences are the compositions of (5.C) and (5.H). Lemma 5.5. There is an isomorphism where the first isomorphism follows from Lemma A.6. This finishes the proof. The following is a generalization of [KO, Theorem 8.5], which we prove by a similar argument as in loc. cit. Lemma 5.6. The following diagram commutes. Thus it is enough to show that there is a natural isomorphism By Lemma 5.6, there is an isomorphism Proof of Corollary 5.3. For simplicity, write Therefore, the assertion follows from Theorem 5.2. This paper is available on arxiv under CC0 1.0 DEED license. [1] Although [HSh] only discusses complexes, there are similar functors and semi-orthogonal decompositions for matrix factorizations by [BFK2], and so a similar argument as in [HSh] works in our setting. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References 5. Applications to Calabi-Yau complete intersections Therefore, (5.A) and (5.B) gives an equivalence Proposition 5.1. The restriction of (5.F Proposition 5.1. The restriction of (5.F to a magic window and the functor (5.G) are equivalences. are equivalences. Since the bottom functor is an equivalence by Theorem A.5, so is (5.G). for derived factorization categories is an equivalence by Theorem A.5. The following shows that the equivalences of magic windows generating the group action (5.D) correspond to mutation functors between noncommutative matrix factorizations. Proof . We only show that the left square commutes, since the commutativity of the right one follows from a similar argument. Consider the following diagram Proof commutes, where the vertical equivalences are the compositions of (5.C) and (5.H). commutes, where the vertical equivalences are the compositions of and Lemma 5.5. There is an isomorphism Lemma 5.5. There is an isomorphism where the first isomorphism follows from Lemma A.6. This finishes the proof. The following is a generalization of [KO, Theorem 8.5], which we prove by a similar argument as in loc. cit. Lemma 5.6. The following diagram commutes. Lemma 5.6. The following diagram commutes. Thus it is enough to show that there is a natural isomorphism By Lemma 5.6, there is an isomorphism Proof of Corollary 5.3. For simplicity, write Proof of Corollary 5.3. Therefore, the assertion follows from Theorem 5.2. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv [1] Although [HSh] only discusses complexes, there are similar functors and semi-orthogonal decompositions for matrix factorizations by [BFK2], and so a similar argument as in [HSh] works in our setting.

Mutations of noncommutative crepant resolutions:  Applications to Calabi-Yau complete intersections

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: ক্যালাবি-ইয়াউ সম্পূর্ণ ইন্টারসেকশনে আবেদন

Mutationen nichtkommutativer Crepant-Auflösungen: Anwendungen auf vollständige Calabi-Yau-Schnittpunkte

Mutaciones de resoluciones crepantes no conmutativas: aplicaciones a intersecciones completas Calabi-Yau

Mutations de résolutions crépantes non commutatives : applications aux intersections complètes de Calabi-Yau

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: कैलाबी-याउ पूर्ण प्रतिच्छेदन पर अनुप्रयोग

非可換クレパント分解の突然変異：カラビ・ヤウ完全交差への応用

비가환적 크레판트 해결의 돌연변이: Calabi-Yau 완전 교차점에 적용

Mutações de resoluções crepantes não comutativas: Aplicações a interseções completas de Calabi-Yau

Мутации некоммутативных крепантных резольвент: приложения к полным пересечениям Калаби-Яу

Değişmeyen crepant çözümlerin mutasyonları: Calabi-Yau tam kavşaklarına uygulamalar

Đột biến của độ phân giải crepant không giao hoán: Ứng dụng cho giao lộ hoàn chỉnh Calabi-Yau

非交换 crepant 解析的突变：应用于 Calabi-Yau 完全交集

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References Appendix B. List of Notation This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References Appendix B. List of Notation This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Mutations of noncommutative crepant resolutions: Appendix B. List of Notation

ননকমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: পরিশিষ্ট বি. নোটেশনের তালিকা

Mutationen nichtkommutativer Crepant-Resolutionen: Anhang B. Liste der Notation

Mutaciones de resoluciones crepantes no conmutativas: Apéndice B. Lista de notación

Mutations de résolutions crépantes non commutatives : Annexe B. Liste des notations

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: परिशिष्ट बी. संकेतन की सूची

非可換クレパント分解の変形: 付録 B. 表記法の一覧

비가환적 crepant 해상도의 돌연변이: 부록 B. 표기법 목록

Mutações de resoluções crepantes não comutativas: Apêndice B. Lista de notação

Мутации некоммутативных крепантных резольвент: Приложение Б. Список обозначений

Değişmeyen crepant çözünürlüklerin mutasyonları: Ek B. Gösterim Listesi

Đột biến của độ phân giải crepant không giao hoán: Phụ lục B. Danh sách ký hiệu

非交换 crepant 解析的突变：附录 B。符号列表

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References 4. Main results 4.1. Wall crossing and tilting equivalence. This section shows that wall-crossings of magic windows correspond to equivalences that are induced by tilting modules. Proof. By Teleman’s quantization theorem [Tel], for all k ∈ Z, the natural restriction map induces an isomorphism of equivalences is commutative. Proof. (1) The adjunction gives an isomorphism Therefore we only need to prove that the right hand sides of (4.E) and (4.F) are isomorphic functors. But this follows from a natural isomorphism Lemma 4.8. Notation is same as above. (2) This also follows from Lemma 3.19 and the fact that µδ,δ′ is a bijection. (3) This is a consequence of (2). For each F ∈ F(δ,δ′) Theorem 4.9. Notation is same as above. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References 4. Main results 4.1. Wall crossing and tilting equivalence. This section shows that wall-crossings of magic windows correspond to equivalences that are induced by tilting modules. Wall crossing and tilting equivalence. Proof. By Teleman’s quantization theorem [Tel], for all k ∈ Z, the natural restriction map induces an isomorphism of equivalences is commutative. of equivalences is commutative. Proof . (1) The adjunction gives an isomorphism Proof Therefore we only need to prove that the right hand sides of (4.E) and (4.F) are isomorphic functors. But this follows from a natural isomorphism Lemma 4.8. Notation is same as above. Lemma 4.8. Notation is same as above. (2) This also follows from Lemma 3.19 and the fact that µδ,δ′ is a bijection. (3) This is a consequence of (2). For each F ∈ F(δ,δ′) Theorem 4.9. Notation is same as above. Theorem 4.9. Notation is same as above. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Mutations of noncommutative crepant resolutions: Main results

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: প্রধান ফলাফল

Mutationen nichtkommutativer Crepant-Auflösungen: Hauptergebnisse

Mutaciones de resoluciones crepantes no conmutativas: principales resultados

Mutations de résolutions crépantes non commutatives : principaux résultats

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: मुख्य परिणाम

非可換クレパント分解の変異: 主な結果

비가환적 크레판트 해상도의 돌연변이: 주요 결과

Mutações de resoluções crepantes não comutativas: Principais resultados

Мутации некоммутативных крепантных резолюций: основные результаты

Değişmeyen crepant çözünürlüklerin mutasyonları: Ana sonuçlar

Đột biến của độ phân giải crepant không giao hoán: Kết quả chính

非交换 crepant 解析的突变：主要结果

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References References [BFK1] M. Ballard, D. Favero and L. Katzarkov, A category of kernels for equivariant factorizations and its implications for Hodge theory. Publ. Math. Inst. Hautes Etudes Sci. ´ 120, 1–111 (2014). 36, 38 [BFK2] M. Ballard, D. Favero and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math. 746, 235–303 (2019). 32, 35 [BDFIK] M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov, Resolutions in factorization categories. Adv. Math. 295, 195–249 (2016). 34, 36 [BLS] D. Bergh, V. A. Lunts, O. M. Schn¨urer1, Geometricity for derived categories of algebraic stacks, Selecta Math. (N.S.) 22 (2016), no. 4, 2535–2568. 9 [Bri] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. 1 [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. 7, 12 [Che] J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Differential Geom. 61 (2002), no. 2, 227–261. 1 [Har1] W. Hara, Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops. Adv. Math.318(2017), 355–410. 2, 4, 31 [Har2] W. Hara, On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists. Matematiche (Catania) 77(2022), no.2, 329–371. 2, 4, 10, 30, 31 [Hal] D. Halpern-Leistner, The derived category of a GIT quotient. J. Amer. Math. Soc. 28 (2015), no. 3, 871–912. 23 [HSa] D. Halpern-Leistner and S. V. Sam, Combinatorial constructions of derived equivalences. J. Amer. Math. Soc. 33, no. 3, 871–912 (2020). 1, 2, 3, 12, 13, 14, 16, 21, 31, 32, 33 [HSh] D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory. Adv. Math. 303, 1264–1299 (2016). 34, 35 [HN] A. Higashitani and Y. Nakajima, Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions. Selecta Math. (N.S.) 25(2019), no.5, Paper No. 78, 25 pp. 2, 4 [Hir1] Y. Hirano, Equivalences of derived factorization categories of gauged Landau-Ginzburg models. Adv. Math. 306, 200–278 (2017). 36 [Hir2] Y. Hirano, Derived Kn¨orrer periodicity and Orlov’s theorem for gauged Landau-Ginzburg models. Compos. Math. 153, no. 5, 973–1007 (2017). 32 [Hir3] Y. Hirano, Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations . SIGMA Symmetry Integrability Geom. Methods Appl. 17, Paper No. 055, 43 pp (2021). 5, 36, 37, 38 [HW1] Y. Hirano and M. Wemyss, Faithful actions from hyperplane arrangements, Geom. Topol. 22 (2018), no. 6, 3395–3433. 1 [HW2] Y. Hirano and M. Wemyss, Stability conditions for 3-fold flops, arXiv:1907.09742. 1, 3 [HR] J. Hall, D. Rydh, Perfect complexes on algebraic stacks, Compos. Math. 153 (2017), no. 11, 2318–2367. 9 [Isi] M. U. Isik, Equivalence of the derived category of a variety with a singularity category, Int. Math. Res. Not. IMRN (2013), no. 12, 2787–2808. 32 [IR] O. Iyama and I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras. Am. J. Math. 130 (4), 1087–1149 (2008). 5 [IW1] O. Iyama and M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities. Invent. Math. 197 (2014), no. 3, 521–586. 1, 2, 6, 7, 8, 11 [IW2] O. Iyama and M. Wemyss, Tits cones intersections and applications, preprint. 3 [Kaw] Y. Kawamata, Flops connect minimal models, Publ. RIMS, 44, 419–423 (2008). 1 [KO] N. Koseki and G. Ouchi, Perverse schobers and Orlov equivalences, Eur. J. Math. 9, no. 2, Paper No. 32, 38 pp (2023). 36 [Nak] Y. Nakajima, Mutations of splitting maximal modifying modules: the case of reflexive polygons. Int. Math. Res. Not. IMRN(2019), no.2, 470–550. 2 [OT] C. Okonek and A. Teleman, Graded tilting for gauged Landau-Ginzubrg models and geometric applications. arXiv:1907.10099. 5, 37 [Pos] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Mem. Amer. Math. Soc. 212 (2011), no. 966. 36 [Shi] I. Shipman, A geometric approach to Orlov’s theorem, Compos. Math. 148, no. 5, 1365-1389 (2012). 32 [SV1] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative resolutions of quotient singularities for reductive groups. Invent. Math. 210, no. 1, 3–67 (2017). 1, 12, 21, 22, 25 [SV2] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities I. Int. Math. Res. Not. IMRN(2020), no.21, 8120–8138. 10 [SV3] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities. II. J. Noncommut. Geom. 14 (2020), no. 1, 73–103. 9 [SV4] S. ˇ Spenko and M. Van den Bergh, ˇ Tilting bundles on hypertoric varieties. Int. Math. Res. Not. IMRN(2021), no.2, 1034–1042. 31 [SV5] S. ˇ Spenko and M. Van den Bergh, J.-P. Bell, ˇ On the noncommutative Bondal-Orlov conjecture for some toric varieties. Math. Z. 300(2022), no.1, 1055–1068. 2 [Sta] The Stacks Project Authors, Stacks Project. https://stacks.math.columbia.edu 12 [Tel] C. Teleman, The quantization conjecture revisited. Ann. of Math. (2) 152 (2000), no. 1, 1–43. 23, 24 [Van1] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. 1 [Van2] M. Van den Bergh, Non-commutative crepant resolutions. The Legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004) 1, 2 [Van3] M. Van den Bergh, Non-commutative crepant resolutions, an overview. arXiv:2207.09703. 1, 6, 26 [Wem] M. Wemyss, Flops and Clusters in the Homological Minimal Model Program, Invent. Math. 211 (2018), no. 2, 435–521. 1, 2, 30, 31 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan Email address: wahei.hara@ipmu.jp Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan Email address: hirano@go.tuat.ac.jp This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References References [BFK1] M. Ballard, D. Favero and L. Katzarkov, A category of kernels for equivariant factorizations and its implications for Hodge theory. Publ. Math. Inst. Hautes Etudes Sci. ´ 120, 1–111 (2014). 36, 38 [BFK2] M. Ballard, D. Favero and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math. 746, 235–303 (2019). 32, 35 [BDFIK] M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov, Resolutions in factorization categories. Adv. Math. 295, 195–249 (2016). 34, 36 [BLS] D. Bergh, V. A. Lunts, O. M. Schn¨urer1, Geometricity for derived categories of algebraic stacks, Selecta Math. (N.S.) 22 (2016), no. 4, 2535–2568. 9 [Bri] T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), no. 3, 613–632. 1 [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. 7, 12 [Che] J.-C. Chen, Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Differential Geom. 61 (2002), no. 2, 227–261. 1 [Har1] W. Hara, Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops. Adv. Math.318(2017), 355–410. 2, 4, 31 [Har2] W. Hara, On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists. Matematiche (Catania) 77(2022), no.2, 329–371. 2, 4, 10, 30, 31 [Hal] D. Halpern-Leistner, The derived category of a GIT quotient. J. Amer. Math. Soc. 28 (2015), no. 3, 871–912. 23 [HSa] D. Halpern-Leistner and S. V. Sam, Combinatorial constructions of derived equivalences. J. Amer. Math. Soc. 33, no. 3, 871–912 (2020). 1, 2, 3, 12, 13, 14, 16, 21, 31, 32, 33 [HSh] D. Halpern-Leistner and I. Shipman, Autoequivalences of derived categories via geometric invariant theory. Adv. Math. 303, 1264–1299 (2016). 34, 35 [HN] A. Higashitani and Y. Nakajima, Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions. Selecta Math. (N.S.) 25(2019), no.5, Paper No. 78, 25 pp. 2, 4 [Hir1] Y. Hirano, Equivalences of derived factorization categories of gauged Landau-Ginzburg models. Adv. Math. 306, 200–278 (2017). 36 [Hir2] Y. Hirano, Derived Kn¨orrer periodicity and Orlov’s theorem for gauged Landau-Ginzburg models. Compos. Math. 153, no. 5, 973–1007 (2017). 32 [Hir3] Y. Hirano, Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations . SIGMA Symmetry Integrability Geom. Methods Appl. 17, Paper No. 055, 43 pp (2021). 5, 36, 37, 38 [HW1] Y. Hirano and M. Wemyss, Faithful actions from hyperplane arrangements, Geom. Topol. 22 (2018), no. 6, 3395–3433. 1 [HW2] Y. Hirano and M. Wemyss, Stability conditions for 3-fold flops, arXiv:1907.09742. 1, 3 [HR] J. Hall, D. Rydh, Perfect complexes on algebraic stacks, Compos. Math. 153 (2017), no. 11, 2318–2367. 9 [Isi] M. U. Isik, Equivalence of the derived category of a variety with a singularity category, Int. Math. Res. Not. IMRN (2013), no. 12, 2787–2808. 32 [IR] O. Iyama and I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras. Am. J. Math. 130 (4), 1087–1149 (2008). 5 [IW1] O. Iyama and M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities. Invent. Math. 197 (2014), no. 3, 521–586. 1, 2, 6, 7, 8, 11 [IW2] O. Iyama and M. Wemyss, Tits cones intersections and applications, preprint. 3 [Kaw] Y. Kawamata, Flops connect minimal models, Publ. RIMS, 44, 419–423 (2008). 1 [KO] N. Koseki and G. Ouchi, Perverse schobers and Orlov equivalences, Eur. J. Math. 9, no. 2, Paper No. 32, 38 pp (2023). 36 [Nak] Y. Nakajima, Mutations of splitting maximal modifying modules: the case of reflexive polygons. Int. Math. Res. Not. IMRN(2019), no.2, 470–550. 2 [OT] C. Okonek and A. Teleman, Graded tilting for gauged Landau-Ginzubrg models and geometric applications. arXiv:1907.10099. 5, 37 [Pos] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Mem. Amer. Math. Soc. 212 (2011), no. 966. 36 [Shi] I. Shipman, A geometric approach to Orlov’s theorem, Compos. Math. 148, no. 5, 1365-1389 (2012). 32 [SV1] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative resolutions of quotient singularities for reductive groups. Invent. Math. 210, no. 1, 3–67 (2017). 1, 12, 21, 22, 25 [SV2] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities I. Int. Math. Res. Not. IMRN(2020), no.21, 8120–8138. 10 [SV3] S. ˇ Spenko and M. Van den Bergh, ˇ Non-commutative crepant resolutions for some toric singularities. II. J. Noncommut. Geom. 14 (2020), no. 1, 73–103. 9 [SV4] S. ˇ Spenko and M. Van den Bergh, ˇ Tilting bundles on hypertoric varieties. Int. Math. Res. Not. IMRN(2021), no.2, 1034–1042. 31 [SV5] S. ˇ Spenko and M. Van den Bergh, J.-P. Bell, ˇ On the noncommutative Bondal-Orlov conjecture for some toric varieties. Math. Z. 300(2022), no.1, 1055–1068. 2 [Sta] The Stacks Project Authors, Stacks Project. https://stacks.math.columbia.edu 12 [Tel] C. Teleman, The quantization conjecture revisited. Ann. of Math. (2) 152 (2000), no. 1, 1–43. 23, 24 [Van1] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. 1 [Van2] M. Van den Bergh, Non-commutative crepant resolutions. The Legacy of Niels Henrik Abel, pp. 749–770. Springer, Berlin (2004) 1, 2 [Van3] M. Van den Bergh, Non-commutative crepant resolutions, an overview. arXiv:2207.09703. 1, 6, 26 [Wem] M. Wemyss, Flops and Clusters in the Homological Minimal Model Program, Invent. Math. 211 (2018), no. 2, 435–521. 1, 2, 30, 31 Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan Email address: wahei.hara@ipmu.jp Email address: wahei.hara@ipmu.jp Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan Email address: hirano@go.tuat.ac.jp Email address: hirano@go.tuat.ac.jp This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Mutations of noncommutative crepant resolutions: References

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: রেফারেন্স

Mutationen nichtkommutativer Crepant-Auflösungen: Referenzen

Mutaciones de resoluciones crepantes no conmutativas: referencias

Mutations de résolutions crépantes non commutatives : références

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: संदर्भ

非可換クレパント解決の突然変異: 参考文献

비가환적 크레판트 해상도의 돌연변이: 참고자료

Mutações de resoluções crepantes não comutativas: Referências

Мутации некоммутативных крепантных резолюций: Литература.

Değişmeyen crepant çözünürlüklerin mutasyonları: Referanslar

Đột biến của độ phân giải crepant không giao hoán: Tài liệu tham khảo

非交换 crepant 解析的突变：参考文献

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References 1. Introduction 1.1. Backgrounds. A crepant resolution is one of the best modifications of singularities. This can be regarded as a higher dimensional analog of minimal resolutions of surface singularities, and in the terminology from minimal model theory, a crepant resolution can be paraphrased as a smooth minimal model of the singularity. As a noncommutative analog of the notion of crepant resolutions, Van den Bergh introduced noncommutative crepant resolutions (= NCCRs) [Van2, Van3]. In both commutative and noncommutative cases, the existence of such a resolution is not always true. There are two large classes of singularities for which the study of NCCRs (and crepant resolutions) is well established. One is the class of quotient singularities arising from quasi-symmetric representations of reductive groups, which was first studied in [SV1], and the other is the class of (3-fold) compound du Val singularities, studied in [Van1, Wem]. To investigate the latter class, Iyama and Wemyss [IW1] introduced an operation called mutation, which produce a new NCCR from the original one. By Kawamata [Kaw], it is known that all minimal models (and hence all crepant resolutions) are connected by iterated flops, and mutations can be regarded as a noncommutative counterpart of flops. Indeed, it is proved in [Wem] that the derived equivalences associated to 3-fold flops, which are established in [Bri, Che], correspond to derived equivalences associated to mutations of NCCRs. This interpretation and the technique of mutations of NCCRs provides the main ingredients for the study of Bridgeland stability conditions for 3-fold flops [HW1, HW2]. The main purpose of this paper is to import such a technology established by [IW1] to deepen the study of NCCRs for quotient singularities arising from quasi-symmetric representations, by studying the combinatorics associated to the representation, and by accessing to the ideas from [HSa, SV1]. 1.2. Exchanges and mutations of modifying modules. The present section, Section 1.3, and Section 1.4 explain the setting of this paper, and recall some terminologies, notations, and known results that are needed to state our results. The precise statements of main results are given in Section 1.5. Let R be a normal equidimensional Gorenstein ring. A finitely generated reflexive Rmodule M is said to be modifying if the endomorphism ring EndR(M) is Cohen-Macaulay as an R-module. A noncommutative crepant resolution (=NCCR) of R is the endomorphism ring Λ = EndR(M) of some modifying R-module M such that the global dimension of Λ is finite. If EndR(M) is an NCCR, we say that M gives an NCCR. The following is one of the central problems about NCCRs. Conjecture 1.1 ([Van2]). Let R be an equidimensional normal Gorenstein ring. Then all crepant resolutions and all NCCRs of R are derived equivalent. In relation to this derived equivalence problem, Iyama and Wemys It is natural to ask if two given NCCRs are connected by (iterated) mutations or not. It is known that, for many types of singularities, their natural NCCRs are actually connected by mutations [Har1, Har2, HN, Nak, SV5, Wem]. One of the main aims of this paper is to present a similar result for NCCRs associated to the quotient of quasi-symmetric representations, which are recalled in the next section. will be constructed, and using this yields The following is our main result. Acknowledgements. W. H. would like to thank Prof. Michael Wemyss for discussions and comments. W. H. was supported by EPSRC grant EP/R034826/1 and by ERC Consolidator Grant 101001227 (MMiMMa). Y. H. was supported by JSPS KAKENHI 19K14502. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References 1. Introduction 1.1. Backgrounds. A crepant resolution is one of the best modifications of singularities. This can be regarded as a higher dimensional analog of minimal resolutions of surface singularities, and in the terminology from minimal model theory, a crepant resolution can be paraphrased as a smooth minimal model of the singularity. 1.1. Backgrounds. As a noncommutative analog of the notion of crepant resolutions, Van den Bergh introduced noncommutative crepant resolutions (= NCCRs) [Van2, Van3]. In both commutative and noncommutative cases, the existence of such a resolution is not always true. There are two large classes of singularities for which the study of NCCRs (and crepant resolutions) is well established. One is the class of quotient singularities arising from quasi-symmetric representations of reductive groups, which was first studied in [SV1], and the other is the class of (3-fold) compound du Val singularities, studied in [Van1, Wem]. To investigate the latter class, Iyama and Wemyss [IW1] introduced an operation called mutation, which produce a new NCCR from the original one. By Kawamata [Kaw], it is known that all minimal models (and hence all crepant resolutions) are connected by iterated flops, and mutations can be regarded as a noncommutative counterpart of flops. Indeed, it is proved in [Wem] that the derived equivalences associated to 3-fold flops, which are established in [Bri, Che], correspond to derived equivalences associated to mutations of NCCRs. This interpretation and the technique of mutations of NCCRs provides the main ingredients for the study of Bridgeland stability conditions for 3-fold flops [HW1, HW2]. The main purpose of this paper is to import such a technology established by [IW1] to deepen the study of NCCRs for quotient singularities arising from quasi-symmetric representations, by studying the combinatorics associated to the representation, and by accessing to the ideas from [HSa, SV1]. 1.2. Exchanges and mutations of modifying modules. The present section, Section 1.3, and Section 1.4 explain the setting of this paper, and recall some terminologies, notations, and known results that are needed to state our results. The precise statements of main results are given in Section 1.5. Exchanges and mutations of modifying modules. Let R be a normal equidimensional Gorenstein ring. A finitely generated reflexive Rmodule M is said to be modifying if the endomorphism ring EndR(M) is Cohen-Macaulay as an R-module. A noncommutative crepant resolution (=NCCR) of R is the endomorphism ring Λ = EndR(M) of some modifying R-module M such that the global dimension of Λ is finite. If EndR(M) is an NCCR, we say that M gives an NCCR. The following is one of the central problems about NCCRs. Conjecture 1.1 ([Van2]). Let R be an equidimensional normal Gorenstein ring. Then all crepant resolutions and all NCCRs of R are derived equivalent. In relation to this derived equivalence problem, Iyama and Wemys Conjecture 1.1 It is natural to ask if two given NCCRs are connected by (iterated) mutations or not. It is known that, for many types of singularities, their natural NCCRs are actually connected by mutations [Har1, Har2, HN, Nak, SV5, Wem]. One of the main aims of this paper is to present a similar result for NCCRs associated to the quotient of quasi-symmetric representations, which are recalled in the next section. will be constructed, and using this yields The following is our main result. Acknowledgements . W. H. would like to thank Prof. Michael Wemyss for discussions and comments. W. H. was supported by EPSRC grant EP/R034826/1 and by ERC Consolidator Grant 101001227 (MMiMMa). Y. H. was supported by JSPS KAKENHI 19K14502. Acknowledgements This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Mutations of noncommutative crepant resolutions: Abstract and Intro

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: অ্যাবস্ট্রাক্ট এবং ইন্ট্রো

Mutationen nichtkommutativer Crepant-Auflösungen: Zusammenfassung und Einführung

Mutaciones de resoluciones crepantes no conmutativas: resumen e introducción

Mutations de résolutions crépantes non commutatives : résumé et introduction

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: सार और परिचय

非可換クレパント解像度の突然変異: 概要と序論

비가환적 크레판트 해결의 돌연변이: 초록 및 소개

Mutações de resoluções crepantes não comutativas: Resumo e introdução

Мутации некоммутативных крепантных резолюций: Аннотация и введение

Değişmeyen crepant çözünürlüklerin mutasyonları: Özet ve Giriş

Đột biến của độ phân giải crepant không giao hoán: Tóm tắt và Giới thiệu

非交换性 crepant 解析的突变：摘要和简介

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References Appendix A. Matrix factorizations This appendix recalls definitions and fundamental properties of derived factorization categories. See [Pos, BFK1, BDFIK, Hir1, Hir3] for more details. where W in the left LG model denotes f ∗W by abuse of notation, and the functor (A.A) defines the right derived functor The following shows an equivariant and factorization version of a tilting equivalence. Lemma A.6 ([BFK1, Proposition 3.20][1]). Assume that the sections s and t ∗ are regular. Then there are isomorphisms This paper is available on arxiv under CC0 1.0 DEED license. [1] There is a typo in the latter assertion in loc. cit. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References Appendix A. Matrix factorizations This appendix recalls definitions and fundamental properties of derived factorization categories. See [Pos, BFK1, BDFIK, Hir1, Hir3] for more details. where W in the left LG model denotes f ∗W by abuse of notation, and the functor (A.A) defines the right derived functor The following shows an equivariant and factorization version of a tilting equivalence. Lemma A.6 ([BFK1, Proposition 3.20][1]). Assume that the sections s and t ∗ are regular. Then there are isomorphisms Lemma A.6 This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv [1] There is a typo in the latter assertion in loc. cit.

Mutations of noncommutative crepant resolutions: Appendix A. Matrix factorizations

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: পরিশিষ্ট A. ম্যাট্রিক্স ফ্যাক্টরাইজেশন

Mutationen nichtkommutativer Crepant-Auflösungen: Anhang A. Matrixfaktorisierungen

Mutaciones de resoluciones crepantes no conmutativas: Apéndice A. Factorizaciones matriciales

Mutations de résolutions crépantes non commutatives : Annexe A. Factorisations matricielles

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: परिशिष्ट A. मैट्रिक्स फैक्टराइजेशन

非可換クレパント分解の突然変異: 付録 A. 行列分解

비가환적 크레판트 해법의 변형: 부록 A. 행렬 인수분해

Mutações de resoluções crepantes não comutativas: Apêndice A. Fatorações de matrizes

Мутации некоммутативных крепантных резольвент: Приложение А. Факторизации матриц

Değişmeyen crepant çözünürlüklerin mutasyonları: Ek A. Matris çarpanlarına ayırma

Đột biến của độ phân giải crepant không giao hoán: Phụ lục A. Phân tích nhân tử ma trận

非交换 crepant 解析的突变：附录 A. 矩阵分解

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References 3. Quasi-symmetric representation and GIT quotient 3.1. Quasi-symmetric representations and magic windows. This section recalls fundamental properties of derived categories of GIT quotients arising from quasi-symmetric representations, which are established in [HSa] and [SV1]. We freely use notation from Section 1.6. and then it associates the GIT quotient stack [Xss(ℓ)/G]. Proposition 3.10 ([HSa, Proposition 6.2]). There is an equivalence of groupoids Proposition 3.13 ([HSa, Proposition 6.5]). There is an equivalence extending the equivalence in Proposition 3.10. (3) This follows from (2). The following is elementary, but we give a proof for the convenience of the reader Proof. If W is trivial, the results are obvious. Thus, assume that W ̸= 1 The following result proves that this map is bijective. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References 3. Quasi-symmetric representation and GIT quotient 3.1. Quasi-symmetric representations and magic windows. This section recalls fundamental properties of derived categories of GIT quotients arising from quasi-symmetric representations, which are established in [HSa] and [SV1]. We freely use notation from Section 1.6. Quasi-symmetric representations and magic windows. and then it associates the GIT quotient stack [Xss(ℓ)/G]. Proposition 3.10 ([HSa, Proposition 6.2]). There is an equivalence of groupoids Proposition 3.10 There is an equivalence of groupoids Proposition 3.13 ([HSa, Proposition 6.5]). There is an equivalence Proposition 3.13 There is an equivalence extending the equivalence in Proposition 3.10. extending the equivalence in Proposition 3.10. (3) This follows from (2). The following is elementary, but we give a proof for the convenience of the reader Proof. If W is trivial, the results are obvious. Thus, assume that W ̸= 1 The following result proves that this map is bijective. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Mutations of noncommutative crepant resolutions: Quasi-symmetric representation and GIT quotient

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: কোয়াসি-সিমেট্রিক রিপ্রেজেন্টেশন এবং জিআইটি ভাগফল

Mutationen nichtkommutativer Crepant-Auflösungen: Quasi-symmetrische Darstellung und GIT-Quotient

Mutaciones de resoluciones crepantes no conmutativas: representación cuasisimétrica y cociente GIT

Mutations de résolutions crépantes non commutatives : représentation quasi-symétrique et quotient GIT

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: अर्ध-सममित प्रतिनिधित्व और जीआईटी भागफल

非可換クレパント分解の突然変異: 準対称表現と GIT 商

비가환적 크레판트 분해능의 돌연변이: 준대칭 표현 및 GIT 지수

Mutações de resoluções crepantes não comutativas: representação quase simétrica e quociente GIT

Мутации некоммутативных крепантных резольвент: квазисимметричное представление и фактор GIT

Değişmeyen crepant çözünürlüklerin mutasyonları: Yarı simetrik gösterim ve GIT bölümü

Đột biến của độ phân giải crepant không giao hoán: Biểu diễn gần như đối xứng và thương số GIT

非交换 crepant 解析的突变：准对称表示和 GIT 商

Authors:
(1) Wahei Hara;
(2) Yuki Hirano. Table of Links Abstract and Intro
Exchanges and Mutations of modifying modules
Quasi-symmetric representation and GIT quotient
Main results
Applications to Calabi-Yau complete intersections
Appendix A. Matrix factorizations
Appendix B. List of Notation
References 2. Exchanges and Mutations of modifying modules 2.1. Noncommutative crepant resolution. The present section recalls the definition of some basic notions that are studied in this article. (1) A reflexive R-module M is called a modifying module if EndR(M) is a (maximal) Cohen-Macaulay R-module. (2) We say that a reflexive module M gives a noncommutative crepant resolution (=NCCR) Λ = EndR(M) if M is modifying and the algebra Λ has finite global dimension. Remark 2.4. Note that our definition of NCCR is different from the one in [Van3] or [IW1]. However, if R is d-sCY, our definition is equivalent to other definitions. See [Van3, Lemma 4.2] or [IW1, Lemma 2.23]. from K ∈ addL such that the induced morphism α ◦ (−): Hom(N, K) → Hom(N, M) is surjective. If L = N, we just call α a right (addL)-approximation of M. A right (add L)N - approximation α: K → M of M is said to be minimal if any endomorphism φ ∈ End(K) satisfying α◦φ = α is an automorphism, and we say that α is reduced if any direct summand K′ of K does not contained in Ker(α). Note that if a right approximation is minimal, it is reduced, and in the case when R is complete local, the converse also holds. Definition 2.6. Let R be a normal d-sCY, and let M, N, L ∈ ref R. Lemma 2.7. Notation is same as above (1) If L ′ ∈ addL, there is an inclusion which remains to be true when restricting to reduced exchanges. (2) If N′ ∈ add N, there is an inclusion which remains to be true when restricting to reduced exchanges. (3) For another full subcategory S ′ ⊆ ref R, there is an inclusion If R is complete local, the similar inclusion also holds for reduced exchanges. Proof. (1), (2) and the first assertion in (3) are obvious. The second assertion in (3) follows from the fact that, if R is complete local, two approximations α: K → M and α ′ : K′ → M′ are reduced if and only if α ⊕ α ′ : K ⊕ K′ → M ⊕ M′ is reduced. Proof. Assume that Hom(N, M ⊕ N) is Cohen-Macaulay, and consider an exact sequence 0 → F Ker α → FK → FM → 0. Now applying the functor Hom(−, FR) to this sequence together with the reflexive equivalence proves that the dual sequence 0 → M∗ → K∗ → (Ker α) is exact. 0 → Hom(FM, FN) → Hom(FK, FN) → Hom(F Ker α, FN) → 0 remains to be exact. Since all modules in the original sequence are reflexive, the reflexive equivalence and the duality yield an isomorphism and similar isomorphisms for K and Ker α, which imply the exactness of the sequence 0 → Hom(N ∗ , M∗ ) → Hom(N ∗ , K∗ ) → Hom(N ∗ ,(Ker α) ∗ ) → 0. Thus the dual morphism K∗ → (Ker α) ∗ is a right (add L ∗ )N∗ -approximation with the kernel M∗ , which proves the first assertion. The second assertion follows from a similar argument. The following says that exchanging a direct summand of a modifying module gives a new modifying module in nice situations. Lemma 2.10. Let M ∈ ref R. The following equivalence holds. M ∈ CM R ⇐⇒ M∗ ∈ CM R Proof. We may assume that R is local. Since M is reflexive, it is enough to show the direction (⇒). Since R is Gorenstein, its injective dimension is finite. Thus the result follows from [BH, Proposition 3.3.3 (b)]. Lemma 2.11. Let R be a Gorenstein normal ring, and let M, N ∈ ref R. Then Proof. It is enough to prove the direction (⇒). Assume that Hom(M, N) ∈ CM R. Then Lemma 2.10 implies that Hom(M, N) ∗ ∈ CM R. But by Lemma [IW1, Lemma 2.9], there is an isomorphism Hom(M, N) ∗ ∼= Hom(N, M), which shows that Hom(N, M) ∈ CM R. The proof for the case when m < 0 is similar. Remark 2.13. Since a right approximation is not unique in general, neither is right/left mutation. However, right/left mutation is unique up to additive closure [IW1, Lemma 6.2], and if R is complete local, minimal mutations are unique up to isomorphism. Theorem 2.14 ([IW1, Proposition 6.5, Theorem 6.8, Theorem 6.10]). Let M ∈ ref R be a modifying R-module. 2.3. Tilting bundles and mutations. This section discusses tilting bundles over algebraic stacks. We start from recalling some basic facts on the derived categories of algebraic stacks. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Wahei Hara; (2) Yuki Hirano. Authors: Authors: (1) Wahei Hara; (2) Yuki Hirano. Table of Links Abstract and Intro Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Main results Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix B. List of Notation References Abstract and Intro Abstract and Intro Exchanges and Mutations of modifying modules Exchanges and Mutations of modifying modules Quasi-symmetric representation and GIT quotient Quasi-symmetric representation and GIT quotient Main results Main results Applications to Calabi-Yau complete intersections Applications to Calabi-Yau complete intersections Appendix A. Matrix factorizations Appendix A. Matrix factorizations Appendix B. List of Notation Appendix B. List of Notation References References 2. Exchanges and Mutations of modifying modules 2.1. Noncommutative crepant resolution. The present section recalls the definition of some basic notions that are studied in this article. Noncommutative crepant resolution. (1) A reflexive R-module M is called a modifying module if EndR(M) is a (maximal) Cohen-Macaulay R-module. (2) We say that a reflexive module M gives a noncommutative crepant resolution (=NCCR) Λ = EndR(M) if M is modifying and the algebra Λ has finite global dimension. Remark 2.4. Note that our definition of NCCR is different from the one in [Van3] or [IW1]. However, if R is d-sCY, our definition is equivalent to other definitions. See [Van3, Lemma 4.2] or [IW1, Lemma 2.23]. Remark 2.4. from K ∈ addL such that the induced morphism α ◦ (−): Hom(N, K) → Hom(N, M) is surjective. If L = N, we just call α a right (addL)-approximation of M. A right (add L)N - approximation α: K → M of M is said to be minimal if any endomorphism φ ∈ End(K) satisfying α◦φ = α is an automorphism, and we say that α is reduced if any direct summand K′ of K does not contained in Ker(α). Note that if a right approximation is minimal, it is reduced, and in the case when R is complete local, the converse also holds. Definition 2.6. Let R be a normal d-sCY, and let M, N, L ∈ ref R. Definition 2.6. Lemma 2.7. Notation is same as above Lemma 2.7. Notation is same as above (1) If L ′ ∈ addL, there is an inclusion If L ′ ∈ addL, there is an inclusion which remains to be true when restricting to reduced exchanges. (2) If N′ ∈ add N, there is an inclusion If N′ ∈ add N, there is an inclusion which remains to be true when restricting to reduced exchanges. which remains to be true when restricting to reduced exchanges. (3) For another full subcategory S ′ ⊆ ref R, there is an inclusion For another full subcategory S ′ ⊆ ref R, there is an inclusion If R is complete local, the similar inclusion also holds for reduced exchanges. If R is complete local, the similar inclusion also holds for reduced exchanges. Proof . (1), (2) and the first assertion in (3) are obvious. The second assertion in (3) follows from the fact that, if R is complete local, two approximations α: K → M and α ′ : K′ → M′ are reduced if and only if α ⊕ α ′ : K ⊕ K′ → M ⊕ M′ is reduced. Proof Proof . Assume that Hom(N, M ⊕ N) is Cohen-Macaulay, and consider an exact sequence Proof 0 → F Ker α → FK → FM → 0. Now applying the functor Hom(−, FR) to this sequence together with the reflexive equivalence proves that the dual sequence 0 → M∗ → K∗ → (Ker α) is exact. 0 → Hom(FM, FN) → Hom(FK, FN) → Hom(F Ker α, FN) → 0 remains to be exact. Since all modules in the original sequence are reflexive, the reflexive equivalence and the duality yield an isomorphism and similar isomorphisms for K and Ker α, which imply the exactness of the sequence 0 → Hom(N ∗ , M∗ ) → Hom(N ∗ , K∗ ) → Hom(N ∗ ,(Ker α) ∗ ) → 0. Thus the dual morphism K∗ → (Ker α) ∗ is a right (add L ∗ )N∗ -approximation with the kernel M∗ , which proves the first assertion. The second assertion follows from a similar argument. The following says that exchanging a direct summand of a modifying module gives a new modifying module in nice situations. Lemma 2.10. Let M ∈ ref R. The following equivalence holds. Lemma 2.10. Let M ∈ ref R. The following equivalence holds. M ∈ CM R ⇐⇒ M∗ ∈ CM R Proof. We may assume that R is local. Since M is reflexive, it is enough to show the direction (⇒). Since R is Gorenstein, its injective dimension is finite. Thus the result follows from [BH, Proposition 3.3.3 (b)]. Lemma 2.11. Let R be a Gorenstein normal ring, and let M, N ∈ ref R. Then Lemma 2.11. Proof . It is enough to prove the direction (⇒). Assume that Hom(M, N) ∈ CM R. Then Lemma 2.10 implies that Hom(M, N) ∗ ∈ CM R. But by Lemma [IW1, Lemma 2.9], there is an isomorphism Hom(M, N) ∗ ∼= Hom(N, M), which shows that Hom(N, M) ∈ CM R. Proof The proof for the case when m < 0 is similar. Remark 2.13. Since a right approximation is not unique in general, neither is right/left mutation. However, right/left mutation is unique up to additive closure [IW1, Lemma 6.2], and if R is complete local, minimal mutations are unique up to isomorphism. Theorem 2.14 ([IW1, Proposition 6.5, Theorem 6.8, Theorem 6.10]). Let M ∈ ref R be a modifying R-module. 2.3. Tilting bundles and mutations. This section discusses tilting bundles over algebraic stacks. We start from recalling some basic facts on the derived categories of algebraic stacks. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

Mutations of noncommutative crepant resolutions: Exchanges & Mutations of modifying modules

নন-কমিউটেটিভ ক্রেপ্যান্ট রেজোলিউশনের মিউটেশন: পরিবর্তনকারী মডিউলগুলির বিনিময় এবং মিউটেশন

Mutationen nichtkommutativer krepanter Auflösungen: Austausch und Mutationen modifizierender Module

Mutaciones de resoluciones crepantes no conmutativas: intercambios y mutaciones de módulos modificadores

Mutations de résolutions crépantes non commutatives : Echanges & Mutations de modules modificateurs

गैर-विनिमेय क्रेपेंट संकल्पों के उत्परिवर्तन: संशोधित मॉड्यूल के आदान-प्रदान और उत्परिवर्तन

非可換クレパント解決の突然変異: 修正モジュールの交換と突然変異

비가환적 크레판트 해결의 돌연변이: 수정 모듈의 교환 및 돌연변이

Mutações de resoluções crepantes não comutativas: Trocas e mutações de módulos modificadores

Мутации некоммутативных крепантных резолюций: обмены и мутации модифицирующих модулей

Değişmeyen crepant çözünürlüklerin mutasyonları: Değiştirici modüllerin Değişimleri ve Mutasyonları

Đột biến của độ phân giải crepant không giao hoán: Trao đổi & Đột biến của các mô-đun sửa đổi

非交换性 crepant 解析的突变：修改模块的交换和突变

I was inspired to write this blog post after reading a paper called "The Oracle Problem — Why Decentralising Everything is More Difficult than it Sounds" by Victor Hogrefe. As the title suggests, Victor argues that there are no good solutions for handling oracles on the blockchain. It may have been true six years ago when the paper was written, so kudos to Victor for proposing some potential ones back then. However, what struck me the most was that the author failed to consider that the oracles themselves can be decentralized. In this post, I'll explain why it's actually "not that difficult" to decentralize everything and how this issue has already been solved with the help of decentralized oracles. But first, let's take a small step back and learn what oracles are exactly. The Role Of Oracles Smart contracts are designed in a way to provide deterministic results. Their job is to take input, execute logic, and modify the state of the blockchain irreversibly. But the majority of events still happen off-chain. Allowing smart contracts to interact with these events has always been a double-edged sword. On one hand, this significantly increases the potential blockchain use cases but on the other —  poses serious security risks and trust issues. Oracles are here to help. They act as a bridge between blockchains and the external world and enable smart contracts to execute code based on real-world events and conditions. For example, to be able to bet on the outcome of a tennis match between Roger Federer and Novak Djokovic, you need a way to tell the smart contract who won. Since we don't live in the Metaverse yet, a smart contract can't know the outcome of the match — someone has to tell it who the winner is, so that the contract can payout to the winners. The big question is: who will be the one to tell the smart contract who won? We would need to trust this person or entity to be correct and honest... But wait a minute... Trust? That's not how things work in Web3. We don't want to trust. We want trustless. Decentralized Oracles In the original paper I mentioned in the beginning, the author overlooks the possibility of oracles being decentralized and is rather focused on discussing different algorithmic solutions, weaknesses, and potential attack vectors. What he doesn't consider is that the oracles themselves can be decentralized. Decentralized oracles, unlike the normal ones, rely on multiple independent nodes to fetch and validate information. This means there's no single point of failure, making it much harder to compromise the system. By using various mechanisms, such as multiple data sources, multiple oracles, or the need to have a stake in the system, decentralized oracles can prevent different sorts of malicious attacks, data corruption, or bribery. One such mechanism has been implemented by the UMA Protocol. UMA's Optimistic Oracles UMA is a leader in the oracles space, alongside Chainlink, which is the largest oracle protocol in the world. UMA's oracle system consists of two main components: the Optimistic Oracle and the Data Verification Mechanism (DVM). The principle is straightforward: anyone can submit an answer on-chain, and a dispute arises only if the answer is incorrect. The first layer of UMA's oracle system is the Optimistic Oracle. This layer is designed to quickly verify data optimistically, meaning that all submitted statements are true by default unless challenged. The second layer is the DVM (Data Verification Mechanism), which serves as a fallback for the Optimistic Oracle by resolving disputes if they occur. The entire system is efficient and cheap because it aims to prevent disputes in the first place. Here’s how it works: A smart contract requests a piece of data and sets a dispute period.
A proposer posts a bond and submits the data.
If no one disputes the data within the dispute period, the data is finalized, and the proposer gets their bond back.
If a dispute arises, the disputer posts a bond equal to the proposer’s bond and escalates the dispute to the DVM.
UMA token holders resolve the dispute within 48 hours. If the disputer is correct, they receive the proposer's bond as a reward, and vice versa. Use Cases Apart from the obvious use cases for oracles in DeFi, UMA's optimistic oracles offer much more flexibility, as practically anything can be submitted as a data point. Here are some projects that utilize UMA's Optimistic Oracles: Polymarket Polymarket is a decentralized predictions market, one of the most prominent DeFi apps lately. On Polymarket, you can place your bets on pretty much anything — from the winner of the Presidential Elections in 2024 to how many tweets Elon Musk will post this month Sherlock Sherlock is a risk management platform designed for DeFi. They run extensive DeFi audit contests where many security researchers compete for a bug bounty. Cozy Cosy is an open-source insurance protocol that allows users to provide and receive protection against predefined conditions, like a loss of funds or smart contract hacks. The use cases are quite diverse already and with the upcoming RWA narrative, the amount of projects using UMA's OOs will only get bigger. RWA Tokenization RWA tokenization is one of the most promising opportunities in crypto. It has the potential to unlock multi-trillion dollar markets and onboard millions of people into DeFi. Major institutions are launching tokenized funds, and the role of Oracles in this market is as important as it's ever been. Blackrock's BUIDL fund uses Chainlink to track real-world assets like US Treasury bill prices, dollar yields, etc. It allows investors to purchase tokens representing shares in the fund with "safe" assets and offers unmatched liquidity. As the sector grows, the demand for fast and accurate Oracle infrastructure and reliable data will grow significantly. Protocols like Chainlink, UMA, and Pyth are well-positioned to capture the majority of this market share. Conclusion We brought smart contracts to the world; now, it's time to bring the world into smart contracts. Allowing smart contracts to interact with the outside world seems like the next logical step in blockchain development. While DeFi remains one of the most promising and straightforward ways to onboard the next billion users, there are many ways in which oracles can be used: they can provide real-world data for various applications, from financial products and insurance to gaming and supply chain management. Decentralized oracles ensure that this data is accurate and trustworthy without relying on a single point of failure, opening a world of new possibilities for the technology. I was inspired to write this blog post after reading a paper called " The Oracle Problem — Why Decentralising Everything is More Difficult than it Sounds" by Victor Hogrefe. The Oracle Problem As the title suggests, Victor argues that there are no good solutions for handling oracles on the blockchain. It may have been true six years ago when the paper was written, so kudos to Victor for proposing some potential ones back then. good solutions However, what struck me the most was that the author failed to consider that the oracles themselves can be decentralized. In this post, I'll explain why it's actually "not that difficult" to decentralize everything and how this issue has already been solved with the help of decentralized oracles. But first, let's take a small step back and learn what oracles are exactly. The Role Of Oracles Smart contracts are designed in a way to provide deterministic results. Their job is to take input, execute logic, and modify the state of the blockchain irreversibly. But the majority of events still happen off-chain. Allowing smart contracts to interact with these events has always been a double-edged sword. On one hand, this significantly increases the potential blockchain use cases but on the other —  poses serious security risks and trust issues. Oracles are here to help. They act as a bridge between blockchains and the external world and enable smart contracts to execute code based on real-world events and conditions. For example, to be able to bet on the outcome of a tennis match between Roger Federer and Novak Djokovic, you need a way to tell the smart contract who won. Since we don't live in the Metaverse yet, a smart contract can't know the outcome of the match — someone has to tell it who the winner is, so that the contract can payout to the winners. know The big question is: who will be the one to tell the smart contract who won? We would need to trust this person or entity to be correct and honest... But wait a minute... Trust? trust Trust? That's not how things work in Web3. We don't want to trust. We want trustless. trustless. Decentralized Oracles In the original paper I mentioned in the beginning, the author overlooks the possibility of oracles being decentralized and is rather focused on discussing different algorithmic solutions, weaknesses, and potential attack vectors. What he doesn't consider is that the oracles themselves can be decentralized. Decentralized oracles, unlike the normal ones, rely on multiple independent nodes to fetch and validate information. This means there's no single point of failure, making it much harder to compromise the system. By using various mechanisms, such as multiple data sources, multiple oracles, or the need to have a stake in the system, decentralized oracles can prevent different sorts of malicious attacks, data corruption, or bribery. One such mechanism has been implemented by the UMA Protocol. UMA's Optimistic Oracles UMA is a leader in the oracles space, alongside Chainlink, which is the largest oracle protocol in the world. UMA's oracle system consists of two main components: the Optimistic Oracle and the Data Verification Mechanism (DVM). The principle is straightforward: anyone can submit an answer on-chain, and a dispute arises only if the answer is incorrect. The first layer of UMA's oracle system is the Optimistic Oracle. This layer is designed to quickly verify data optimistically, meaning that all submitted statements are true by default unless challenged. The second layer is the DVM (Data Verification Mechanism), which serves as a fallback for the Optimistic Oracle by resolving disputes if they occur. The entire system is efficient and cheap because it aims to prevent disputes in the first place. Here’s how it works: A smart contract requests a piece of data and sets a dispute period. A proposer posts a bond and submits the data. If no one disputes the data within the dispute period, the data is finalized, and the proposer gets their bond back. If a dispute arises, the disputer posts a bond equal to the proposer’s bond and escalates the dispute to the DVM. UMA token holders resolve the dispute within 48 hours. If the disputer is correct, they receive the proposer's bond as a reward, and vice versa. A smart contract requests a piece of data and sets a dispute period. A proposer posts a bond and submits the data. If no one disputes the data within the dispute period, the data is finalized, and the proposer gets their bond back. If a dispute arises, the disputer posts a bond equal to the proposer’s bond and escalates the dispute to the DVM. UMA token holders resolve the dispute within 48 hours. If the disputer is correct, they receive the proposer's bond as a reward, and vice versa. Use Cases Apart from the obvious use cases for oracles in DeFi, UMA's optimistic oracles offer much more flexibility, as practically anything can be submitted as a data point. anything Here are some projects that utilize UMA's Optimistic Oracles: Polymarket Polymarket is a decentralized predictions market, one of the most prominent DeFi apps lately. On Polymarket, you can place your bets on pretty much anything — from the winner of the Presidential Elections in 2024 to how many tweets Elon Musk will post this month Sherlock Sherlock is a risk management platform designed for DeFi. They run extensive DeFi audit contests where many security researchers compete for a bug bounty. Cozy Cosy is an open-source insurance protocol that allows users to provide and receive protection against predefined conditions, like a loss of funds or smart contract hacks. The use cases are quite diverse already and with the upcoming RWA narrative, the amount of projects using UMA's OOs will only get bigger. RWA Tokenization RWA tokenization is one of the most promising opportunities in crypto. It has the potential to unlock multi-trillion dollar markets and onboard millions of people into DeFi. Major institutions are launching tokenized funds, and the role of Oracles in this market is as important as it's ever been. Blackrock's BUIDL fund uses Chainlink to track real-world assets like US Treasury bill prices, dollar yields, etc. It allows investors to purchase tokens representing shares in the fund with " safe " assets and offers unmatched liquidity. safe As the sector grows, the demand for fast and accurate Oracle infrastructure and reliable data will grow significantly. Protocols like Chainlink, UMA, and Pyth are well-positioned to capture the majority of this market share. Conclusion We brought smart contracts to the world; now, it's time to bring the world into smart contracts. Allowing smart contracts to interact with the outside world seems like the next logical step in blockchain development. While DeFi remains one of the most promising and straightforward ways to onboard the next billion users, there are many ways in which oracles can be used: they can provide real-world data for various applications, from financial products and insurance to gaming and supply chain management. Decentralized oracles ensure that this data is accurate and trustworthy without relying on a single point of failure, opening a world of new possibilities for the technology.

The Oracle Solution: Why Decentralizing Everything Is Less Difficult Than it Sounds

ওরাকল সমাধান: কেন সবকিছুর বিকেন্দ্রীকরণ শোনার চেয়ে কম কঠিন

Die Oracle-Lösung: Warum die Dezentralisierung des Ganzen weniger schwierig ist, als es klingt

La solución de Oracle: por qué descentralizar todo es menos difícil de lo que parece

La solution Oracle : pourquoi tout décentraliser est moins difficile qu'il n'y paraît

ऑरेकल समाधान: सब कुछ विकेंद्रीकृत करना जितना लगता है उससे कम कठिन क्यों है

Oracle ソリューション: すべてを分散化することが思ったほど難しくない理由

Oracle 솔루션: 모든 것을 분산화하는 것이 생각보다 덜 어려운 이유

A solução Oracle: por que descentralizar tudo é menos difícil do que parece

Решение Oracle: почему децентрализация всего проще, чем кажется

Oracle Çözümü: Her Şeyi Merkezi Olmayan Hale Getirmek Neden Sanıldığından Daha Kolaydır?

Giải pháp của Oracle: Tại sao việc phân cấp mọi thứ lại ít khó khăn hơn bạn tưởng

预言机解决方案：为何去中心化一切并没有听起来那么困难

Developing Next-Gen Data Solutions: SingleStore, MinIO, and the Modern Datalake Stack

নেক্সট-জেন ডেটা সলিউশন তৈরি করা: একক স্টোর, মিনিও এবং আধুনিক ডেটালেক স্ট্যাক

Entwicklung von Datenlösungen der nächsten Generation: SingleStore, MinIO und der moderne Datalake-Stack

Desarrollo de soluciones de datos de próxima generación: SingleStore, MinIO y Modern Datalake Stack

Développer des solutions de données de nouvelle génération : SingleStore, MinIO et la pile Datalake moderne

अगली पीढ़ी के डेटा समाधान विकसित करना: सिंगलस्टोर, मिनियो और आधुनिक डेटालेक स्टैक

次世代データソリューションの開発: SingleStore、MinIO、最新のデータレイク スタック

차세대 데이터 솔루션 개발: SingleStore, MinIO 및 최신 Datalake 스택

Desenvolvendo soluções de dados de última geração: SingleStore, MinIO e Modern Datalake Stack

Разработка решений для обработки данных следующего поколения: SingleStore, MinIO и современный стек Datalake

Yeni Nesil Veri Çözümlerinin Geliştirilmesi: SingleStore, MinIO ve Modern Datalake Stack

Phát triển các giải pháp dữ liệu thế hệ tiếp theo: SingleStore, MinIO và Datalake Stack hiện đại

开发下一代数据解决方案：SingleStore、MinIO 和现代 Datalake 堆栈

By Ben Xiang, SVP, Product & Engineering, Veritone Artificial Intelligence (AI), particularly Large Language Models (LLMs), has emerged as a transformative force in business technology. However, the true value of these advanced AI tools lies in their strategic application to address specific business needs. The first–and critically important–step for businesses is to effectively evaluate their needs and align them with appropriate AI solutions, maximizing the impact of AI integration. The place to begin is by examining business operations through a thorough evaluation of your current challenges and opportunities. This process involves assessing areas where efficiency, accuracy, or innovation could be enhanced. Assessing Current Challenges and Opportunities Start by conducting a comprehensive analysis of your business processes to identify bottlenecks or inefficiencies. Are there repetitive language tasks that consume valuable human resources? Are there obvious “low-hanging fruit” opportunities for innovation in customer engagement or product development that require extensive language processing? For instance, consider a customer service department inundated with routine queries. This scenario often leads to long wait times and overworked staff. Identifying such challenges can highlight the potential for LLMs to automate responses, thus improving efficiency and customer satisfaction. LLMs excel in automating routine tasks and enhancing more complex ones, making customer support a prime area where AI can manage routine inquiries, allowing human agents to focus on more intricate issues. Similarly, in content creation, LLMs can produce high-quality content rapidly, freeing up time for more strategic human activities. For example, an e-commerce company might struggle with creating unique product descriptions for thousands of items. An LLM can generate these descriptions automatically, ensuring consistency and saving significant time and effort. Evaluating Data Availability The success of LLM implementation hinges on access to substantial and relevant data. Businesses must assess whether they generate enough data to feed and train the LLM effectively. Data such as customer interactions, product descriptions, or research reports are invaluable. Consider a healthcare provider with vast amounts of patient data. By leveraging this data, an LLM can assist in diagnosing conditions or providing personalized patient care recommendations, thus enhancing service quality and operational efficiency. Considering Scalability and Impact When choosing use cases, focus on areas where LLMs can significantly improve outcomes or scale operations. For instance, LLMs can process vast amounts of text data in analytics, offering deeper insights and enhancing decision-making. A financial services firm, for instance, might use LLMs to analyze market trends and generate reports. This capability not only speeds up the process but also provides more comprehensive insights, aiding better investment decisions. Common Use Cases of LLMs Across Industries LLMs are versatile and can be applied in numerous sectors. Here are some impactful use cases demonstrating their diverse capabilities: Document Summarization In academic and corporate research settings, LLMs can process extensive texts to extract key information and themes. This significantly reduces the time required to digest large volumes of information, enhancing productivity and decision-making. For example, a law firm could use LLMs to summarize lengthy legal documents, making it easier for lawyers to review key points quickly and efficiently. Creative Writing Assistance Authors can leverage LLMs to generate ideas, suggest narrative structures, and even draft content. This helps overcome writer’s block and enhances creativity, allowing writers to produce more refined and imaginative work. Email and Communication Management LLMs streamline email sorting, drafting, and scheduling, preserving context and personalizing communication. This boosts productivity by managing routine communications efficiently, freeing up time for more critical tasks. Personalized Content Recommendation LLMs analyze user preferences to suggest relevant content, enhancing user engagement on streaming services and online platforms through tailored content suggestions. Technical Support Automation LLMs generate technical support guides and troubleshooting tips, enhancing customer support in technology sectors by providing instant and accurate technical assistance. Copywriting and Advertising LLMs craft creative marketing copy and promotional material, which are crucial for marketing agencies and businesses and help create compelling and targeted advertising content. Scripting and Code Generation LLMs assist developers by generating code scripts, aiding in debugging, streamlining software development processes, and reducing errors. Resume Screening and HR Automation LLMs automate the screening of resumes and job applications, making the hiring process more efficient and fair by integrating with applicant tracking systems and utilizing bias-free evaluation algorithms. Essential Features of LLM Applications Each use case requires specific LLM features to be effective. Here are key features to consider: Conversational Response: Essential for chatbots and virtual assistants, enabling natural, fluid dialogue with users.
Coding and Math Capabilities: Crucial for code generation and mathematical problem-solving in technical applications.
Reasoning and Comprehension: Vital for logical analysis, problem-solving, and understanding complex texts.
Word Knowledge and Fast Response: Important for tasks requiring factual accuracy and real-time interactions.
Grounded Knowledge and Fine-Tuning: Ensures accuracy and relevance, particularly in professional and educational settings.
Extended Context and Content Moderation: Maintains coherence over long interactions and ensures safe, reliable usage. By strategically aligning your business needs with the right LLM applications, you can harness the power of AI to transform operations, enhance efficiency, and drive innovation. The key lies in a thoughtful evaluation of your challenges and opportunities, coupled with a deep understanding of the capabilities of LLMs, ensuring a marriage of business needs with the right AI solutions. About Ben Xiang Ben Xiang is SVP of Corporate Development & Strategy and General Manager of aiWARETM for Veritone, Inc. Ben is responsible for overseeing Veritone’s AI Solutions group, product development and Veritone’s proprietary AI operating system. In addition, he also leads corporate development and strategy. Prior to Veritone, Ben held multiple roles with Ingram Micro as the global executive for the IoT, AI, and Mixed-Reality business as well as roles in strategy and corporate development. He has extensive experience in global business operations and cross-border M&A. About Veritone Veritone (NASDAQ: VERI) designs human-centered AI solutions. Serving customers in the talent acquisition, media, entertainment, and public sector industries, Veritone's software and services empower individuals at the world's largest and most recognizable brands to run more efficiently, accelerate decision-making, and increase profitability. Veritone's leading enterprise AI platform, aiWARETM, orchestrates an ever-growing ecosystem of machine learning models, transforming data sources into actionable intelligence. By blending human expertise with AI technology, Veritone advances human potential to help organizations solve problems and achieve more than ever before, enhancing lives everywhere. By Ben Xiang, SVP, Product & Engineering, Veritone By Ben Xiang, SVP, Product & Engineering, Veritone Veritone Veritone Artificial Intelligence (AI), particularly Large Language Models (LLMs), has emerged as a transformative force in business technology. However, the true value of these advanced AI tools lies in their strategic application to address specific business needs. The first–and critically important–step for businesses is to effectively evaluate their needs and align them with appropriate AI solutions, maximizing the impact of AI integration. The place to begin is by examining business operations through a thorough evaluation of your current challenges and opportunities. This process involves assessing areas where efficiency, accuracy, or innovation could be enhanced. transformative force Assessing Current Challenges and Opportunities Assessing Current Challenges and Opportunities Start by conducting a comprehensive analysis of your business processes to identify bottlenecks or inefficiencies. Are there repetitive language tasks that consume valuable human resources? Are there obvious “low-hanging fruit” opportunities for innovation in customer engagement or product development that require extensive language processing? For instance, consider a customer service department inundated with routine queries. This scenario often leads to long wait times and overworked staff. Identifying such challenges can highlight the potential for LLMs to automate responses, thus improving efficiency and customer satisfaction. LLMs excel in automating routine tasks and enhancing more complex ones, making customer support a prime area where AI can manage routine inquiries, allowing human agents to focus on more intricate issues. Similarly, in content creation, LLMs can produce high-quality content rapidly, freeing up time for more strategic human activities. For example, an e-commerce company might struggle with creating unique product descriptions for thousands of items. An LLM can generate these descriptions automatically, ensuring consistency and saving significant time and effort. Evaluating Data Availability Evaluating Data Availability The success of LLM implementation hinges on access to substantial and relevant data. Businesses must assess whether they generate enough data to feed and train the LLM effectively. Data such as customer interactions, product descriptions, or research reports are invaluable. Consider a healthcare provider with vast amounts of patient data. By leveraging this data, an LLM can assist in diagnosing conditions or providing personalized patient care recommendations, thus enhancing service quality and operational efficiency. Considering Scalability and Impact Considering Scalability and Impact When choosing use cases, focus on areas where LLMs can significantly improve outcomes or scale operations. For instance, LLMs can process vast amounts of text data in analytics, offering deeper insights and enhancing decision-making. A financial services firm, for instance, might use LLMs to analyze market trends and generate reports. This capability not only speeds up the process but also provides more comprehensive insights, aiding better investment decisions. Common Use Cases of LLMs Across Industries Common Use Cases of LLMs Across Industries LLMs are versatile and can be applied in numerous sectors . Here are some impactful use cases demonstrating their diverse capabilities: numerous sectors Document Summarization Document Summarization In academic and corporate research settings, LLMs can process extensive texts to extract key information and themes. This significantly reduces the time required to digest large volumes of information, enhancing productivity and decision-making. For example, a law firm could use LLMs to summarize lengthy legal documents, making it easier for lawyers to review key points quickly and efficiently. Creative Writing Assistance Creative Writing Assistance Authors can leverage LLMs to generate ideas, suggest narrative structures, and even draft content. This helps overcome writer’s block and enhances creativity, allowing writers to produce more refined and imaginative work. Email and Communication Management Email and Communication Management LLMs streamline email sorting, drafting, and scheduling, preserving context and personalizing communication. This boosts productivity by managing routine communications efficiently, freeing up time for more critical tasks. Personalized Content Recommendation Personalized Content Recommendation LLMs analyze user preferences to suggest relevant content, enhancing user engagement on streaming services and online platforms through tailored content suggestions. Technical Support Automation Technical Support Automation LLMs generate technical support guides and troubleshooting tips, enhancing customer support in technology sectors by providing instant and accurate technical assistance. Copywriting and Advertising Copywriting and Advertising LLMs craft creative marketing copy and promotional material, which are crucial for marketing agencies and businesses and help create compelling and targeted advertising content. Scripting and Code Generation Scripting and Code Generation LLMs assist developers by generating code scripts, aiding in debugging, streamlining software development processes, and reducing errors. Resume Screening and HR Automation Resume Screening and HR Automation LLMs automate the screening of resumes and job applications, making the hiring process more efficient and fair by integrating with applicant tracking systems and utilizing bias-free evaluation algorithms. Essential Features of LLM Applications Essential Features of LLM Applications Each use case requires specific LLM features to be effective. Here are key features to consider: Conversational Response: Essential for chatbots and virtual assistants, enabling natural, fluid dialogue with users. Coding and Math Capabilities: Crucial for code generation and mathematical problem-solving in technical applications. Reasoning and Comprehension: Vital for logical analysis, problem-solving, and understanding complex texts. Word Knowledge and Fast Response: Important for tasks requiring factual accuracy and real-time interactions. Grounded Knowledge and Fine-Tuning: Ensures accuracy and relevance, particularly in professional and educational settings. Extended Context and Content Moderation: Maintains coherence over long interactions and ensures safe, reliable usage. Conversational Response: Essential for chatbots and virtual assistants, enabling natural, fluid dialogue with users. Coding and Math Capabilities: Crucial for code generation and mathematical problem-solving in technical applications. Reasoning and Comprehension: Vital for logical analysis, problem-solving, and understanding complex texts. Word Knowledge and Fast Response: Important for tasks requiring factual accuracy and real-time interactions. Grounded Knowledge and Fine-Tuning: Ensures accuracy and relevance, particularly in professional and educational settings. Extended Context and Content Moderation: Maintains coherence over long interactions and ensures safe, reliable usage. By strategically aligning your business needs with the right LLM applications, you can harness the power of AI to transform operations, enhance efficiency, and drive innovation. The key lies in a thoughtful evaluation of your challenges and opportunities, coupled with a deep understanding of the capabilities of LLMs, ensuring a marriage of business needs with the right AI solutions. About Ben Xiang About Ben Xiang About Ben Xiang Ben Xiang is SVP of Corporate Development & Strategy and General Manager of aiWARETM for Veritone, Inc. Ben is responsible for overseeing Veritone’s AI Solutions group, product development and Veritone’s proprietary AI operating system. In addition, he also leads corporate development and strategy. Prior to Veritone, Ben held multiple roles with Ingram Micro as the global executive for the IoT, AI, and Mixed-Reality business as well as roles in strategy and corporate development. He has extensive experience in global business operations and cross-border M&A. About Veritone About Veritone About Veritone Veritone (NASDAQ: VERI) designs human-centered AI solutions. Serving customers in the talent acquisition, media, entertainment, and public sector industries, Veritone's software and services empower individuals at the world's largest and most recognizable brands to run more efficiently, accelerate decision-making, and increase profitability. Veritone's leading enterprise AI platform, aiWARETM, orchestrates an ever-growing ecosystem of machine learning models, transforming data sources into actionable intelligence. By blending human expertise with AI technology, Veritone advances human potential to help organizations solve problems and achieve more than ever before, enhancing lives everywhere. Veritone Veritone

How to Get Started Harnessing the Power of AI: Aligning Business Needs with the Right Solutions

Temporary Lifetime Extension: Mistakes and Solutions

This guide provides a comprehensive overview and step-by-step instructions for integrating AWS Karpenter with Amazon Elastic Kubernetes Service (EKS). It demonstrates how to set up Karpenter, configure it for efficient autoscaling, and verify its functionality in managing compute resources based on real-time application demands. By the end of this guide, you will have a fully functional setup of Karpenter on EKS, optimized for dynamic workload management. Autoscaling with AWS Karpenter on EKS AWS Karpenter represents a significant evolution in autoscaling for Kubernetes environments. Developed by Amazon Web Services, it optimizes scaling clusters efficiently and intelligently. Karpenter stands out with its rapid adjustment to workload demands by provisioning the right types and quantities of instances within minutes. One key benefit of Karpenter is its application-aware scaling. Unlike conventional autoscalers that focus on cluster state and metrics, Karpenter considers the specific needs of the applications running on the cluster. This ensures resources are scaled appropriately, aligning closely with the actual requirements of the workloads. How Karpenter Complements EKS Amazon Elastic Kubernetes Service (EKS) is a widely-used managed Kubernetes service that simplifies running Kubernetes on AWS. Integrating Karpenter with EKS enhances scalability and resource optimization. Karpenter dynamically adjusts compute resources based on workload demands, beneficial for scenarios with fluctuating workloads, such as e-commerce platforms or data processing applications. Additionally, Karpenter's flexibility in handling various AWS instance types allows EKS clusters to use the most cost-effective and suitable resources for their workloads. Prerequisites Before setting up AWS Karpenter for EKS, ensure all necessary prerequisites are in place. This guide is tailored for a Linux environment. AWS Account and EKS Cluster AWS Account: Ensure you have an active AWS account.
Amazon EKS Cluster: You need an existing EKS cluster. Follow the EKS Getting Started Guide for setup. IAM Permissions IAM User with Necessary Permissions: Ensure the IAM user has permissions to manage EKS clusters, EC2 instances, and IAM roles.
IAM Role for Karpenter: Create an IAM role for Karpenter to manage EC2 instances. Tools and Configurations AWS CLI: Install the AWS Command Line Interface (CLI).
kubectl: Install kubectl, the command-line tool for Kubernetes.
Helm: Install Helm, a package manager for Kubernetes. Configure AWS CLI Run aws configure to set your credentials and default region. Update kubeconfig Update your kubeconfig file with the EKS cluster information: aws eks update-kubeconfig --name <Your-EKS-Cluster-Name> Verify Setup Verify access to your EKS cluster using kubectl: kubectl get nodes Environment Variables Set up the following environment variables: export KARPENTER_NAMESPACE=kube-system
export KARPENTER_VERSION=v0.33.0
export K8S_VERSION=1.28
export AWS_PARTITION="aws"
export CLUSTER_NAME="${USER}-karpenter-demo"
export AWS_DEFAULT_REGION="us-east-1"
export AWS_ACCOUNT_ID="$(aws sts get-caller-identity --query Account --output text)"
export TEMPOUT=$(mktemp) Setting Up Karpenter Setting up Karpenter in your EKS cluster involves installing the Karpenter software and configuring the provisioner. Installing Karpenter in EKS Run the following command to install Karpenter: helm upgrade --install karpenter oci://public.ecr.aws/karpenter/karpenter \
  --version "${KARPENTER_VERSION}" --namespace "${KARPENTER_NAMESPACE}" \
  --create-namespace \
  --set "settings.clusterName=${CLUSTER_NAME}" \
  --set "settings.interruptionQueue=${CLUSTER_NAME}" \
  --set controller.resources.requests.cpu=1 \
  --set controller.resources.requests.memory=1Gi \
  --set controller.resources.limits.cpu=1 \
  --set controller.resources.limits.memory=1Gi \
  --wait Node Pool Create NodePool using commands from the official Karpenter guide: https://karpenter.sh/docs/getting-started/getting-started-with-karpenter/#5-create-nodepool Testing and Verification Monitor Cluster Resources: Use tools like Kubernetes Dashboard to track resource usage.
Simulate Load: Test the integration by simulating increased load and observing Karpenter's response: kubectl create deployment nginx-load-generator --image=nginx:1.19.0 --replicas=5 --port=80 --requests='cpu=100m,memory=100Mi' --limits='cpu=200m,memory=200Mi' Monitor Scaling Activities: Use Kubernetes commands and AWS Management Console to monitor scaling activities: kubectl get nodes Check Karpenter logs for insights: kubectl logs -f -n "${KARPENTER_NAMESPACE}" -l app.kubernetes.io/name=karpenter -c controller Scale down: kubectl scale deployment nginx-load-generator --replicas 0 Additional Considerations Best Practice Tip: Regularly review and optimize instance types.
Common Troubleshooting Tip: Handling unscheduled pods. Conclusion We've successfully integrated AWS Karpenter with Amazon EKS, enhancing autoscaling capabilities. This guide covered the essentials from setup to configuration, demonstrating Karpenter's ability to dynamically adjust resources based on real-time application demands. Continue exploring Karpenter's capabilities to keep your Kubernetes deployments agile and efficient. Useful Resources Karpenter Official Documentation
Karpenter Best Practices This guide provides a comprehensive overview and step-by-step instructions for integrating AWS Karpenter with Amazon Elastic Kubernetes Service (EKS). It demonstrates how to set up Karpenter, configure it for efficient autoscaling, and verify its functionality in managing compute resources based on real-time application demands. By the end of this guide, you will have a fully functional setup of Karpenter on EKS, optimized for dynamic workload management. Autoscaling with AWS Karpenter on EKS AWS Karpenter represents a significant evolution in autoscaling for Kubernetes environments. Developed by Amazon Web Services, it optimizes scaling clusters efficiently and intelligently. Karpenter stands out with its rapid adjustment to workload demands by provisioning the right types and quantities of instances within minutes. One key benefit of Karpenter is its application-aware scaling. Unlike conventional autoscalers that focus on cluster state and metrics, Karpenter considers the specific needs of the applications running on the cluster. This ensures resources are scaled appropriately, aligning closely with the actual requirements of the workloads. How Karpenter Complements EKS Amazon Elastic Kubernetes Service (EKS) is a widely-used managed Kubernetes service that simplifies running Kubernetes on AWS. Integrating Karpenter with EKS enhances scalability and resource optimization. Karpenter dynamically adjusts compute resources based on workload demands, beneficial for scenarios with fluctuating workloads, such as e-commerce platforms or data processing applications. Additionally, Karpenter's flexibility in handling various AWS instance types allows EKS clusters to use the most cost-effective and suitable resources for their workloads. Prerequisites Before setting up AWS Karpenter for EKS, ensure all necessary prerequisites are in place. This guide is tailored for a Linux environment. AWS Account and EKS Cluster AWS Account: Ensure you have an active AWS account. Amazon EKS Cluster: You need an existing EKS cluster. Follow the EKS Getting Started Guide for setup. AWS Account : Ensure you have an active AWS account. AWS Account Amazon EKS Cluster : You need an existing EKS cluster. Follow the EKS Getting Started Guide for setup. Amazon EKS Cluster IAM Permissions IAM User with Necessary Permissions: Ensure the IAM user has permissions to manage EKS clusters, EC2 instances, and IAM roles. IAM Role for Karpenter: Create an IAM role for Karpenter to manage EC2 instances. IAM User with Necessary Permissions : Ensure the IAM user has permissions to manage EKS clusters, EC2 instances, and IAM roles. IAM User with Necessary Permissions IAM Role for Karpenter : Create an IAM role for Karpenter to manage EC2 instances. IAM Role for Karpenter Tools and Configurations AWS CLI: Install the AWS Command Line Interface (CLI). kubectl: Install kubectl, the command-line tool for Kubernetes. Helm: Install Helm, a package manager for Kubernetes. AWS CLI : Install the AWS Command Line Interface (CLI). AWS CLI kubectl : Install kubectl, the command-line tool for Kubernetes. kubectl Helm : Install Helm, a package manager for Kubernetes. Helm Configure AWS CLI Run aws configure to set your credentials and default region. aws configure Update kubeconfig Update your kubeconfig file with the EKS cluster information: aws eks update-kubeconfig --name <Your-EKS-Cluster-Name> aws eks update-kubeconfig --name <Your-EKS-Cluster-Name> Verify Setup Verify access to your EKS cluster using kubectl : kubectl kubectl get nodes kubectl get nodes Environment Variables Set up the following environment variables: export KARPENTER_NAMESPACE=kube-system
export KARPENTER_VERSION=v0.33.0
export K8S_VERSION=1.28
export AWS_PARTITION="aws"
export CLUSTER_NAME="${USER}-karpenter-demo"
export AWS_DEFAULT_REGION="us-east-1"
export AWS_ACCOUNT_ID="$(aws sts get-caller-identity --query Account --output text)"
export TEMPOUT=$(mktemp) export KARPENTER_NAMESPACE=kube-system
export KARPENTER_VERSION=v0.33.0
export K8S_VERSION=1.28
export AWS_PARTITION="aws"
export CLUSTER_NAME="${USER}-karpenter-demo"
export AWS_DEFAULT_REGION="us-east-1"
export AWS_ACCOUNT_ID="$(aws sts get-caller-identity --query Account --output text)"
export TEMPOUT=$(mktemp) Setting Up Karpenter Setting up Karpenter in your EKS cluster involves installing the Karpenter software and configuring the provisioner. Installing Karpenter in EKS Run the following command to install Karpenter: helm upgrade --install karpenter oci://public.ecr.aws/karpenter/karpenter \
  --version "${KARPENTER_VERSION}" --namespace "${KARPENTER_NAMESPACE}" \
  --create-namespace \
  --set "settings.clusterName=${CLUSTER_NAME}" \
  --set "settings.interruptionQueue=${CLUSTER_NAME}" \
  --set controller.resources.requests.cpu=1 \
  --set controller.resources.requests.memory=1Gi \
  --set controller.resources.limits.cpu=1 \
  --set controller.resources.limits.memory=1Gi \
  --wait helm upgrade --install karpenter oci://public.ecr.aws/karpenter/karpenter \
  --version "${KARPENTER_VERSION}" --namespace "${KARPENTER_NAMESPACE}" \
  --create-namespace \
  --set "settings.clusterName=${CLUSTER_NAME}" \
  --set "settings.interruptionQueue=${CLUSTER_NAME}" \
  --set controller.resources.requests.cpu=1 \
  --set controller.resources.requests.memory=1Gi \
  --set controller.resources.limits.cpu=1 \
  --set controller.resources.limits.memory=1Gi \
  --wait Node Pool Create NodePool using commands from the official Karpenter guide: https://karpenter.sh/docs/getting-started/getting-started-with-karpenter/#5-create-nodepool https://karpenter.sh/docs/getting-started/getting-started-with-karpenter/#5-create-nodepool Testing and Verification Monitor Cluster Resources: Use tools like Kubernetes Dashboard to track resource usage. Simulate Load: Test the integration by simulating increased load and observing Karpenter's response: Monitor Cluster Resources : Use tools like Kubernetes Dashboard to track resource usage. Monitor Cluster Resources Simulate Load : Test the integration by simulating increased load and observing Karpenter's response: Simulate Load kubectl create deployment nginx-load-generator --image=nginx:1.19.0 --replicas=5 --port=80 --requests='cpu=100m,memory=100Mi' --limits='cpu=200m,memory=200Mi' kubectl create deployment nginx-load-generator --image=nginx:1.19.0 --replicas=5 --port=80 --requests='cpu=100m,memory=100Mi' --limits='cpu=200m,memory=200Mi' Monitor Scaling Activities: Use Kubernetes commands and AWS Management Console to monitor scaling activities: Monitor Scaling Activities : Use Kubernetes commands and AWS Management Console to monitor scaling activities: Monitor Scaling Activities kubectl get nodes kubectl get nodes Check Karpenter logs for insights: kubectl logs -f -n "${KARPENTER_NAMESPACE}" -l app.kubernetes.io/name=karpenter -c controller kubectl logs -f -n "${KARPENTER_NAMESPACE}" -l app.kubernetes.io/name=karpenter -c controller Scale down: Scale down : Scale down kubectl scale deployment nginx-load-generator --replicas 0 kubectl scale deployment nginx-load-generator --replicas 0 Additional Considerations Best Practice Tip: Regularly review and optimize instance types. Common Troubleshooting Tip: Handling unscheduled pods. Best Practice Tip : Regularly review and optimize instance types. Best Practice Tip Common Troubleshooting Tip : Handling unscheduled pods. Common Troubleshooting Tip Conclusion We've successfully integrated AWS Karpenter with Amazon EKS, enhancing autoscaling capabilities. This guide covered the essentials from setup to configuration, demonstrating Karpenter's ability to dynamically adjust resources based on real-time application demands. Continue exploring Karpenter's capabilities to keep your Kubernetes deployments agile and efficient. Useful Resources Karpenter Official Documentation Karpenter Best Practices Karpenter Official Documentation Karpenter Official Documentation Karpenter Best Practices Karpenter Best Practices

Understanding AWS Karpenter: A Modern Autoscaling Solution

**TORRANCE, United States / California, May 22nd, 2024/CyberNewsWire/--**AI SPERA, a leader in Cyber Threat Intelligence (CTI) solutions, announced today that its proprietary search engine, Criminal IP, is now available on the AWS Marketplace. This integration ensures efficient software procurement and deployment, aligning seamlessly with customers' existing cloud architectures. After meeting specific technical and security standards set by AWS, the SaaS-based Criminal IP search engine ensures reliability and seamless integration with AWS services. The AWS Marketplace, a significant platform primarily used in the US, provides Criminal IP with access to a vast global customer base, enhancing its visibility and credibility. This listing demonstrates the critical role of AWS Marketplace in the software's adoption and success. <Criminal IP, a comprehensive threat intelligence tool, is now available on the AWS Marketplace> Criminal IP excels in threat detection, empowering cybersecurity with unparalleled intelligence. Criminal IP is the industry's leading IP address intelligence tool, leveraging AI and machine learning to provide unparalleled visibility into the risks associated with internet-connected devices. It offers comprehensive solutions for fraud detection, attack surface management, and threat hunting. Additionally, Criminal IP offers seamless API integration, allowing effortless incorporation of threat intelligence data into existing services and security systems such as SOAR and SIEM. With a rich repository of cyber threat intelligence data, including risk classification, geographical insights, vulnerable asset graphs, and more, Criminal IP empowers organizations to stay ahead in the ever-evolving landscape of cybersecurity. Seamless Integration and Payment Flexibility Between AWS Marketplace and Criminal IP Criminal IP's presence on the AWS Marketplace brings several conveniences for users. The interconnected tokens of AWS and Criminal IP seamlessly exchange information, allowing users to leverage both platforms' strengths without encountering data silos or compatibility issues. Additionally, customers enjoy consistent plans and subscription options on Criminal IP, regardless of whether transactions are initiated through Criminal IP or the AWS Marketplace. This uniformity extends to credit usage monitoring for specific features and APIs, accessible directly from the dashboard, promoting transparency and ease of management. <Payments on AWS Marketplace seamlessly reflect on Criminal IP> “The most important aspect of entering the AWS Marketplace was to ensure easier compatibility between AWS Cloud and 'Criminal IP' threat intelligence. We paid a lot of attention to interoperability with AWS products and credit management systems,” stated Byungtak Kang, CEO of AI SPERA. “We will continue to pursue Marketplace registration to secure global customers and increase interoperability with various clouds in the future.” Explore the detailed features of the newly listed Criminal IP on the AWS Marketplace, as well as Criminal IP ASM, an Automated Attack Surface Management SaaS solution that monitors all internet-connected assets and vulnerabilities. About AI Spera AI SPERA, a leader in Cyber Threat Intelligence (CTI) solutions, significantly expanded its reach with the launch of its flagship solution, Criminal IP, in 2023. Since then, the company has established technical and business collaborations with over 40 renowned global security firms, including VirusTotal, Cisco, Tenable, Sumo Logic, and Quad9. Available in five languages (English, French, Arabic, Korean, and Japanese), the search engine ensures a powerful service for users worldwide. In addition to the CTI search engine, the company also offers Criminal IP ASM, a SaaS-based Attack Surface Management Solution available on Azure Marketplace, and Criminal IP FDS, an AI-based Anomaly Detection Solution used for credential stuffing prevention and fraud detection. Contact Michael Sena AI SPERA support@aispera.com This story was distributed as a release by Cyberwire under HackerNoon’s Business Blogging Program. Learn more about the program here. **TORRANCE, United States / California, May 22nd, 2024/CyberNewsWire/--**AI SPERA, a leader in Cyber Threat Intelligence (CTI) solutions, announced today that its proprietary search engine, Criminal IP, is now available on the AWS Marketplace. This integration ensures efficient software procurement and deployment, aligning seamlessly with customers' existing cloud architectures. After meeting specific technical and security standards set by AWS, the SaaS-based Criminal IP search engine ensures reliability and seamless integration with AWS services. The AWS Marketplace, a significant platform primarily used in the US, provides Criminal IP with access to a vast global customer base, enhancing its visibility and credibility. This listing demonstrates the critical role of AWS Marketplace in the software's adoption and success. <Criminal IP, a comprehensive threat intelligence tool, is now available on the AWS Marketplace> Criminal IP excels in threat detection, empowering cybersecurity with unparalleled intelligence. Criminal IP is the industry's leading IP address intelligence tool, leveraging AI and machine learning to provide unparalleled visibility into the risks associated with internet-connected devices. It offers comprehensive solutions for fraud detection, attack surface management, and threat hunting. Additionally, Criminal IP offers seamless API integration, allowing effortless incorporation of threat intelligence data into existing services and security systems such as SOAR and SIEM. With a rich repository of cyber threat intelligence data, including risk classification, geographical insights, vulnerable asset graphs, and more, Criminal IP empowers organizations to stay ahead in the ever-evolving landscape of cybersecurity. Seamless Integration and Payment Flexibility Between AWS Marketplace and Criminal IP Criminal IP's presence on the AWS Marketplace brings several conveniences for users. The interconnected tokens of AWS and Criminal IP seamlessly exchange information, allowing users to leverage both platforms' strengths without encountering data silos or compatibility issues. Additionally, customers enjoy consistent plans and subscription options on Criminal IP, regardless of whether transactions are initiated through Criminal IP or the AWS Marketplace. This uniformity extends to credit usage monitoring for specific features and APIs, accessible directly from the dashboard, promoting transparency and ease of management. subscription options subscription options <Payments on AWS Marketplace seamlessly reflect on Criminal IP> “The most important aspect of entering the AWS Marketplace was to ensure easier compatibility between AWS Cloud and 'Criminal IP' threat intelligence. We paid a lot of attention to interoperability with AWS products and credit management systems,” stated Byungtak Kang, CEO of AI SPERA. “We will continue to pursue Marketplace registration to secure global customers and increase interoperability with various clouds in the future.” Explore the detailed features of the newly listed Criminal IP on the AWS Marketplace, as well as Criminal IP ASM, an Automated Attack Surface Management SaaS solution that monitors all internet-connected assets and vulnerabilities. About AI Spera AI SPERA, a leader in Cyber Threat Intelligence (CTI) solutions, significantly expanded its reach with the launch of its flagship solution, Criminal IP, in 2023. Since then, the company has established technical and business collaborations with over 40 renowned global security firms, including VirusTotal, Cisco, Tenable, Sumo Logic, and Quad9. technical and business collaborations technical and business collaborations Available in five languages (English, French, Arabic, Korean, and Japanese), the search engine ensures a powerful service for users worldwide. In addition to the CTI search engine, the company also offers Criminal IP ASM, a SaaS-based Attack Surface Management Solution available on Azure Marketplace, and Criminal IP FDS, an AI-based Anomaly Detection Solution used for credential stuffing prevention and fraud detection. Contact Michael Sena AI SPERA support@aispera.com This story was distributed as a release by Cyberwire under HackerNoon’s Business Blogging Program. Learn more about the program here. This story was distributed as a release by Cyberwire under HackerNoon’s Business Blogging Program. Learn more about the program here . here here

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