Title: Symplectic geometry and inscription problems

Speaker: Josh Greene (Boston College)

Date, Time, Location: Wednesday, December 4th @4:30PM in Math Hall 520

Abstract:

The square peg problem was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve in the plane contains the vertices of a square, and it is still open to this day. I will survey the approaches to this problem and its relatives using symplectic geometry. This talk is based on joint work with Andrew Lobb.

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Title: Some new perspectives in the Langlands program

Speaker: Professor Matthew Emerton (University of Chicago)

Date, Time, Location: Wednesday, November 20th @4:30PM in Math Hall 520

Abstract:

The goal of this talk is to explain some recent (so-called “categorical”) perspectives on the Langlands program. I will gently lead up to these new developments, beginning with background and examples aimed at introducing some of the ideas of the Langlands program to non-experts.

The underlying theme of the talk will be a common one in representation theory: if we have a commuting action of two groups (or algebras) on a vector space, then we can use the resulting decomposition into irreducible representations to (attempt to) induce a correspondence between irreducible representations of one group (or algebra) and the other. The Langlands program for the group GL_2 over the field Q of rational numbers (the “first non-abelian case” of the Langlands program) implements this idea by taking the vector spaces to be the first cohomology groups of modular curves (certain finite volume quotients of the complex upper half-plane). These curves have an inordinate amount of symmetry, as a result of which the cohomology gets a commuting action of three different objects: p-adic Lie groups, so-called Hecke algebras, and the absolute Galois group of Q. The rich interaction of these actions gives rise to a vast amount of number theory; starting with some key examples, I will pursue one aspect of this story, building up to a statement of the categorical Langlands correspondence for the p-adic Lie group GL_2(Q_p).

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Title: Exploring Stability in Geometric and Functional Inequalities

Speaker: Alessio Figalli (ETH)

Date, Time, Location: Wednesday, November 13th @4:30PM in Math Hall 520

Abstract: In the realms of analysis and geometry, geometric and functional inequalities are of paramount significance, influencing a variety of problems. Traditionally, the focus has been on determining precise constants and identifying minimizers. More recently, there has been a growing interest in investigating the stability of these inequalities. The central question we aim to explore is: “If a function nearly achieves equality in a known functional inequality, can we demonstrate, in a quantitative way, its proximity to a minimizer?” In this colloquium, I will first overview this topic and discuss some recent results.

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We are deeply saddened to inform you that our colleague Richard Hamilton passed away on Sunday, September 29, 2024.

Richard was a giant of mathematics who has inspired generations of geometers. His groundbreaking work on the Ricci flow has opened up a new field of geometric flows, making it possible to attack fundamental questions in geometry and topology that seemed intractable before. Richard’s work was recognized with many awards, including the Oswald Veblen Prize of the American Mathematical Society; the Clay Research Award; the Leroy P. Steele Prize; the Shaw Prize; and, most recently, the Basic Science Lifetime Award in Mathematics.

Richard has served Columbia for a quarter of a century. His topics course on Ricci flow was legendary. Richard loved mathematics and pursued his passion throughout his life.

Richard was a kind and generous human being, a deep thinker with an extraordinary sense of humor. He warmly welcomed everyone who shared his love of mathematics, freely sharing his own insights and ideas.

We will miss him dearly.

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Title: The stability-compactness method and qualitative properties of nonlinear elliptic PDEs

Speaker: Henri Berestycki (University of Maryland, College Park & EHESS, Paris)

Date, Time, Location: Wed. October 16th @ 4:30 PM in 520 Math Hall

Abstract: Nonlinear elliptic equations describe the stationary states of numerous systems in physics, biology and medicine. Qualitative properties such as monotonicity, symmetry, stability or uniqueness are essential features of their study. In this talk, I will present a new general framework to approach this kind of questions, focusing on uniqueness. It rests on decomposing the domain into one region with a certain compactness feature and another supporting a form of spectral stability. This approach has proved to be unexpectedly versatile and in fact encompasses past works on the subject such as the general moving plane method originating in the study of minimal surfaces by Alexandrov. The results concern positive solutions of nonlinear elliptic PDEs in general unbounded domains with Dirichlet boundary conditions and for various types of reaction terms. This represents a series of joint works with Cole Graham and with Cole Graham and Jun-Cheng Wei.

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Title: Matrix Rigidity

Speaker: Josh Alman (Columbia CS)

Date, Time, Location: Wed. Sept 25 @ 4:30 PM in 520 Math Hall

Abstract: A matrix is called rigid if one must change many of its entries before it becomes a low-rank matrix. Leslie Valiant introduced the notion in 1977 as a tool to prove lower bounds on the number of arithmetic operations needed to compute linear transformations like the discrete Fourier transform. Since then, many connections have been demonstrated between matrix rigidity and other topics in the theory of computation.

Unfortunately, proving that matrices of interest are rigid has shown to be a major challenge. In fact, it remains an open problem to prove that any explicit family of matrices is rigid. By contrast, it is known that a random matrix is rigid with high probability.

In this talk, I’ll give a brief overview of matrix rigidity and its uses in computer science. I’ll then discuss some recent results, including proofs that matrices like Hadamard and Fourier matrices, which were previously conjectured to be rigid, are in fact not rigid. The talk will not assume the audience has a background in computer science.

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Please join us in congratulating Professor Richard Hamilton on being awarded the Chinese Basic Science Lifetime Award in Mathematics by the International Congress of Basic Science.  (more)

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