# Spring 2023

Date Speaker Title Abstract
Jan 17 Sam Ballas Smooth manifolds through the lens of algebraic geometry

One of the tenets algebraic geometry is that one should try to understand a space by understanding the functions on that space. In many situations one can associate to a space a certain class of functions endowed with an algebraic structure. Furthermore, it is often possible to recover the original space from its space of functions. A simple example of this phenomenon is seen in finite dimensional vector spaces, where one can associate to a vector space its space of linear maps to the ground field (i.e. its dual space) and given the dual space one can recover the original vector space (up to isomorphism).

In this talk, we will focus on smooth manifolds. In this setting one can associate to a manifold its algebra of smooth real valued functions. In this context, we will focus on two questions: how can one recover a manifold from its algebra of smooth functions and which algebras arise as smooth functions on some manifold.

Jan 24 Sam Ballas Smooth manifolds through the lens of algebraic geometry: II Last time we saw how given a sufficiently nice algebra we could construct a "dual space" on which the algebra was a space of real valued functions. We endowed this dual space with a topology and showed that for the algebra of smooth functions on R^n the dual is the R^n with its standard topology. This time we will examine some special properties of smooth functions and ultimately arrive at the algebraic properties an algebra must possess for its dual space to be a smooth manifold.
Jan 31 Sam Ballas Smooth manifolds through the lens of algebraic geometry: III Last time we introduced the notion of a smooth algebra, and showed that the algebra of smooth functions on a smooth manifold is an object of this type. Today we will describe how to recover a smooth manifold from a smooth algebra and how algebra homomorphism between smooth algebras give rise to smooth functions between the corresponding manifold. Said more succinctly, this provides an equivalence of categories between the category of smooth manifolds with smooth maps and the category of smooth algebras with algebra homomorphisms.
Feb 7
Feb 14 Florian Stecker Projective triangle groups and Anosov representations I will talk about groups generated by three projective reflections, configured so that the product of any two of them has finite order. There is only a 1-dimensional space of these groups, and some of them (called Anosov representations) show nice hyperbolic plane like dynamics. We identify when this happens and where this property suddenly breaks down. This is joint work with Gye-Seon Lee and Jaejeong Lee.
Feb 21
Feb 28 Phil Bowers Geometry and Probability I Type Problems: I will begin with the classical conformal type problem for simply connected Riemann surfaces and then walk through several discrete versions of the problem in the settings of circle packings, equilateral surfaces, negatively curved graphs, and Riemannian manifolds. I will then pivot to probability theory and consider type problems in the settings of random walks on graphs and Brownian motion on manifolds, and describe the connections between the geometric and the probabilistic settings.
Mar 7 Phil Bowers Geometry and Probability II
Mar 21 Phil Bowers Geometry and Probability III Conformally Invariant Critical Processes in the Plane: Brownian motion as the scaling limit of random walks has been understood since Donsker’s 1952 proof. In 2D, Brownian motion has the added benefit that it is conformally invariant. This is the archetype of the movement from the discrete setting to the continuous setting. Since the 1930’s physicists have modeled various processes in statistical mechanics on discrete lattices and then attempted to find scaling limits to continuous processes. Though they have developed a trove of conjectures based on numerical results and have applied QFT calculations to derive formulae, they have been hampered by the lack of mathematical tools to verify rigorously their conjectures and QFT calculations. This changed in 2000 when Oded Schramm defined SLE and used it to develop mathematical tools to attack these problems in statistical mechanics.
Mar 28
Apr 4
Apr 11
Apr 18 Anindya Chanda Mixing, Counting, and Equidistribution in Geometry and Dynamics In this talk we will introduce the concepts of mixing, counting and equidistribution in geometry and dynamics. The talk will be based on a pioneering paper by Alex Eskin and Curt McMullen.
Apr 25 Tom Needham Topological Data Analysis and 2-Categories The field of Topological Data Analysis (TDA) provides a collection of techniques for processing and analyzing datasets using tools from algebraic topology. Much of the theory can be expressed categorically; in particular, there is a family of pseudometrics, called interleaving distances, which are used to compare certain functors arising in data science applications. In this talk, I’ll present a generalization of this concept which is formulated in the language of 2-categories, with a view toward proving new Lipschitz stability results in TDA and providing a conceptual connection between TDA and well-known concepts in geometric shape analysis. This is joint work-in-progress with Patrick McFaddin (Fordham University).

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