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DTSTART;TZID=America/New_York:20260204T170000
DTEND;TZID=America/New_York:20260204T180000
DTSTAMP:20260424T171915
CREATED:20250409T160357Z
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SUMMARY:Millennium Prize Problems Lecture - Barry Mazur: About the Birch and Swinnerton–Dyer Conjecture
DESCRIPTION:Date: February 4\, 2026 \nTime: 5:00–6:00 pm \nLocation: Harvard Science Center Hall C\, 1 Oxford St.\, Cambridge MA \nSpeaker: Barry Mazur\, Harvard University \nTitle: About the Birch and Swinnerton–Dyer Conjecture \nAbstract: \nIn the 1950s Bryan Birch and Peter Swinnerton–Dyer made computations that suggested a striking connection between a basic global invariant of an elliptic curve E over the field of rational numbers (namely\, the rank of its group of rational points) and certain asymptotics of its local arithmetic invariants (i.e.\, the number of its rational points over finite fields). \nThis initial observation has evolved into their conjecture. My lecture will be an introduction to the general ideas behind its ever-expanding development. \nRead more about the Birch and Swinnerton–Dyer Conjecture at the Clay Math website. \n  \nOrganizers: Martin Bridson\, Clay Mathematics Institute | Dan Freed\, Harvard University and CMSA | Mike Hopkins\, Harvard University \nBarry Mazur joined the Harvard University faculty in 1959 as a Junior Fellow in the Society of Fellows and advanced through the ranks to become the Gerhard Gade University Professor of Mathematics\, a position he has held since 1998. During his tenure at Harvard\, he has mentored 60 doctoral students and served as a pivotal figure in bridging topology and number theory\, notably through his classification of the possible torsion subgroups of elliptic curves over the rational numbers (Mazur’s torsion theorem)\, which identifies exactly 15 possible finite groups. This theorem\, detailed in his 1977 paper “Modular curves and the Eisenstein ideal\,” provided crucial insights into the Taniyama-Shimura conjecture and laid groundwork for Andrew Wiles’s 1994 proof of Fermat’s Last Theorem. \nHis broader research includes seminal works on étale homotopy theory (co-authored with Michael Artin in 1969)\, the arithmetic moduli of elliptic curves (with Nicholas M. Katz in 1985)\, and the Iwasawa main conjecture (proved with Andrew Wiles in 1984)\, as well as advancements in p-adic L-functions and the formulation of the Fontaine-Mazur conjecture on Galois representations. Mazur’s influence extends to public communication of mathematics; he has authored books like Imagining Numbers (2003)\, exploring historical perspectives on complex numbers. \nAmong his numerous honors\, Mazur received the Cole Prize in Number Theory from the American Mathematical Society in 1982\, the Chauvenet Prize in 1994 for expository writing\, the Leroy P. Steele Prize for Lifetime Achievement in 2000\, and election to the National Academy of Sciences in 1982. In 2011 (presented in 2013)\, he was awarded the National Medal of Science by President Barack Obama for his pioneering work in these fields.Most recently\, in 2022\, he received the Chern Medal from the International Mathematical Union\, recognizing his profound discoveries and mentorship. \n  \n\n                   \n\nMillennium Prize Problems Lecture Series \n 
URL:https://cmsa.fas.harvard.edu/event/clay_2426/
LOCATION:Harvard Science Center Hall D\, 1 Oxford Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Millennium Prize Problems Lecture,Special Lectures
ATTACH;FMTTYPE=image/jpeg:https://cmsa.fas.harvard.edu/media/Mazur_AD.hallc_.web_.jpg
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20251203T170000
DTEND;TZID=America/New_York:20251203T180000
DTSTAMP:20260424T171915
CREATED:20250409T160258Z
LAST-MODIFIED:20251205T171720Z
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SUMMARY:Millennium Prize Problems Lecture - Madhu Sudan: P vs NP Problem
DESCRIPTION:Pamphlet (pdf) \nSlides (pdf) \nDate: December 3\, 2025 \nTime: 5:00–6:00 pm \nLocation: Harvard Science Center Hall D\, 1 Oxford St.\, Cambridge MA \nSpeaker: Madhu Sudan\, Harvard University \nTitle: The P vs. NP problem: An Existential Question for Mathematics \nAt the beginning of the twentieth century\, in response to questions raised by Hilbert\, illustrious mathematicians such as Godel\, Church and Turing formalized the notion of theorems and proofs. Proofs were automatically verifiable while theorems are logical propositions for which proofs exist. The formal definition of a computer\, a definition that had strong influence on the later development of the technology\, was a by-product of the effort to define the phrase “automatically verifiable”! \nWhile the resulting theory had major implications already\, one notion was however missing in the early definitions. Proofs were meant to be easily verifiable\, while determining the truth of a proposition/conjecture (arguably a core task of mathematics) was not necessarily so. But what is “easiness” and how is it to be defined? While this was already hinted at by Godel in the 50s\, the notion was finally formalized in seminal works of Cook\, Levin and Karp in the early 70s. Central notions here included the adoption of the notion that polynomial time algorithms are (the only) tractable ones\, and the realization that algorithms seeking to remove the existential quantifier in the definition of a “theorem” lead naively to exponential time algorithms. But are there no sophisticated algorithms to search for proofs? This is the profound “Is P = NP?” question. \nIn this talk we will introduce the question and explain implications of resolutions of this question to the modern computing infrastructure\, to mathematics and other sciences. We will briefly describe the state of progress on this question and recent progress on weaker forms of this question. Finally we will also aim to connect this question\, and why one may believe that P != NP (proof search can not be automated) even in the face of accumulating evidence on the ability of computers to solve more and more complex mathematical problems\, which seem to implement brute force search in less than polynomial time. \n  \nRead more about the P vs NP Problem at the Clay Math website. \n  \nOrganizers: Martin Bridson\, Clay Mathematics Institute | Dan Freed\, Harvard University and CMSA | Mike Hopkins\, Harvard University \n\n                   \n\nMillennium Prize Problems Lecture Series
URL:https://cmsa.fas.harvard.edu/event/clay_12325/
LOCATION:Harvard Science Center Hall D\, 1 Oxford Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Millennium Prize Problems Lecture,Special Lectures
ATTACH;FMTTYPE=image/jpeg:https://cmsa.fas.harvard.edu/media/Sudan_web-ad_CROP-scaled.jpg
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20251112T170000
DTEND;TZID=America/New_York:20251112T180000
DTSTAMP:20260424T171915
CREATED:20250311T134920Z
LAST-MODIFIED:20251201T154039Z
UID:10003658-1762966800-1762970400@cmsa.fas.harvard.edu
SUMMARY:Millennium Prize Problems Lecture - Pierre Deligne: What is the Hodge conjecture?
DESCRIPTION:  \n \nDate: November 12\, 2025 \nTime: 5:00–6:00 pm \nLocation: Harvard Science Center Hall D\, 1 Oxford St.\, Cambridge MA \nSpeaker: Pierre Deligne\, Institute for Advanced Study \nTitle: What is the Hodge conjecture? \nAbstract: The Hodge conjecture is about projective non-singular complex algebraic varieties. It characterizes the cohomology classes coming from algebraic cycles. I will explain these terms\, tell why the conjecture is so hard to attack\, and why we care. \n  \nSeries Pamphlet (pdf) \nRead more about the Hodge Conjecture at the Clay Math website. \nOrganizers: Martin Bridson\, Clay Mathematics Institute | Dan Freed\, Harvard University and CMSA | Mike Hopkins\, Harvard University \n\n                   \n\nMillennium Prize Problems Lecture Series
URL:https://cmsa.fas.harvard.edu/event/clay_111225/
LOCATION:Harvard Science Center Hall D\, 1 Oxford Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Millennium Prize Problems Lecture,Special Lectures
ATTACH;FMTTYPE=image/jpeg:https://cmsa.fas.harvard.edu/media/Deligne_web-ad-scaled.jpg
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20251015T170000
DTEND;TZID=America/New_York:20251015T180000
DTSTAMP:20260424T171915
CREATED:20250311T134919Z
LAST-MODIFIED:20251021T134849Z
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SUMMARY:Millennium Prize Problems Lecture - Sourav Chatterjee: Yang-Mills and the foundations of quantum field theory
DESCRIPTION:Millennium Prize Problems Lecture  \nDate: October 15\, 2025 \nTime: 5:00–6:00 pm \nLocation: Harvard Science Center Hall D\, 1 Oxford St.\, Cambridge MA \nSpeaker: Sourav Chatterjee\, Stanford University \nTitle: Yang-Mills and the foundations of quantum field theory \nAbstract: Yang-Mills theories are the building blocks of the Standard Model of quantum mechanics\, which is the best available model for our universe at the quantum scale. Yet\, these theories do not have a rigorous mathematical foundation. Physical calculations are based on perturbation theory\, but there are various phenomena that are believed to be out of the reach of perturbative arguments. Building a mathematical foundation is\, therefore\, important even from the physics point of view. A program with this objective\, known as “constructive field theory”\, was initiated in the 1960s. In spite of many successes\, the program has not reached its original goal. Completing this program is the Clay Millennium Prize problem of Yang-Mills existence and mass gap. I will give a general introduction to the main questions\, and an overview of exciting recent progress that has rejuvenated the quest for a solution in the last ten years. \nRead more about the Yang-Mills Existence and Mass Gap at the Clay Math website. \nOrganizers: Martin Bridson\, Clay Mathematics Institute | Dan Freed\, Harvard University and CMSA | Mike Hopkins\, Harvard University \n\n                   \n\nMillennium Prize Problems Lecture Series
URL:https://cmsa.fas.harvard.edu/event/clay_101425/
LOCATION:Harvard Science Center Hall D\, 1 Oxford Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Millennium Prize Problems Lecture,Special Lectures
ATTACH;FMTTYPE=image/jpeg:https://cmsa.fas.harvard.edu/media/Chatterjee_web_ad.2-scaled.jpg
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250917T170000
DTEND;TZID=America/New_York:20250917T180000
DTSTAMP:20260424T171915
CREATED:20250311T134916Z
LAST-MODIFIED:20251010T115024Z
UID:10003656-1758128400-1758132000@cmsa.fas.harvard.edu
SUMMARY:Millennium Prize Problems Lecture - Michael Freedman: The Poincaré Conjecture and Mathematical Discovery  
DESCRIPTION:Millennium Prize Problems Lecture\nDate: September 17\, 2025 \nLocation: Harvard Science Center Hall D & via Zoom Webinar \nTime: 5:00–6:00 pm \nSpeaker: Michael Freedman\, Harvard CMSA and Logical Intelligence  \nTitle: The Poincaré Conjecture and Mathematical Discovery   \nAbstract: The AI age requires us to re-examine what mathematics is about. The Seven Millenium Problems provide an ideal lens for doing so. Five of the seven are core mathematical questions\, two are meta-mathematical – asking about the scope of mathematics. The Poincare conjecture represents one of the core subjects\, manifold topology. I’ll explain what it is about\, its broader context\, and why people cared so much about finding a solution\, which ultimately arrived through the work of R. Hamilton and G. Perelman. Although stated in manifold topology\, the proof requires vast developments in the theory of parabolic partial differential equations\, some of which I will sketch. Like most powerful techniques\, the methods survive their original objectives and are now deployed widely in both three- and four-dimensional manifold topology.  \n  \nRead more about the Poincaré Conjecture at the Clay Math website. \nOrganizers: Martin Bridson\, Clay Mathematics Institute | Dan Freed\, Harvard University and CMSA | Mike Hopkins\, Harvard University \n\n                   \n\nMillennium Prize Problems Lecture Series
URL:https://cmsa.fas.harvard.edu/event/clay_91725/
LOCATION:Harvard Science Center Hall D\, 1 Oxford Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Millennium Prize Problems Lecture,Special Lectures
ATTACH;FMTTYPE=image/jpeg:https://cmsa.fas.harvard.edu/media/Freedman_web_ad.jpg
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