## Upcoming MAMS Seminar Series

#### Analysis and Probability

Tuesday January 25: Michael Roysden (Kent State University)

Title: Measure theoretic Rogers-Shephard and Zhang inequalities

Abstract: This talk will detail two recent papers concerning Rogers-Shephard inequalities and Zhang inequalities for various classes of measures. The covariogram of a measure plays an essential role in both of these inequalities, which includes a variational formula resulting in a measure theoretic version of the projection body, an object which has recently gained a lot of attention.

Tuesday February 15, 2022: Stanislaw Szarek (CWRU)

Title:  Löwner-John ellipsoid of a convex body

Abstract: We will prove a somewhat modern version of the 1948 result of F. John concerning the maximal volume ellipsoid contained in a convex body, and sketch some applications.  This will be an educational talk, the only hard prerequisite being  linear algebra

Tuesday February 22, 2022: Stanislaw Szarek (CWRU)

Title:  Löwner-John ellipsoid of a convex body II

Abstract: We will prove a somewhat modern version of the 1948 result of F. John concerning the maximal volume ellipsoid contained in a convex body, and sketch some applications.  This will be an educational talk, the only hard prerequisite being  linear algebra.

Tuesday March 1, 2022: Dylan Langharst (Kent State)

Title: Measure Theoretic Minkowski’s Existence Theorem and Projection Bodies

Abstract:

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a measure theoretic Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with density, denoted by $\Lambda^\prime$: for $\nu$ a finite, even Borel measure on the unit sphere and $\mu\in\Lambda^\prime$, there exists a symmetric convex body $K$ such that

$$d\nu(u)=c_{\mu,K}dS_{\mu,K}(u),$$

where $c_{\mu,K}$ is a quantity that depends on $\mu$ and $K$ and $dS_{\mu,K}(u)$ is the surface area-measure of $K$ with respect to $\mu$. Examples of measures in $\Lambda^\prime$ are homogeneous measures (with $c_{\mu,K}=1$) and probability measures with continuous densities (e.g. the Gaussian measure). We will also consider measure dependent projection bodies $\Pi_\mu K$ by classifying them and studying the isomorphic Shephard problem: if $\mu$ and $\nu$ are even, homogeneous measures with density and $K$ and $L$ are symmetric convex bodies such that

$\Pi_{\mu} K \subset \Pi_{\nu} L$, then can one find an optimal quantity $\mathcal{A}>0$ such that $\mu(K)\leq \mathcal{A}\nu(L)$? Among other things, we show that, in the case where $\mu=\nu$ and $L$ is a projection body, $\mathcal{A}=1$

Tuesday March 15, 2022:  Mathias Sonnleitner (University Graz)

Titel:

Isotropic discrepancy: the gap between order and chaos

Abstract:

Isotropic discrepancy is a measure of equidistribution of a point set

based on convex sets and is related to cubature rules for numerical

integration. We present known bounds on the isotropic discrepancy of

optimal point sets on the unit cube and compare probabilistic

constructions with lattice point sets. The latter arise from the

intersection of the unit cube with lattices and their isotropic

discrepancy is comparable to a geometric quantity of the lattice. We

find that there is a gap between these structured point sets and random

point sets. The talk is based on joint work with F. Pillichshammer.

Tuesday March 22, 2022: David  Grzybowski (CWRU)

Title: A CLT for Traces of Powers of Random Unitary Matrices

Abstract: This talk will explain an application of Stein’s method to prove a central limit theorem for traces of powers of Haar-distributed matrices from the unitary, special orthogonal, and unitary symplectic groups. We use unitary Brownian motion to construct a family of exchangeable pairs and apply an infinitesimal version of Stein’s method to prove that a vector of such traces converges to a vector of independent Gaussian random variables. We also prove a rate of convergence in the Wasserstein distance

Tuesday March 29, 2022: Mark Meckes (CWRU)

Title: MCMC volume estimation and the geometry of high-dimensional convex bodies

Abstract: I will first discuss the computational challenges involved in finding the volume of a high-dimensional convex set, then how MCMC algorithms (going back to Dyer, Frieze, and Kannan, and improved by many authors since) give the best known approach to this problem.  I will end with a brief discussion of how a geometric conjecture motivated by rigorous performance guarantees for these MCMC algorithms has been one of the main driving forces of work in high-dimensional geometry for the last 25 years.

Tuesday April 5, 2022: Elisabeth Werner (CWRU)

TITLE:  On the $L_p$ Brunn Minkowski theory

ABSTRACT:  The Brunn Minkowski theory, sometimes  also  called the theory of

mixed volumes,  is the very core of convex geometric analysis.

It centers around the study of geometric invariants and geometric measures

associated with convex bodies. A cornerstone of this theory is the classical Steiner formula.

An extension of the classical Brunn Minkowski theory,  the $L_p$ Brunn Minkowski theory

has emerged  and has evolved rapidly over the last years. It  is now a central part of

modern convex geometry.

The $L_p$ Brunn Minkowski theory focuses on the study of affine invariants associated

with convex bodies.  We show an analogue of the classical Steiner formula in the context

of the $L_p$ Brunn Minkowski theory.

The classical Steiner formula is a special case of this more general $L_p$ Steiner formula.

### FALL 2022

11/9/2022
Title: A PDE-Based Analysis of the Symmetric Two-Armed Bernoulli Bandit
Abstract: The multi-armed bandit is a classic sequential prediction problem. At each round, the predictor (player) selects a probability distribution from a finite collection of distributions (arms) with the goal of minimizing the difference (regret) between the player’s rewards sampled from the selected arms and the rewards of the arm with the highest expected reward. The player’s choice of the arm and the reward sampled from that arm are revealed to the player, and this prediction process is repeated until the final round. Our work addresses a version of the two-armed bandit problem where the arms are distributed independently according to Bernoulli distributions and the sum of the means of the arms is one (the symmetric two-armed Bernoulli bandit). In a regime where the gap between these means goes to zero and the number of prediction periods approaches infinity, we obtain the leading order terms of the expected regret and pseudoregret for this problem by associating each of them with a solution of a linear parabolic partial differential equation. Our results improve upon the previously known results; specifically we explicitly compute the leading order term of the optimal regret and pseudoregret in three different scaling regimes for the gap. Additionally, we obtain new non-asymptotic bounds for any given time horizon. This is joint work with Robert Kohn available at https://arxiv.org/abs/2202.05767.

10/19/22
Title: Patterns, algorithms and your friends
Speaker: Dr. Emily Evans, Department of Mathematics, Brigham Young University
Abstract: In this talk, I will give a survey of methods used to determine resistance distance in networks.  These techniques will range from linear algebra to geometric simplices and from numerical optimization to projection methods. For each method discussed an illustrative example of the technique will be provided.  This talk will also feature interesting numerical patterns and real-world applications.  Finally, I will close my talk with a number of conjectures related to resistance distance.

10/5/22
Title: Crowdsourced Ionospheric Observations: Data and Discussion
Speaker Name/Affiliation: Kristina Collins, PhD Candidate, Electrical Engineering, CWRU
Abstract: Understanding ionospheric variability remains a frontier topic in the space physics community. This variability is key not only to understanding ionospheric dynamics in its own right, but also as a means to understanding the coupled geospace system as a whole, which includes the ionosphere’s connection to both space above and the neutral atmosphere below.  Sources of variability from space include solar flares that last minutes, substorms that last a few hours, and ionospheric and geomagnetic storms that can last days. Sources of variability from below include traveling ionospheric disturbances (TIDs) associated with atmospheric gravity waves (AGWs), which may be caused by tornadoes, tsunamis, or high latitude sources.
Ionospheric variability due to atmospheric coupling produces measurable effects in Doppler shift of HF (high frequency, 3-30 MHz) skywave signals, which are straightforward to measure with low-cost equipment and are conducive to citizen science campaigns. The Personal Space Weather Station network is a modular network of community-maintained, open-source receivers, which measure Doppler shift in the precise carrier signals of time standard stations WWV, WWVH and CHU. Here, data from the first prototype of the Low-Cost Personal Space Weather Station (https://doi.org/10.1016/j.ohx.2022.e00289) are presented for a period of time spanning late 2019 to early 2022 (www.doi.org/10.5281/zenodo.6622111). Software tools for the visualization and analysis of this living dataset are provided (https://github.com/HamSCI/hamsci_psws) and exemplars of short- and long-term variability and events will be presented and discussed.
This presentation will be followed by an open discussion of what applied mathematical methods may be used to leverage this dataset.

9/21/22
Title: Hypermodels, Sparsity and Approximate Bayesian Computing
Speaker: Dr. Erkki Somersalo, CWRU

Abstract: In numerous applications involving an underdetermined large scale inverse problem, sparsity of the solution is a desired property. In the Bayesian framework of inverse problems, sparsity requirement of the solution may be implemented by properly defining the prior distribution. While Gaussian priors are not well suited for promoting sparsity, certain hierarchical, conditionally Gaussian models have been demonstrated to be efficient. In this talk, we review a general class of conditionally Gaussian hypermodels that provide a flexible framework for promoting sparsity of the solution, and discuss approximate iterative methods that can be used both for finding single sparse solutions as well as for approximate sampling of the posterior distributions. The motivation for this work comes from an ongoing work on brain imaging using magnetoencephalography data.