Upcoming MAMS Seminar Series
Fall 2024
12/3/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Zachary Leclaire (Case Western Reserve University)
Title: Geometric Brascamp-Lieb Inequality & Applications
Abstract: The Brascamp-Lieb Inequality generalizes familiar results like Hölder’s Inequality. In this presentation, we’ll discuss a notable special case, the “geometric” version, and explore its applications to the volume ratios of convex bodies and the reverse isoperimetric inequality.
Speaker: Sam Quin (Case Western Reserve University)
Title: Bernstein Inequalities
Abstract: We aim to introduce and prove some Bernstein inequalities, which give tail bounds on sums of independent random variables. We first prove a Bernstein inequality for absolutely bounded random variables. We then define sub-gaussian and sub-exponential random variables and prove their respective Bernstein inequalities, time permitting.
11/16/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Diliya Yalikun (Case Western Reserve University)
Title: Convolution body and volume maximization
Abstract: We will look at the convolution body and its volume under n-dimensional Lebesgue measure. We will show an unexpected result due to Julian Haddad, which states that the volume of the convolution body in the plane is not maximized by ellipsoids.
11/19/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Isaac Hubbard
Title: The Gaussian Isoperimetric Inequality and Concentration of Measure
Abstract: We prove the Gaussian isoperimetric inequality, which states that Gaussian surface area is maximized among sets of fixed Gaussian measure by half-spaces. As an intermediate step, we will prove a functional form, called Bobkov’s inequality, for Lipschitz functions. We will also derive concentration of Gaussian measure.
11/12/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Fabio Izasa (Case Western Reserve University)
Title: Approximating Continuous Functions with ReLU Neural Networks
Abstract: This presentation explores the theoretical underpinnings of Rectified Linear Unit (ReLU) neural networks as universal approximators of continuous functions and discusses practical considerations in their implementation for function approximation tasks.
10/29/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Effrosini Chasioti (Case Western Reserve University)
Title: The Prekopa-Leindler Inequality
Abstract: We will present the functional version of Brunn Minkowski Inequality i.e Prekopa-Leindler Inequality. The Prékopa–Leindler inequality is an integral inequality which is related to the reverse Young’s inequality, the Brunn–Minkowski inequality and many other important and classical inequalities in analysis. We will see how it’s connected with the Brunn-Minkowski Inequality and if the time permits we may see some of its applications.
10/15/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Mengchun Cai (Case Western Reserve University)
Title: Trace-Norm and Convergence Rate for some Unitary Ensembles
Abstract: We give some convergence rate for the circular unitary ensemble (CUE) and the Laguerre unitary ensemble (LUE) by their trace-norms and determinant point structures.
10/8/2024, Tue. 2:30-3:30 in Glennan 400
Speaker: Semyon Alesker (Tel Aviv University)
Title: The valuation theory and geometric inequalities
Abstract: Valuations are finitely additive measures on convex compact sets. In the last two decades a number of structures (e.g. product and convolution) with non-trivial properties were discovered on the space of valuations. One such recently discovered property is an analogue of the classical Hodge-Riemann bilinear relations known in algebraic/Kaehler geometry. In special cases, they lead to new inequalities for convex bodies, to be discussed in the talk. No familiarity with valuations theory and algebraic/Kaehler geometry is assumed.
10/1/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Colin Tang (Carnegie Mellon University)
Title: Simplex slicing: an asymptotically-sharp lower bound
Abstract: We show that for the regular n-simplex, the 1-codimensional central slice that’s parallel to a facet will achieve the minimum area (up to a 1-o(1) factor) among all 1-codimensional central slices, thus improving the previous best known lower bound (Brzezinski 2013) by a factor of $2\sqrt{3}/e \approx 1.27$. In addition to the standard technique of interpreting geometric problems as problems about probability distributions and standard Fourier-analytic techniques, we rely on a new idea, mainly \emph{changing the contour of integration} of a meromorphic function.
9/24/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Brandon Oliva (Case Western Reserve University)
Title: Quantitative Metric Density and Connectivity for sets of positive measure
Abstract: We show that in doubling, geodesic metric measure spaces (including, for example, Euclidean space), sets of positive measure have a certain large-scale metric density property. As an application, we prove that a set of positive measure in the unit cube of IR^d can be decomposed into a controlled number of subsets that are “well-connected” within the original set, along with a “garbage set” of arbitrarily small measure. Our results are quantitative, i.e., they provide bounds independent of the particular set under consideration.
9/17/2024, Tue. 2:30-3:30 in Clark 210
Speaker: Dr. Stanisław Szarek (CWRU)
Title: Löwner-John ellipsoid of a convex body
Abstract: We will prove a 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 prerequisite is linear algebra.
9/10/2024, Tue. 2:30-3:30 in Clark 309
Speaker: Dr. Mathias Sonnleitner (University of Passau, Germany)
Title: Covering completely symmetric convex bodies
Abstract: A completely symmetric convex body is invariant under reflections or permutations of coordinates. We can bound its metric entropy numbers and consequently its mean width using sparse approximation. We provide an extension to quasi-convex bodies and present an application to unit balls of Lorentz spaces, where we can provide a complete picture of the rich behavior of entropy numbers. These spaces are compatible with sparse approximation and arise from interpolation of Lebesgue sequence spaces, for which a similar result is by now classical. Based on joint work with J. Prochno and J. Vybiral.
Fall 2024
11/13/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. Abdul-Nasah Soale
Title: Detecting influential observations in single-index Fréchet regression
Abstract: Regression with random data objects is becoming increasingly common in modern data analysis. Unfortunately, this novel regression method is not immune to the trouble caused by unusual observations. A metric Cook’ss distance extending the original Cook’s distances of Cook (1977) to regression between metric-valued response objects and Euclidean predictors is proposed. The performance of the metric Cook’s distance is demonstrated in regression across four different response spaces in an extensive experimental study. Two real data applications involving the analyses of distributions of COVID-19 transmission in the State of Texas and the analyses of the structural brain connectivity networks are provided to illustrate the utility of the proposed method in practice.
10/23/2024, Wed. 4-5 pm in Sears 439
Speaker: Professor Matthew Wascher
Title: The effects of individual-level behavioral responses on SIS epidemic persistence
Abstract: The contact process (SIS epidemic model) has long been studied as a model for the spread of infectious disease through a population. One important question concerns the long-term behavior of the epidemic–does it result in a large outbreak, or does the infection die out quickly? There is a large literature on the effects of the underlying population structure on this long-term behavior. However, the role of individual-level behavioral responses to the epidemic is less studied. In this talk, I will introduce the contact process, some key ideas used to analyze it, and a few notable results. I will then discuss my recent work on modified versions of the contact process that include individual-level behavioral responses to the epidemic. I will present some results on how individual-level behavioral responses can influence the long-term behavior of an epidemic and discuss why analyzing these models is mathematically challenging.
10/16/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. Weihong Guo (Department of Mathematics, Applied Mathematics and Statistics, CWRU)
Title: Assemble Learnable Mumford-Shah Type Model with Multi-Grid Technique for Image Segmentation
Abstract: The main objective of image segmentation is to divide an image into homogeneous regions for further analysis. This is a significant and crucial task in many applications such as medical imaging. Deep learning (DL) methods have been proposed and widely used for image segmentation. However, these methods usually require a large amount of manually segmented data as training data and suffer from poor interpretability (known as the black box problem). In this talk, we will present a variational model-based segmentation network with better generalizability and interpretability. This approach allows for the incorporation of learnable prior information into the network structure design. Moreover, the multi-grid framework enables multi-scale feature extraction and offers a mathematical explanation for the effectiveness of the U-shaped network structure in producing good image segmentation results. Due to the proposed network originating from a variational model, it can also handle small training sizes. Our experiments on the REFUGE dataset, the White Blood Cell image dataset, and 3D thigh muscle magnetic resonance (MR) images demonstrate that even with smaller training datasets, our method yields better segmentation results compared to related state of the art segmentation methods. This is a joint work with Jun Liu and Junying Meng from Beijing Normal University, China.
9/11/2024, Wed. 4-5 pm in Sears 439
Speaker: Dr. Howard Levinson (Assistant Professor of Computer Science from Oberlin College)
Title: The Inverse (re)Born Series
Abstract: Inverse scattering problems are powerful experiments to recover internal properties of an opaque object. One of the main difficulties in solving these problems is inherent nonlinearity. In this talk, I will describe a nonlinear reconstruction method, the inverse Born series, along with some recent improvements for practical computation.
8/28/2024, Wed. 4-5 pm in Wickenden 321
Speaker: Venky Krishnan (TIFR Centre for ApplicableMathematics, Bangalore, India)
Title: Unique continuation results for certain generalized Radon transforms
Abstract: In the first part of the talk, we study unique continuation results for ray transform of symmetric tensor fields. We show that if the ray transform of asymmetric tensor field f vanishes along all lines passing through a non-empty open set and if the Saint-Venant operator acting on f vanishes on the same open set, then f is potential. In the second half of the talk, we study spherical mean transforms in odd dimensions and show that unique continuation results cannot hold. This is based on joint works – Gaik Ambartsoumian, Divyansh Agrawal, Suman Kumar Sahoo and Nisha Singhal.