Viewing archives for 2025

The P vs NP Problem: an existential question for mathematics

Made Sudan’s Millennium Prize Problem lecture presented at Harvard Science Center, December 3, 2025

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Speaker: Madhu Sudan (Harvard)

What is the Hodge Conjecture

Pierre Deligne’s Millennium Prize Problem lecture presented at Harvard Science Center, November 12, 2025

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Speaker: Pierre Deligne (IAS)

Yang-Mills and the Foundations of Quantum Field Theory

Sourav Chatterjee’s Millennium Prize Problem lecture presented at Harvard Science Center, October 15, 2025

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Speaker: Sourav Chatterjee (Stanford)

Progress on zeta and L-functions motivated by the Riemann hypothesis

I will discuss some of the developments over the last twenty five years in the analytic theory of L-functions, motivated by RH.

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Speaker: Kannan Soundararajan (Stanford)

The Hodge Conjecture: geometry and analysis

William Hodge made his famous conjecture in 1950. It predicted a sharp connection between geometry and analysis. Arguably, the evidence is very limited. But the conjecture has repeatedly led to new ideas and developments, including some spectacular recent advances.

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Speaker: Burt Totaro (UCLA)

Ricci flow after Perelman

In a remarkable achievement, Perelman used Ricci flow to prove Thurston’s Geometrization Conjecture, a central conjecture in 3-manifold topology which includes the Poincare Conjecture as a special case. This breakthrough inspired twenty years of rapid progress in many directions, leading to the resolution of long-standing conjectures in geometry and topology, and opening new vistas in geometric analysis. Beginning with a recap of Perelman’s work, the lecture will survey these developments, and conclude with a discussion of some central open questions.

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Speaker: Bruce Kleiner (NYU)

3-manifolds after Perelman: topology, geometry, and effective rigidity

Our understanding of 3-manifolds has advanced considerably since Perelman’s proof of Thurston’s Geometrization Conjecture, with many original approaches to the conjecture becoming new exciting theorems in their own right. In the hyperbolic setting, however, determining precisely how a topological description of a 3-manifold determines its geometry has remained a central challenge. I’ll review examples and describe some progress toward elucidating geometric information.

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Speaker: Jeff Brock (Yale)

P vs NP

I will review the history and meaning of the P vs NP problem. I will then discuss how much we know, and how little we know, after studying it intensively for the last 50 years.

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Speaker: Avi Wigderson (IAS)

Yang-Mills and the Mass Gap

We will give an introduction to the probabilistic formulation of the Yang-Mills problem and discuss some of the reasons why it is so hard. We will then discuss some of the progress made in this direction over the past five years or so and some hurdles that still need to be overcome.

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Speaker: Martin Hairer (EPFL and Imperial)

The Birch–Swinnerton-Dyer Conjecture: a millennium prize problem at 25

This talk will recall the Birch—Swinnerton-Dyer Conjecture and describe highlights of the progress towards it, both pre- and post-millennium.

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Speaker: Chris Skinner (Princeton)

A report on the Navier-Stokes problem

We will discuss advances and continuing challenges in the mathematical analysis of the Navier–Stokes equations and related problem.

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Speaker: Vladimir Šverak (Minnesota)

The Poincaré Conjecture and Mathematical Discovery

Michael Freedman’s Millennium Prize Problem lecture presented at Harvard Science Center, September 17, 2025

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Speaker: Michael Freedman (Harvard)