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“Elliptic Curves, Heegner Points, and Beyond”

Special Seminar

Come join us Wednesday, February 14, 2018 at 12 pm in RM 507, Professor Chao Li (Columbia University) will be giving a special lecture about “Elliptic Curves, Heegner Points, and Beyond”.

Abstract

An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which led to the celebrated conjecture of Birch and Swinnerton-Dyer. We will start with a history of this problem and then discuss our recent work (with D. Kriz) on certain families of elliptic curves (such as y^2=x^3-d), which in particular proves a conjecture of Goldfeld. Our approach uses Heegner points and their deep connection with L-functions. We will explain these key ingredients and ideas. If time permits, we will also illustrate a framework to go beyond the case of Heegner points and to build a connection with L-functions of more general symplectic motives.

Mathematics Hall, Room 507

Wednesday, February 14, 2018 at 12 pm

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“The Threshold Theorem for the Hyperbolic Yang-Mills Equation”

Special Seminar

Come join us Monday February 12, 2018 at 12 pm in RM 507, Professor Sung-Jin Oh (KIAS) will be giving a special lecture about “The Threshold Theorem for the Hyperbolic Yang-Mills Equation”.

Abstract

In this lecture, I will present the recent proof (joint with D. Tataru) of the threshold theorem for the energy critical hyperbolic Yang-Mills equation in (4+1) dimensions. This theorem provides a sharp criterion for global existence and scattering in terms of the energy of the initial data. Moreover, we prove that failure of global existence/scattering is characterized by “bubbling” of a solution to the harmonic Yang-Mills equation.

Our proof lies at the intersection of many recent developments, such as null form estimates and function spaces; parametrix construction via pseudodifferential gauge renormalization; induction on energy; monotonicity formulae arising from the normalized scaling vector field, etc. Also of note is the use of the associated parabolic flow, namely the Yang-Mills heat flow, to construct a high quality global gauge (called the caloric gauge), extending the idea of Tao for the harmonic map heat flow.

Mathematics Hall, Room 507

Monday February 12, 2018 at 12 pm

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“Equivariant Commutative Ring Spectra”

Special Topology Seminar

Come join us Monday February 12, 2018 at 4:30 pm in RM 520, Professor Andrew Blumberg (UT Austin) will be giving a special lecture about “Equivariant Commutative Ring Spectra”.

Abstract

One of the central developments in algebraic topology in the last 30 years has been the study of the chromatic filtration on the stable homotopy category.  The chromatic perspective relates the fine structure of the stable category to the algebraic geometry of formal groups.  In particular, an important role in the story is played by an elliptic cohomology theory called topological modular forms (tmf).

Calculations with forms of tmf that incorporate information about level structures suggest that the framework of equivariant derived algebraic geometry would be extremely useful.  To develop this framework, it is necessary to understand equivariant commutative ring spectra.  This turns out to be a subtle problem; in this talk I will explain the situation and what the answers look like.

Mathematics Hall, Room 520

Monday February 12, 2018 at 4:30 pm

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“Estimates for Modular Forms Using Exponential Sums”

Special Seminar

Come join us Friday February 9, 2018 at 12 pm in RM 507, Professor Will Sawin (ETH Zürich) will be giving a special lecture about “Estimates for Modular Forms Using Exponential Sums”.

Abstract

Together with Emmanuel Kowalski and Philippe Michel, I proved two estimates on sums of Fourier coefficients of modular forms — one on the second moment of critical value of the L-function twisted by Dirichlet characters, and the other on the level of distribution. We did this using bounds for exponential sums proved by geometric and topological methods. I will explain these techniques and how they can be used for these two problems.

Mathematics Hall, Room 507

Friday February 9, 2018 at 12 pm

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“Rare Behavior in Models of Random Geometry”

Special Seminar

Come join us Wednesday February 7, 2018 at 12 pm in RM 507, Professor Shirshendu Ganguly (Berkeley) will be giving a special lecture about “Rare Behavior in Models of Random Geometry”.

Abstract

Models of random geometry have long been investigated in contexts such as real life networks, fluid flow in porous media, and interface dynamics in statistical physics. To develop a refined understanding of such models, one often needs to study not only typical fluctuation theory but also the realm of atypical events. In this talk we describe such a program for two classical models of random geometry: percolation on the complete graph and random distortions of the Euclidean lattice. In particular, we will consider the rare events that a sparse random network has an atypical number of some local structure and that a geodesic in a random metric space has atypical length; both of the corresponding random variables being non-linear functions of the underlying randomness. The geometry associated to typical instances of these rare events is an important topic of inquiry: it can involve merely local objects, or more global ones. We will discuss recent resolutions of certain long standing questions concerning such phenomena, and connections to other areas of mathematics including random matrix theory, information theory, and algebraic and extremal combinatorics.

Mathematics Hall, Room 507

Wednesday February 7, 2018 at 12 pm

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“Counting in Algebraic Geometry and Modular Forms”

Special Seminar

Come join us Monday February 5, 2018 at 12 pm in RM 507, Professor Georg Oberdeck (MIT) will be giving a special lecture about “Counting in Algebraic Geometry and Modular Forms”.

Abstract

It is generally expected that the enumerative geometry of Calabi-Yau geometries
should be closely related to the theory of modular forms. A basic example
(which I will recall) is the count of uniruled divisors on hyper-Kaehler
manifolds by Fourier coefficients of Jacobi forms. Another interesting but less
understood case is the count of algebraic curves and sheaves on Calabi-Yau
threefolds. I will discuss recent work with A. Pixton and J. Shen which, for a
particularly Calabi-Yau threefold, proves the modularity of its curve counting
partition function using geometric arguments.

Mathematics Hall, Room 507

Monday February 5, 2018 at 12 pm

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“Homology Cobordism and Pin(2)-monopole Floer Homology”

Special Seminar

Come join us Friday February 2, 2018 at 12 pm in RM 507, Professor Francesco Lin (Princeton University) will be giving a special lecture about “Homology Cobordism and Pin(2)-monopole Floer Homology”.

Abstract

It is a classical fact that every closed oriented three-manifold Y arises as the boundary of a compact four-manifold W. It is then a natural question to ask how simple such a W can be, once Y is fixed. In the present talk, I will describe a package of gauge-theoretic in variants called Pin(2)-monopole Floer homology, and discuss its relevance to this problem. I will focus in particular on how it can be used to provide an alternative disproof of the long-standing Triangulation Conjecture (recently settles by Manolescu), and its further applications to several related questions.

Mathematics Hall, Room 507

Friday February 2, 2018 at 12 pm

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“Invertibility and Spectrum of Random Matrices: A Convex-Geometric Approach”

Special Seminar

Come join us Wednesday January 31, 2018 at 12 pm in RM 507, Professor Konstantin Tikhomirov (Princeton) will be giving a special lecture about “Invertibility and Spectrum of Random Matrices: A Convex-Geometric Approach”.

Abstract

Convex-geometric methods, involving random projection operators and coverings, have been successfully used in the study of the largest and smallest singular value, delocalization of eigenvectors, and in establishing the limiting spectral distribution for certain random matrix models. Among further applications of those methods in computer science and statistics are restricted invertibility (dimension reduction), as well as approximation of covariance matrices of multidimensional distributions. Conversely, random linear operators play a very important role in geometric functional analysis. In this talk, I will discuss some recent results (by my collaborators and myself) within the theory of random matrices, focusing on invertibility of square non-Hermitian random matrices (with applications to numerical analysis and the study of the limiting spectral distribution of directed d-regular graphs) and approximation of co-variance matrices (in particular, a strengthening of the Bai-Yin theorem.

Mathematics Hall, Room 507

Wednesday January 31, 2018 at 12 pm

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“Legendrian Topology and its Applications”

Special Seminar

Come join us Monday January 29, 2018 at 12 pm in RM 507, Professor Roger Casals(University College London, Topology) will be giving a special lecture about “Legendrian Topology and its Applications”.

Abstract

This is an invitation to contact geometry and its applications. First, I will introduce Legendrian submanifolds and explain new ways of using them to understand complex affine manifolds. Then I will present current progress in the study of higher-dimensional Legendrains which connects to combinatorics, via the study of planar graphs, and gauge theory, through moduli spaces of flat connections.

Mathematics Hall, Room 507

Monday January 29, 2018 at 12 pm

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“The Symplectic Isotopy Problem & Lagrangian Skeleta”

Special Seminar

Come join us Friday January 26, 2018 at 12 pm in RM 507, Professor Laura Starkston (Stanford) will be giving a special lecture about “The Symplectic Isotopy Problem & Lagrangian Skeleta”.

Abstract

Symplectic manifolds decompose into a symplectic divisor and an
exact Weinstein manifold. We will discuss both sides of this essential
decomposition.

On the divisor side, we will focus on symplectic surfaces in 4-manifolds,
particularly the longstanding symplectic isotopy problem. We will leverage
singularities and study symplectic versions of line arrangements and
rational cuspidal curves. On the Weinstein side, we will see how to encode
the symplectic geometry of a 2n-dimensional manifold using the topology of
an n-dimensional singular complex: the Lagrangian skeleton.

Mathematics Hall, Room 507

Friday January 26, 2018 at 12 pm

 

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