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)
Home — Clay Research Conference
I will discuss some of the developments over the last twenty five years in the analytic theory of L-functions, motivated by RH.
Speaker: Kannan Soundararajan (Stanford)
Home — Clay Research Conference
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.
Speaker: Burt Totaro (UCLA)
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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.
Speaker: Bruce Kleiner (NYU)
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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.
Speaker: Jeff Brock (Yale)
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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.
Speaker: Avi Wigderson (IAS)
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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.
Speaker: Martin Hairer (EPFL and Imperial)
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This talk will recall the Birch—Swinnerton-Dyer Conjecture and describe highlights of the progress towards it, both pre- and post-millennium.
Speaker: Chris Skinner (Princeton)
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We will discuss advances and continuing challenges in the mathematical analysis of the Navier–Stokes equations and related problem.
Speaker: Vladimir Šverak (Minnesota)
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Abstract: The complex singular cohomology groups of a projective manifold can be described in at least three ways via the de Rham Theorem and the Hodge Decomposition. By taking into account integral cohomology, we obtain three different descriptions of the singular cohomology groups with coefficients in the non-zero complex numbers GL1. Now, replace GL1 with a complex algebraic reductive group G, e.g. GLn. The Non Abelian Hodge Theory of Corlette, Simpson et al. establishes a natural homeomorphism between three distinct complex algebraic varieties parametrizing three different kinds of structures on the projective manifold associated with the reductive group: representations of the fundamental group into G, flat algebraic G-connections, G-Higgs bundles. The case G=GL1, which is Abelian, recaptures the de Rham and Hodge Decomposition. The three complex algebraic varieties on Non Abelian Hodge Theory have naturally isomorphic cohomology groups. However, by taking into account their distinct structures as algebraic varieties, the cohomology groups carry additional distinct structures. The P=W Conjecture seeks to relate two of these structures, at least in the case of compact Riemann surfaces. This talk is devoted to introducing the audience to this circle of ideas and related developments.
Speaker: Mark Andrea de Cataldo (Stony Brook University)
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Abstract: Shimura varieties are highly symmetric algebraic varieties that play an important role in the Langlands program. In the first part of this talk, I will try to give you a sense of what they are like, with a focus on their different kinds of symmetries. In the second part of the talk, I will survey a recent class of results about the vanishing of the cohomology of Shimura varieties with torsion coefficients. To give you a sense of the breadth of the subject, I will mention both connections to the geometric Langlands program and applications to long-standing problems, such as the modularity of elliptic curves.
Speaker: Ana Caraiani (Imperial College London)
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Abstract: I will explain developments in the study of cycles on the moduli space of abelian varieties with connections to the moduli space of curves, the cohomology of the Lagrangian Grassmannian, and the quantum cohomology of the Hilbert scheme of points of the plane.
Speaker: Rahul Pandharipande (ETH Zürich)
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Speaker: Philippe Michel (EPFL)