September 25, 2022

There are no events currently scheduled for September 25th.

September 26, 2022

• 8:50 - 9:00 am EDT

  Welcome

  11th Floor Lecture Hall

  • Session Chair
  • Brendan Hassett, ICERM/Brown University
• 9:00 - 9:45 am EDT

  Haagerup's phase transition at polydisc slicing

  11th Floor Lecture Hall

  • Speaker
  • Tomasz Tkocz, Carnegie Mellon University

  • Session Chair
  • Alexander Koldobskiy, University of Missouri-Columbia

  Abstract

  We show a probabilistic extension of the Oleszkiewicz-Pełczyński polydisc slicing result. The Haagerup-type phase transition occurs exactly when the p-norm recovers volume, in contrast to the real case. Based on joint work with Chasapis and Singh.

  • Video Available Soon
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EDT

  On the minimal dispersion on the cube

  11th Floor Lecture Hall

  • Speaker
  • Galyna Livshyts, Georgia Tech

  • Session Chair
  • Alexander Koldobskiy, University of Missouri-Columbia

  Abstract

  We discuss a randomized construction of a point configuration, which gives a bound for the minimal dispersion on the cube. The bound is close to optimal, and in some regime it is optimal for the Poisson point process. Joint work with Alexander Litvak.

  • Video Available Soon
• 11:30 am - 12:15 pm EDT

  From intersection bodies to dual centroid bodies: a stochastic approach to isoperimetry

  11th Floor Lecture Hall

  • Speaker
  • Peter Pivovarov, University of Missouri

  • Session Chair
  • Alexander Koldobskiy, University of Missouri-Columbia

  Abstract

  I will discuss a family of affine isoperimetric inequalities for bodies that interpolate between intersection bodies and dual Lp centroid bodies. The focus will be a common framework for the Busemann intersection inequality and the Lutwak--Zhang inequality. The approach depends on new empirical versions of these inequalities. Based on joint work with R. Adamczak, G. Paouris and P. Simanjuntak.
• 12:30 - 2:30 pm EDT

  Lunch/Free Time
• 2:30 - 3:15 pm EDT

  Shortest closed curve to inspect a sphere

  11th Floor Lecture Hall

  • Speaker
  • Mohammad Ghomi, Georgia Institute of Technology

  • Session Chair
  • Kateryna Tatarko, University of Waterloo

  Abstract

  We show that in Euclidean 3-space any closed curve which contains the unit sphere in its convex hull has length at least 4pi, and characterize the case of equality, which settles a conjecture of Zalgaller. Furthermore, we establish an estimate for the higher dimensional version of this problem by Nazarov, which is sharp up to a multiplicative constant. Finally we discuss connections with sphere packing problems, and other optimization questions for convex hull of space curves. This is joint work with James Wenk.

  • Video Available Soon
• 3:30 - 4:00 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 4:00 - 4:45 pm EDT

  Dual curvature measures for log-concave functions

  11th Floor Lecture Hall

  • Speaker
  • Yiming Zhao, Syracuse University

  • Session Chair
  • Kateryna Tatarko, University of Waterloo

  Abstract

  Dual curvature measures for convex bodies were introduced by Huang-Lutwak-Yang-Zhang in 2016. In this talk, we will discuss how this can be naturally extended to the set of log-concave functions. We will also discuss the Minkowski problem for these measures. This is joint work with Yong Huang, Jiaqian Liu, and Dongmeng Xi.

  • Video Available Soon
• 5:00 - 6:30 pm EDT

  Reception

  10th Floor Collaborative Space

September 27, 2022

• 9:00 - 9:45 am EDT

  TBA

  11th Floor Lecture Hall

  • Virtual Speaker
  • Sergii Myroshnychenko, Lakehead University

  • Session Chair
  • Dmitry Ryabogin, Kent State University
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EDT

  Full Field Photoacoustic Tomography with Variable Sound Speed

  11th Floor Lecture Hall

  • Speaker
  • Ngoc Do, Missouri State university

  • Session Chair
  • Dmitry Ryabogin, Kent State University

  Abstract

  Photoacoustic tomography (PAT) is a non-invasive imaging modality that requires recovering the initial data of the wave equation from certain measurements of the solution outside the object. In the standard PAT, the measured data consist of time-dependent signals measured on an observation surface. In contrast, the measured data from the recently invented full-field detection technique provide the solution of the wave equation on a spatial domain at a single instant in time. While reconstruction using classical PAT data has been extensively studied, not much is known for the full field PAT problem. I will discuss the mathematical foundations of the latter problem for variable sound speed and its uniqueness, stability, and exact inversion method using time-reversal. Our results demonstrate the suitability of both the full field approach and the proposed time-reversal technique for high resolution photoacoustic imaging.

  • Video Available Soon
• 11:30 - 11:40 am EDT

  Smooth selection of convex sets

  Lightning Talks - 11th Floor Lecture Hall

  • Speaker
  • Fushuai Jiang, University of Maryland

  • Session Chair
  • Dmitry Ryabogin, Kent State University

  Abstract

  We consider a generalization of the classical Whitney extension problem. Let $E\subset \mathbb{R}^n$ be a compact set and let $K(x) \subset \mathbb{R}^d$ be a convex set for each $x \in E$. I will describe a procedure to determine whether or not there exists a $C^m$ selection of $K$, i.e., if there exists $\phi \in C^m(\mathbb{R}^n, \mathbb{R}^d)$ such that $\phi(x)\in K(x)$ for every $x \in E$. This is based on the joint work with Kevin Luli and Kevin O'Neill.
• 11:40 - 11:50 am EDT

  Measure Theoretic Minkowski's Existence Theorem

  Lightning Talks - 11th Floor Lecture Hall

  • Speaker
  • Dylan Langharst, Kent State University

  • Session Chair
  • Dmitry Ryabogin, Kent State University

  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).
• 11:50 am - 12:00 pm EDT

  Harmonic analysis and geometric configurations in fractals

  Lightning Talks - 11th Floor Lecture Hall

  • Speaker
  • Alex McDonald, The Ohio State University

  • Session Chair
  • Dmitry Ryabogin, Kent State University

  Abstract

  An active area of research is to determine when a set of sufficient Hausdorff dimension contains finite point configurations of some geometric type. In this talk, I will discuss how techniques from harmonic analysis are used to study such problems.
• 12:00 - 12:10 pm EDT

  Valuations on convex functions with compact domain

  Lightning Talks - 11th Floor Lecture Hall

  • Speaker
  • Jacopo Ulivelli, La Sapienza, University of Rome

  • Session Chair
  • Dmitry Ryabogin, Kent State University

  Abstract

  We provide a Homogenous decomposition Theorem for continuous and translation invariant valuations on convex functions with compact domain. As a consequence of an extension argument, these valuations are the same for super coercive convex functions, a case settled by Colesanti, Ludwig and Mussnig. Joint work with Jonas Knoerr.
• 12:10 - 12:20 pm EDT

  On Gaussian projection type inequalities

  Lightning Talks - 11th Floor Lecture Hall

  • Speaker
  • Sudan Xing, University of Alberta

  • Session Chair
  • Dmitry Ryabogin, Kent State University

  Abstract

  We provide an overview of projection bodies in Gaussian probability space for sets of finite Gaussian perimeter and their corresponding applications in functions of Bounded variation space. On the one hand, we study the properties of Gaussian projection bodies for sets of finite Gaussian perimeter under Ehrhard symmetrization and establish a Gaussian projection type inequality. The inequality concludes that Ehrhard symmetrization contracts the Minkowski sum of the Gaussian projection bodies for set of finite Gaussian perimeter $E$ and its reflection $E^v$. On the other hand, we investigate the functional ``lifting" of Ehrhard symmetrization and establish the affine Gaussian P\'olya-Szeg\"o type inequalities in terms of the functional Ehrhard symmetrization. This is based on a joint work with Prof. Youjiang Lin.
• 12:30 - 2:30 pm EDT

  Lunch/Free Time
• 2:30 - 3:15 pm EDT

  Quasianalyticity and support in geometric tomography

  11th Floor Lecture Hall

  • Speaker
  • Dmitry Faifman, Tel Aviv University

  • Session Chair
  • Luis Rademacher, University of California, Davis

  Abstract

  Section and projection functions of convex bodies are not arbitrary functions; in fact, other than in dimension and codimension one, they span a rather small subspace of all functions on the grassmannian, which exhibits a quasianalytic-type property. This phenomenon holds for a class of integral operators on grassmannians, and more generally for certain representations of the general linear group. As corollaries, we will see sharper versions of Alexandrov's projections theorem, Funk's sections theorem, and Klain's injectivity theorem for even valuations.

  • Video Available Soon
• 3:30 - 4:00 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 4:00 - 4:45 pm EDT

  Curvature measures and soap bubbles beyond convexity

  11th Floor Lecture Hall

  • Speaker
  • Daniel Hug, Karlsruhe Institute of Technology (KIT)

  • Session Chair
  • Luis Rademacher, University of California, Davis

  Abstract

  A fundamental result in differential geometry states that if a smooth hypersurface in a Euclidean space encloses a bounded domain and one of its mean curvature functions is constant, then it is a Euclidean sphere. This statement has been referred to as the soap bubble theorem. Major contributions are due to Alexandrov (1958) and Korevaar--Ros (1988). While the smoothness assumption is seemingly natural at first thought, based on the notion of curvatures measures of convex bodies Schneider (1979) established a characterization of Euclidean spheres among general convex bodies by requiring that one of the curvature measures is proportional to the boundary measure. We describe extensions in two directions: (1) The role of the Euclidean ball is taken by a nice gauge body (Wulff shape) and (2) the problem is studied in a non-convex and non-smooth setting. Thus we obtain characterization results for finite unions of Wulff shapes (bubbling) within the class of mean-convex sets or even for general sets with positive reach. Several related results are established. They include the extension of the classical Steiner--Weyl tube formula to arbitrary closed sets in a uniformly convex normed vector space, formulas for the derivative of the localized volume function of a compact set and general versions of the Heintze--Karcher inequality. (Based on joint work with Mario Santilli)

  • Video Available Soon

September 28, 2022

• 9:00 - 9:45 am EDT

  On the L^p dual Minkowski problem for −1 < p < 0

  11th Floor Lecture Hall

  • Speaker
  • Stephanie Mui, New York University

  • Session Chair
  • Monika Ludwig, Technische Universität Wien

  Abstract

  The L^p dual curvature measure was introduced by Lutwak, Yang, and Zhang in 2018. The associated Minkowski problem, known as the L^p dual Minkowski problem, asks about the existence of a convex body with prescribed L^p dual curvature measure. This question unifies the previously disjoint L^p Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the L^p dual Minkowski problem for the case of q < p + 1, −1 < p < 0, and p≠q for even measures.

  • Video Available Soon
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EDT

  Inequalities for L_p Steiner coefficients

  11th Floor Lecture Hall

  • Speaker
  • Elisabeth Werner, Case Western Reserve University

  • Session Chair
  • Monika Ludwig, Technische Universität Wien

  Abstract

  We show isoperimetric inequalities for weighted L_p affine surface areas which appear in the recently established L_p Steiner formula of the L_p Brunn Minkowski theory. We show that they are related to f-divergences of the cone measures of the convex body and its polar, namely the Kullback-Leibler divergence and the Renyi-divergence. Based on joint work with Kateryna Tatarko.

  • Video Available Soon
• 11:30 am - 12:15 pm EDT

  Randomized Petty projection inequality

  11th Floor Lecture Hall

  • Speaker
  • Kateryna Tatarko, University of Waterloo

  • Session Chair
  • Monika Ludwig, Technische Universität Wien
• 12:25 - 12:30 pm EDT

  Group Photo

  11th Floor Lecture Hall
• 12:30 - 2:30 pm EDT

  Lunch/Free Time
• 2:30 - 3:15 pm EDT

  Infinitesimal characterizations of ellipsoids or balls

  11th Floor Lecture Hall

  • Speaker
  • Alina Stancu, CONCORDIA UNIVERSITY

  • Session Chair
  • Carsten Schuett, CAU Kiel

  Abstract

  We will talk about close (say in Hausdorff metric) convex bodies constructions for which the homothety implies an ellipsoid or a ball. (joint work in progress)

  • Video Available Soon
• 3:30 - 4:00 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 4:00 - 4:45 pm EDT

  The Discrete Gauss Image problem

  11th Floor Lecture Hall

  • Speaker
  • Vadim Semenov, NYU

  • Session Chair
  • Carsten Schuett, CAU Kiel

  Abstract

  The Gauss Image problem is a generalization to the question originally posed by Aleksandrov who studied the existence of the convex body with the prescribed Aleksandrov's integral curvature. A simple discrete case of the Gauss Image Problem can be formulated as follows: given a finite set of directions in Euclidian space and the same number of unit vectors, does there exist a convex polytope in this space containing the origin in its interior with vertices at given directions such that each normal cone at the vertex contains exactly one of the given vectors. In this talk, we are going to discuss the discrete Gauss Image Problem, and its relation to other Minkowski-type problems. Two different proofs of the problem are going to be addressed: A smooth proof based on transportation polytopes and a discrete proof based on Helly’s theorem. This work is based on the recent results of the author.

  • Video Available Soon

September 29, 2022

• 9:00 - 9:45 am EDT

  The extremals of Stanley's inequalities for partially ordered sets

  11th Floor Lecture Hall

  • Speaker
  • Yair Shenfeld, MIT

  • Session Chair
  • Elisabeth Werner, Case Western Reserve University

  Abstract

  The presence of log-concave sequences is prevalent in diverse areas of mathematics ranging from geometry to combinatorics. The ubiquity of such sequences is not completely understood but the last decade has witnessed major progress towards this goal. However, we know very little about the extremals of such sequences: If we have equality somewhere along the sequence, what can be said about the sequence itself? This question is related to optimal structures (e.g. the ball in the isoperimetric inequality) and it is a necessary step towards the improvement and stability of the inequalities themselves. I will talk about the extremals of such sequences coming from the theory of partially ordered sets (posets). R. Stanley showed in the 80's how to associate polytopes to posets and, using this correspondence (via the Alexandrov-Fenchel inequality), he proved that sequences which count the number of linear extensions of posets are log-concave. The extremals of these sequences were unknown however, with even conjectures lacking. I will explain the resolution of this problem and the complete characterization of the extremals. The extremals turn out to be complicated and rich structures which exhibit new phenomena depending on the geometry of the associated polytopes. Towards the resolution of this problem we developed new tools that shed brighter light on the relation between the geometry of polytopes and the combinatorics of partially ordered sets. Joint work with Zhao Yu Ma.
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EDT

  Fractional polar projection bodies

  11th Floor Lecture Hall

  • Speaker
  • Monika Ludwig, Technische Universität Wien

  • Session Chair
  • Elisabeth Werner, Case Western Reserve University

  Abstract

  Affine fractional isoperimetric inequalities are established that are stronger (and directly imply) the Euclidean fractional isoperimetric inequalities. These inequalities are fractional versions of the Petty projection inequality. Using the functional version of fractional polar projection bodies, affine fractional Sobolev inequalities are established that are stronger that the fractional Sobolev inequalities of Almgren and Lieb and imply (in the limit) the affine Sobolev inequality by Gaoyong Zhang. Joint work with Julián Haddad (Universidade Federal de Minas Gerais)

  • Video Available Soon
• 11:30 am - 12:15 pm EDT

  Mean oscillation bounds on geometric rearrangements

  11th Floor Lecture Hall

  • Speaker
  • Almut Burchard, University of Toronto

  • Session Chair
  • Elisabeth Werner, Case Western Reserve University

  Abstract

  Symmetric decreasing rearrangement (when applicable) can reduce a geometric variational problem to a radial problem, where the unknown functions depend on the single variable |x|. Classical inequalities for perimeter, gradient norms, and convolution integrals indicate that symmetric decreasing rearrangement reduces the overall oscillation of functions. Less is known about its effect on the mean oscillation of a function. I will discuss recent result (w. Galia Dafni and Ryan Gibara) on inequalities and continuity properties. The question of sharp inequalities remains open.

  • Video Available Soon
• 12:30 - 2:30 pm EDT

  Lunch/Free Time
• 2:30 - 3:15 pm EDT

  Functional Intrinsic Volumes

  11th Floor Lecture Hall

  • Speaker
  • Fabian Mussnig, TU Wien

  • Session Chair
  • Susanna Dann, Universidad de los Andes

  Abstract

  We consider functional intrinsic volumes on convex functions. In many ways these objects behave similarly to the classical intrinsic volumes on convex bodies. However, we will also show where analogies fail. The presented results include characterizations, representations, integral geometry and inequalities and we will see that some classical results can be retrieved from the new ones. Joint work with Andrea Colesanti, Monika Ludwig and Jacopo Ulivelli.

  • Video Available Soon
• 3:30 - 4:00 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 4:00 - 4:45 pm EDT

  TBA

  11th Floor Lecture Hall

  • Speaker
  • Alexander Litvak, University of Alberta

  • Session Chair
  • Susanna Dann, Universidad de los Andes

October 1, 2022

There are no events currently scheduled for October 1st.