Special Colloquium

Speaker: Lue Pan (Princeton University)
Title: The Fontaine-Mazur conjecture in dimension two
Abstract: The Fontaine-Mazur conjecture predicts which two-dimensional p-adic representations of the absolute Galois group of Q arise from modular forms. In this talk, I will explain this conjecture by some concrete examples and report some recent progress which relies on some recent developments of p-adic geometry.

Date and Time: Tuesday, January 28 @ 2:40PM
Location: 520 Mathematics

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Special Colloquium

Speaker: Alex Smith (UCLA)
Title: The distribution of conjugates of an algebraic integer
Abstract: For every odd prime p, the number 2 + 2cos(2 pi/p) is an algebraic integer whose conjugates are all positive numbers; such a number is known as a totally positive algebraic integer. For large p, the average of the conjugates of this number is close to 2, which is small for a totally positive algebraic integer. The Schur-Siegel-Smyth trace problem, as posed by Borwein in 2002, is to show that no sequence of totally positive algebraic integers could best this bound. In this talk, we will resolve this problem in an unexpected way by constructing infinitely many totally positive algebraic integers whose conjugates have an average of at most 1.899. To do this, we will apply a new method for constructing algebraic integers to an example first considered by Serre. We also will explain how our method can be used to find simple abelian varieties with extreme point counts.

Date and Time: Thursday, January 23 @ 2:40PM
Location: 520 Mathematics

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Special Colloquium

Speaker: Alex Petrov (MIT)
Title: Topology of algebraic varieties in positive characteristic
Abstract: One fruitful way to analyze the topology of an algebraic variety over complex numbers is to consider its cohomology groups; at least if one is content with disregarding torsion in cohomology, these groups can be expressed in terms of algebraic differential forms on the variety, via an algebraic analog of de Rham cohomology. When applied to algebraic varieties in positive characteristic, de Rham cohomology exhibits behavior that in several ways is qualitatively different from the situation in characteristic zero. I will discuss some recent progress in understanding these differences which have to do with the failure of Hodge decomposition in positive characteristic. The methods used turn out to also be helpful for studying torsion in cohomology of families of varieties in characteristic zero.

Date and Time: Wednesday, January 22 @ 4:30PM
Location: 520 Mathematics

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Francesco Lin, Associate Professor, earned The Richard Hamilton Best Paper Award in Geometric Analysis at the 2024 Annual International Congress of Chinese Mathematicians for his paper “The Seiberg-Witten Equations and the Length Spectrum of Hyperbolic Three-manifolds”; the conference was held in Shanghai from January 3-6, 2025, where Professor Lin also presented a lecture on Floer theory and the geometry of hyperbolic $3$-manifolds.

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Professor Dusa McDuff will receive the 2025 American Mathematical Society Leroy P. Steele Prize for Lifetime Achievement. […]

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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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