## Upcoming MAMS Seminar Series

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.

**January 31, February 7, February 14: Dr. Erkki Somersalo (CWRU)**

Title: Monte Carlo integration: Ideas and insights

Abstract: Monte Carlo integration represents one of the bread-and-butter methods in Bayesian computing, and it continues to be a challenge in particular in high dimensional inverse problems, e.g., when the quantity of interest is a discretized version of a distributed parameter.

This lecture contains a gentle introduction in the topic, and reviews some of the methods that have recently gained popularity, such as the preconditioned Crank-Nicholson (pCN) algorithm.

**February 21, Dr. Anirban Mondal (CWRU)**

Abstract: The talk will first focus on two quite powerful ideas in the Markov chain Monte Carlo literature — the two-stage Metropolis-Hastings sampler (a.k.a. delayed acceptance Metropolis-Hastings sampler) and the adaptive Metropolis sampler. I will discuss how these algorithms are very useful for high-dimensional posterior sampling in the Bayesian inverse problem setting. In particular, the former addresses the computational issues for repeated evaluation of expensive likelihoods, the latter addresses the step size tuning and related convergence issues for high-dimensional target densities. Then I will introduce a new sampling algorithm – a two-stage adaptive Metropolis algorithm – where we combine the ideas of these two useful samplers. Being a combination of the two, this new sampler is superior to both predecessors in terms of computational efficiency. The proposals of the sampler are dependent on all the previous states so the chain loses its Markov property, but we prove that it retains the desired ergodicity property

**February 28, Dr. Jenny Brynjarsdottir (CWRU)**

Title: Optimal Estimation Versus MCMC for CO2 Retrievals

Abstract: The Orbiting Carbon Observatory-2 (OCO-2) collects infrared spectra from which atmospheric properties are retrieved. OCO-2 operational data processing uses optimal estimation (OE), a state-of-the-art approach to inference of atmospheric properties from satellite measurements. One of the main advantages of the OE approach is computational efficiency, but it only characterizes the first two moments of the posterior distribution of interest. Here we obtain samples from the posterior using a Markov Chain Monte Carlo (MCMC) algorithm and compare this empirical estimate of the true posterior to the OE results. We focus on 600 simulated soundings that represent the variability of physical conditions encountered by OCO-2 between November 2014 and January 2016.

**March 14, Dr. Mark Meckes (CWRU)**

**March 21 Dr. Daniela Calvetti (CWRU)**

**March 28 Dr. David Gurarie (CWRU)**

Title: Quantifying diagnostic uncertainty for Schistosomiasis with implications to its control and elimination

Abstract: Schistosomiasis is one of neglected tropical diseases caused by parasitic worm Schistosome, circulating between human and snail hosts. The disease is widely spread in many tropical and subtropical countries. It is targeted for control and elimination by WHO, and the proposed strategies rely on its monitoring/evaluation at local community level, using available diagnostic tools. The latter proved notoriously difficult due to extreme variability of diagnostic (test) outcomes. To quantify diagnostic variability and explore its implications for control, we utilized an extensive dataset of large-scale control surveillance studies conducted in 3 African countries (SCORE project).

By combining statistical analysis, mathematical models and computer tools, we discovered new patterns and disease markers relevant to schistosomiasis. Furthermore, we develop consistent procedures and computer tools for diagnostic resampling of test data, applicable to any community (outside SCORE countries). The new resampling methodology has many applications. Here we shall focus on dynamic models of Schistosome transmission calibrated with the SCORE data, and its application to WHO control strategies.

The talk will elaborate data-driven pattern discovery, and dynamic modeling of schistosomiasis.