Colloquium Mathematics Department - The Ohio State University (2024)

Mathematics Department Colloquium
The Ohio State University

Year 2023-2024

Time: Thursdays 4:15 pm
Location: EA 170

YouTube channel

Schedule of talks:

November 2
Fabien Morel
Universität München
A^1-topological approach to the study of A^1-connected smooth projective varieties
November 9
Benjamin Weinkove
(Northwestern University)
The porous medium equation
December 1
Miklos Bona
(University of Florida)
Negative results in enumerative combinatorics
February 15
Mahan Mj
Tata Institute of Fundamental Research&nbsp
February 29
Konstantin Tikhomirov
Carnegie Mellon University
Random polytopes and random matrices
March 7
Arnd Scheel
University of Minnesota
Patterning and self-organization beyond Turing: from myxobacteria to flatworms
March 21
Dmitry Kleinbock
Brandeis University
"Simultaneously dense and non-dense" orbits in hom*ogeneous dynamics and Diophantine approximation
March 28
Joey Iverson
Iowa State University
Symmetry and optimality of subspace packings
April 4
Dmitri Nikshych
University of New Hampshire
The three layers of a braided fusion category: Core, Mantle, and Crust
April 11
Antoine Song
Minimal surfaces, hyperbolic surfaces and random permutations
April 25
Lei Chen
University of Maryland
Maps Between Configuration Spaces and Moduli Spaces


(F. Morel): In this talk I will discuss some new approach to the study and classification of A^1-connected smooth projective varieties over a fixed field k, inspired by the classical surgery approach.The property of being A^1-connected was introduced by V. Voevodsky and the author long ago. One main difference with classical topology, is that a smooth projective k-variety of >0 dimension which is A^1-connected has a non trivial A^1-fundamental (sheaf of) group. I will explain the concrete meaning of being A^1-connected, and give examples, also of A^1-fundamental groups.Then I will give a definition of the A^1-cellular chain complex, introduced in a joint work with Anand Sawant, and explain some conjectures and examples of computations. I will explain the case of surfaces, already interesting, and almost entirely understood in a work in progress sequel to our previous work.I will finish by explaining how all this can be hopefully used in understanding these kind of smooth varieties.

(B. Weinkove): The porous medium equation is a nonlinear partial differential equation used to model the diffusion of gas in a porous medium. Although the study of this equation goes back more than a century, a rigorous theory did not begin to appear until the 1950s, and open problems still remain. Unlike the heat equation, solutions propagate with finite, not infinite, speed. There is a moving free boundary. I will give an overview of what’s known and what’s not, and describe results on the concavity of solutions. This talk is for a general mathematics audience and will not assume any knowledge of differential equations.

(M. Bona): When we cannot solve an enumeration problem, we may wonder whether we are just not using the right methods, or the problem is actually difficult. If it is actually difficult, how do we measure its level of difficulty? In this talk, we will survey recent methods to prove that the generating function of a combinatorially defined sequence is not rational or not algebraic in situation when we know very little about that generating function. Recently, these methods have been successfully applied in the area of pattern avoiding permutations, but they are applicable in other settings as well. While we will use techniques from analytic combinatorics, the talk will be self-contained and accessible for a general mathematics audience.

(M. Mj): A group G is said to commensurate a subgroup H, if for all g \in G, H^g \cap H is of finite index in H and H^g, where H^g denotes the conjugate of H by g. The commensuration action of G on H can be studied dynamically. And, we will discuss a range of theorems and conjectures in this context, starting with work of Margulis, and coming to the present day. We will focus on commensurations in a non-arithmetic setting.

(K. Tikhomirov): Random polytopes defined as intersections of independent affine halfspaces have found numerous applications in the local theory of Banach spaces and, more recently, have been considered in the context of average-case and smoothed analysis of the simplex method. In this talk, I will discuss some recent results concerning the width of such polytopes, which are based on techniques from the non-asymptotic theory of random matrices.

(A. Scheel): Turing's idea that diffusion differences between chemical species can drive pattern formation and select wavelengths has been a central building block for the modeling of patterns arising in chemistry and biology, from simple tabletop chemistry such as the CIMA reaction to morphogenesis and the formation of presomites.I will report on two studies of pattern formation that invoke pattern selection mechanisms quite different from Turing's. In the first example, I will show how simple nonlinear run-and-tumble dynamics can reproduce complex functionality from equidistribution over rippling to fruiting body formation in myxobacteria colonies [arXiv:1805.11903,arXiv:1609.05741]. In the second example, I will describe simple models for the astounding ability of planarian flatworms to regenerate completely from small fragments of body tissue, preserving polarity (that is, position of head versus tail) in the recovery [arXiv:1908.04253].

(D. Kleinbock): Consider a hom*ogeneous space X with the action of a diagonal one-parameter subgroup. In a 1996 paper with Margulis I proved that the set of points in X with bounded forward orbit has full Hausdorff dimension. Question: what about points with bounded forward orbit accumulating on a given z\in X? We prove that, barring a certain obvious obstruction, those points also form a set of large dimension. All this is motivated by the subject of improving Dirichlet's Theorem in Diophantine approximation, which I will explain along with some new applications. No background is needed to follow the talk. Joint work with Manfred Einsiedler and Anurag Rao.

(J. Iverson): In many applications, one desires collections of subspaces separated by wide angles. Ideally, the sharpest angle between subspaces should be as wide as possible. Relatively few examples of such "optimal subspace packings" are known, but a lot of them have something in common: they have symmetry coming out of their ears. In this talk, we share progress toward understanding this phenomenon, and we use our understanding to construct new examples of optimal subspaces.Various parts of the talk are joint with Matt Fickus, Enrique Gomez-Leos, John Jasper, and Dustin Mixon.

(D. Nikshych): The main objects of this talk are semisimple categories with tensor products satisfying a commutativity constraint (braiding). Examples of such categories come from classical sources (representations of groups and Hopf algebras) and quantum ones such as affine Lie algebras and exotic subfactors. When one separates the group-theoretical and quantum parts of a braided fusion category C, several interesting invariants appear. One such invariant is the core defined as the localization of C by a maximal Tannakian subcategory. Another is the mantle of C, which is the localization of C by its Tannakian radical (the "crust"). I will explain how to approach the classification of braided fusion categories using these invariants and a certain gauging procedure. This is based on a joint work with Jason Green.

(A. Song): Minimal surfaces and hyperbolic surfaces are both "optimal 2d geometries" which are ubiquitous in differential geometry. The first kind is defined by an extrinsic condition (the mean curvature vanishes), while the second kind is defined by an intrinsic condition (the Gaussian curvature is equal to -1). I will discuss a surprising connection between these two geometries coming from randomness. The main statement is that there exists a sequence of closed minimal surfaces in Euclidean spheres, constructed from random permutations, which converges to the hyperbolic plane. This result is part of an attempt to bridge minimal surfaces and unitary representations. I will introduce this circle of ideas and mention some general questions.

(L. Chen): In this talk, I will survey what is known or conjectured about maps between configuration spaces and moduli spaces of surfaces. Among them, we consider two categories: continuous maps and also holomorphic maps. We will also talk about some results and conjectures about maps between finite covers of those spaces. Unlike the case of the whole spaces, much less is known about their finite covers.

Past Ohio State University Mathematics Department Colloquia

  • 2022-2023

  • 2020-2021

  • 2019-2020

  • 2018-2019

  • 2017-2018

  • 2016-2017

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Colloquium Mathematics Department - The Ohio State University (2024)
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