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Schedule of talks:

Abstracts

(B. Street): The theory of elliptic PDEs stands apart from many other areas of PDEs because sharp results are known for very general linear and fully nonlinear elliptic PDEs. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDEs leans heavily on the Fourier transform and Riemannian geometry. Starting with work of Hormander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced: now known as maximal subellipticity or maximal hypoellipticity. In the intervening years, many authors have adapted results from elliptic PDEs to various special cases of maximally subelliptic PDEs. Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry. The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis. In this talk, we present the sharp regularity theory of general linear and fully nonlinear maximally subelliptic PDEs.

(A. Yong): The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question: Which multiplicities are nonzero? In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing results for the Newell-Littlewood numbers. This is joint work with Shiliang Gao (UIUC), Gidon Orelowitz (UIUC), and Nicolas Ressayre (Universite Claude Bernard Lyon I). The presentation is based on arXiv:2005.09012, arXiv:2009.09904, and arXiv:2107.03152.

(E. Riehl): What does it mean for something to exist uniquely? Classically, to say that a set A has a unique element means that there is an element x of A and any other element y of A equals x. When this assertion is applied to a space A, instead of a mere set, and interpreted in a continuous fashion, it encodes the statement that the space is contractible, i.e., that A is continuously deformable to a point. This talk will explore this notion of contractibility as uniqueness and its role in generalizing from ordinary categories to infinite-dimensional categories.

(D. Jensen): Segerman's 15+4 puzzle is a hinged version of the classic 15-puzzle, in which the tiles can rotate as they slide around. In 1974, Wilson classified the groups of solutions to sliding block puzzles. In this talk, we generalize Wilson's result to puzzles like the 15+4 puzzle, where the tiles can rotate, and the sets of solutions are subgroups of the generalized symmetric groups. Aside from two exceptional cases, we will see that the group of solutions to such a puzzle is always either the entire generalized symmetric group or one of two special subgroups of index two.

(J-L. Guermond): We consider high-order discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part with diffusion and stiff relaxation terms. (A typical problem in this class is the compressible Navier-Stokes equations.) Assuming that this problem admits non-trivial invariant domains, in the talk we discuss approximation techniques in time that preserve these invariant domains. Before going into the details, we are going to give an overview of the literature on the topic.  Emphasis will be put on explicit and explicit Runge Kutta techniques using Butcher's formalism. Then we are going to describe techniques that make every implicit-explicit time stepping scheme invariant-domain preserving and mass conservative. The proposed methodology is agnostic to the space discretization and allows to optimize the time step restrictions induced by the hyperbolic sub-step.

(P. Bourgade): The limiting distributions for maxima of independent random variables have been classified during the first half of last century. This classification does not extend to strong interactions, in particular to the flurry of processes with natural logarithmic (or multiscale) correlations. These include branching random walks, the 2d Gaussian free field and cover times. More recently, Fyodorov, Hiary and Keating (2012) have formulated very precise conjecture about maxima of the characteristic polynomial of random matrices, and the maximum of L-functions on typical interval the critical line, based on their log-correlated nature. In this talk I will describe the classical methods to calculate the maximum of log-correlated fields, and recent progress to confirm the prediction by Fyodorov, Hiary and Keating.

(S. Gukov): It is hard to imagine our modern life without machine learning: AI algorithms help us navigate complex patterns of traffic and financial markets, diagnose medical problems, eliminate biases in judgment, and assist with many other complex tasks. Neural nets have been extensively used in data-intensive branches of experimental and observational sciences. Can they also help in 'pure' mathematical research? In this talk, intended for a broad audience, I will tell you two stories. One story is about the cutting-edge algorithms in machine translation, whereas the other involves questions that until recently were reserved for paper-and-pencil type derivations in pure mathematics. The confluence of the two leads to surprising new results and opens new doors for extending rigorous mathematical proofs into completely new domains that until recently remained entirely out of reach.

(R. Young) The Heisenberg group is the simplest example of a noncommutative nilpotent Lie group. In this talk, we will explore how that noncommutativity affects geometry and analysis in the Heisenberg group. We will describe why good embeddings of $\mathbb{H}$ must be bumpy at many scales, how to study embeddings into $L_1$ by studying surfaces in $\mathbb{H}$, and how to construct a metric space which embeds into $L_1$ and $L_4$ but not in $L_2$. This talk is joint work with Assaf Naor.

(S. Sun) An n dimensional Riemannian metric g defines a holonomy group, which is a subgroup of SO(n) given by parallel transport along all contractible loops (with respect to the Levi-Civita connection). According to the Berger classification we know that if a complete Riemannian metric is not locally symmetric and not locally reducible then its holonomy group is either the entire SO(n) (generic case), or U(n) (Kahler), or is special and belongs to a small list. Riemannian metrics with special holonomy are very interesting geometric objects to study, with many connections to analysis and physics. The simplest model is given by a 4 dimensional hyperkahler metric. We will explain the general background and discuss recent progress on understanding the geometry of hyperkahler 4 manifolds. This is based on joint work with Ruobing Zhang.

(J. Bochi) I will state two problems, both involving some sort of nonlinear averaging. The first one is about the behaviour of elementary symmetric polynomials evaluated at a stationary sequence of random variables. The second one is about average rates of expansion of a linear operator in high dimension. Though the two problems seem unrelated, they have essentially the same solution. That common solution is a limit law discovered (in the context of the first problem) by Halasz and Szekely in 1976, answering a question posed by Kolmogorov. The explanation for this coincidence is that both problems can be stated in terms of the hypergeometric means introduced by Carlson in 1964. I'll explain Carlson's theory of hypergeometric functions and means, how to extend them to infinite dimension, and how to obtain Halasz-Szekely means as a limit of Carlson's means.

(A. Pillay) A (k-)approximate subgroup X of an arbitrary group G is a symmetric subset X of G such that X.X (set of x.y, for x,y in X) is covered by (k) finitely many  translates of X.  The problem is to explain or describe approximate subgroups in terms of homomorphisms to (locally) compact topological groups.  For X finite and k-approximate this is done by Breuillard, Green, Tao. We give a result in the most general case, using the machinery of abstract topological dynamics. (Also obtained by Hrushovski by other methods.) (Joint with K. Krupinski.)

(S. Koch) This talk will feature a tour of complex dynamical systems. We will begin by exploring the dynamics of quadratic polynomials, highlighting a fundamental subset of them known as postcritically finite polynomials. Beginning with work of Milnor and Thurston, these maps form the backbone of dynamical moduli spaces, and in a very strong sense, they represent special points in these spaces. We will study these maps from both topological and algebraic points of view, focusing on some recent results and exciting open problems. All are welcome!

(N. Dobrinen) The Galvin-Prikry theorem states that Borel subsets of the Baire space are Ramsey. Silver extended this to analytic sets, and Ellentuck gave a topological characterization of Ramsey sets in terms of the property of Baire in the Vietoris topology. We present work extending these theorems to several classes of countable homogeneous structures. An obstruction to exact analogues of Galvin-Prikry or Ellentuck is the presence of big Ramsey degrees. We will discuss how different properties of the structures affect which analogues have been proved. Presented is work of the speaker for Q-like structures, and joint work with Zucker for binary finitely constrained FAP classes. A feature of the work with Zucker is showing that we can weaken one of Todorcevic's four axioms guaranteeing a Ramsey space, and still achieve the same conclusion. These axioms are built on prior work of Carlson and Simpson developing topological Ramsey spaces.

(J. Song) Uniform diameter estimates for Kahler metrics are established, which only require an entropy bound for the volume measure without any assumption on the curvature. The proof builds on recent PDE techniques for the complex Monge-Ampere equation. As a consequence, we solve the long-standing problem of uniform diameter bounds and Gromov-Hausdorff convergence of the Kahler-Ricci flow, for both finite-time and long-time solutions.

(T. Johnson-Freyd) The fundamental theorem of algebra, as Hilbert explained, asserts that every consistent system of polynomial equations over R has a solution over C. Together with David Reutter, we have established a "fundamental theorem of higher algebra": we have constructed and analyzed the n-category in which every consistent (and semisimple) system of "n-categorical polynomial equations" has a solution. In this talk, I will explain a bit about our construction, and why a quantum physicist might care.

Past Ohio State University Mathematics Department Colloquia

• 2020-2021

• 2019-2020

• 2018-2019

• 2017-2018

• 2016-2017

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