Date/Time : 09 Oct 2020 09:00
Venue : via Zoom
Speaker : Yilin Wang, MIT

Join Zoom Meeting Meeting ID: 947 9035 5755 Password: 167685  We introduce the Loewner energy for Jordan curves and relate this quantity to ideas and concepts coming from random conformal geometry, geometric function theory, and Teichmuller theory. Many of those concepts arise from mathematical physics, ranging from statistical mechanics, quantum field theory, and string theory. One motivation for the definition of the Loewner energy is that it describes the large deviations of a family of simple random curves called Schramm-Loewner evolutions (SLE).  This provides a probabilistic interpretation of the Loewner energy that allows us to establish several interesting invariance properties. We further derive an equivalent characterization of the Loewner energy using zeta-regularized determinants of Laplacians. This then identifies the family of finite energy curves to the family of so-called Weil-Petersson quasicircles. Moreover, the Loewner energy is shown to equal the Kahler potential, introduced by Takhtajan and Teo, of the unique Kahler metric on the corresponding Weil-Petersson Teichmuller space.

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Date/Time : 25 Sep 2020 15:00
Venue : via Zoom
Speaker : Joel Moreira , University of Warwick

Join Zoom Meeting Meeting ID: 985 3799 6193 Password: 535348 Abstract:  In this talk I will present a survey of arithmetic Ramsey theory. This fascinating subject, which focuses on patterns which arise in arbitrary finite colourings of the natural numbers, combines ideas and tools from diverse areas of mathematics, such as graph theory, Fourier analysis, geometric group theory, and ergodic theory; and has deep connections with number theory and additive combinatorics. The first half of the talk will be dedicated to the history of this subject, and the second to recent progress on a few simple looking but still open problems.

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Date/Time : 11 Sep 2020 09:00
Venue : via Zoom
Speaker : Jiajun Tong, UCLA

Join Zoom Meeting  Meeting ID: 937 9128 5227 Password: 922637 Abstract: In this talk, we first consider 2-D Stokes immersed boundary problem. It models a 1-D closed elastic string immersed and moving in a 2-D Stokes flow, which features an autonomously moving boundary and singular forcing supported on it. We will discuss results on well-posedness of the string motion (joint work with Fanghua Lin). In the second part of the talk, we introduce a regularized version of the problem inspired by the numerical immersed boundary method, where a regularized delta-function is used to mollify the singular forcing and the flow field. We will show global well-posedness of the regularized string motion. As the regularization diminishes, we prove the string dynamics converges to that in the un-regularized case under certain assumptions. Viewing the latter as a benchmark, we derive error estimates under various norms. Our rigorous results not only coincide with existing numerical observations in many aspects, but also reveal a potential way of improving the (numerical) accuracy by choosing the regularization suitably.

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Date/Time : 28 Aug 2020 09:00
Venue : via Zoom
Speaker : Ying Cui , University of Minnesota

Join Zoom Meeting Meeting ID: 972 8455 8347 Password: 733806 Many modern statistical estimation problems are defined by  three major components: a statistical model that postulates the dependence of an output variable on the input features; a loss function measuring the error between the observed output and the model predicted output; and a regularizer that controls the overfitting and/or variable selection in the model.  We study the sampled version of this generic statistical estimation problem where the model parameters are estimated by empirical risk minimization, which involves the minimization of the empirical average of the loss function at the data points weighted by the model regularizer.  In our setup we allow all three component functions to be of the difference-of-convex type and illustrate them with a host of commonly used examples, including those in continuous piecewise affine regression and in deep learning with piecewise affine activation functions.  We describe a non-monotone majorization-minimization (MM) algorithm for solving the unified nonconvex, nondifferentiable optimization problem which is formulated as a specially structured composite dc program of the pointwise max type.  Numerical results are presented to demonstrate the effectiveness of the proposed algorithm and the superiority of continuous piecewise affine regression over the standard linear model. 

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Date/Time : 14 Aug 2020 09:00
Venue : via Zoom
Speaker : Hong Wang , Institute for Advanced Study, Princeton

Join Zoom Meeting Meeting ID: 932 7338 0910 Password: 451089  Let $E$ be an $s$–dimensional set on the plane. How large is its distance set $\Delta(E):= \{ |x-y|, x, y\in E\}$?  Falconer conjectured in 1985 that if $s>1$, then $\Delta(E)$ should have positive Lebesgue measure.  We are going to discuss how this problem is connected to various problems in harmonic analysis, as well as different approaches toward it.

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Date/Time : 31 Jul 2020 09:00
Venue : via Zoom
Speaker : Hao Huang , Emory University

Join Zoom Meeting  Meeting ID: 995 2080 5067 Password: 103513   Extremal Combinatorics studies how large or how small a collection of finite objects could be, if it has to satisfy certain restrictions. In this talk, we will discuss how the eigenvalue interlacing methods could lead to various interesting results in Extremal Combinatorics, including the Erdos-Ko-Rado Theorem and its degree version, an isodiametric inequality for discrete cubes, and the resolution of a thirty-year-old problem in Theoretical Computer Science, the Sensitivity Conjecture.

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Date/Time : 17 Jul 2020 0900
Venue : via Zoom
Speaker : Yao Yao, Georgia Institute of Technology

Join Zoom Meeting Meeting ID: 940 7048 1887 Password: 5470853 For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation. I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan). I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

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Date/Time : 03 Jul 2020 0900
Venue : via Zoom
Speaker : Ruixiang Zhang, University of Wisconsin-Madison

Join Zoom Meeting Meeting ID: 929 6203 4592 Password: 4680298 Informally, Kakeya type problems ask whether tubes with different positions and directions can overlap a lot. One usually expects the answer to be no in an appropriate sense. Thanks to the uncertainty principle, such a quantified non-overlapping theorem would often see powerful applications in analysis problems that have Fourier aspects. Perhaps the most well-known Kakeya type problem is the Kakeya conjecture. It remains widely open in $\Bbb{R}^n (n>2)$ as of today. Nevertheless, in the recent few decades people have been able to prove new Kakeya type theorems that led to improvements or complete solutions to analysis problems that appeared out of reach before. I will give an introduction to Kakeya type problems/theorems and analysis problems that see their applications. Potentially reporting some recent progress joint with Du, Guo, Guth, Hickman, Rogers and Wang.

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Date/Time : 19 Jun 2020 0900
Venue : via Zoom
Speaker : Linhui Shen, Michigan State University

Join Zoom Meeting Meeting ID: 954 0477 7844 Password: 7710846 Let G be a split semi-simple algebraic group over Q. We introduce a natural cluster structure on moduli spaces of G-local systems over surfaces with marked points. As a consequence, the moduli spaces of G-local systems admit natural Poisson structures, and can be further quantized. We will study the principal series representations of such quantum spaces. It will recover many classical topics, such as the q-deformed Toda systems, quantum groups, and the modular functor conjecture for such representations. This talk will mainly be based on joint work with A.B. Goncharov.

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