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Organizing Committee
North Dakota State University
Kent State University
University of Alberta
Kent State University
Abstract
Geometric tomography is the area of Mathematics dealing with the
retrieval of information about solid objects based on the size of
their sections or projections, or other lower dimensional data.
Results from this area often find real-world applications in science
and engineering.
In recent years geometric tomography has seen a rapid period of growth
due to many exciting developments in harmonic analysis. The goal of
the present workshop is to bring together specialists in geometric
tomography, harmonic analysis, and related areas to discuss important
advances and share new ideas.
retrieval of information about solid objects based on the size of
their sections or projections, or other lower dimensional data.
Results from this area often find real-world applications in science
and engineering.
In recent years geometric tomography has seen a rapid period of growth
due to many exciting developments in harmonic analysis. The goal of
the present workshop is to bring together specialists in geometric
tomography, harmonic analysis, and related areas to discuss important
advances and share new ideas.
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Workshop Schedule
Monday, 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.
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.
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.
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.
5:00 - 6:30 PM EDT
Reception
10th Floor Collaborative Space
Tuesday, September 27, 2022
9:00 - 9:45 AM EDT
TBA
11th Floor Lecture Hall
Virtual Speaker
Serhii 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.
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.
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)
Wednesday, 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.
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.
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 (Immediately After Talk)
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)
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.
Thursday, 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)
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.
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.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
On the volume ratio of projections of convex bodies.
11th Floor Lecture Hall
Speaker
Alexander Litvak, University of Alberta
Session Chair
Susanna Dann, Universidad de los Andes
Abstract
At the beginning of the talk we say a few words about two distinguished mathematicians, Y. Gordon and N. Tomczak-Jaegermann, who passed away in June 2022. Then we review several results related to the volume ratio and the modified Banach-Mazur distance between convex bodies. Finally, we show that for every convex body $K$ there exists a symmetric convex body $L$ such that for any two projections $P$ and $Q$ of rank $k>\sqrt{n \ln n}$ the volume ratio between $PK$ and $QL$ is large. This is a joint work with D.Galicer, M.Merzbacher, and D.Pinasco.
Friday, September 30, 2022
9:00 - 9:45 AM EDT
TBA
11th Floor Lecture Hall
Virtual Speaker
Beatrice-Helen Vritsiou, University of Alberta
Session Chair
Maria Alfonseca Cubero, North Dakota State University
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Expansion of random 0/1 polytopes
11th Floor Lecture Hall
Speaker
Luis Rademacher, University of California, Davis
Session Chair
Maria Alfonseca Cubero, North Dakota State University
Abstract
This talk will be about a type of discrete isoperimetric inequality and uses projections of polytopes in a fundamental way. A conjecture of Milena Mihail and Umesh Vazirani states that the edge expansion of the graph of every 0/1 polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a 0/1 polytope in R^d is greater than 1 over some polynomial function of d. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random 0/1 polytope in R^d is at least 1/12d with high probability.
This is joint work with Brett Leroux.
11:30 AM - 12:15 PM EDT
The approximation of almost time- and band-limited functions by their expansion in some orthogonal polynomials bases
11th Floor Lecture Hall
Speaker
Susanna Spektor, Sheridan college institute of technology
Session Chair
Maria Alfonseca Cubero, North Dakota State University
Abstract
In this joint work with Philippe Jaming and Abderrazek Karoui our aim is to investigate the quality of approximation of almost time- and almost band-limited functions by its expansion in two classical orthogonal polynomials bases: the Hermite basis and the ultraspherical polynomials bases (which include Legendre and Chebyshev bases as particular cases). This allows us to obtain the quality of approximation in the $L^2$ Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions.
12:30 - 2:30 PM EDT
Lunch/Free Time
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
