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School of Mathematics

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Now showing 1 - 10 of 436

Fourier Analysis in Geometric Tomography - Part 3

(Georgia Institute of Technology, 2019-12-13) Koldobsky, Alexander

Geometric tomography is the study of geometric properties of solids based on data about sections and projections of these solids. The lectures will include: 1. An outline of proofs of two of the main features of the Fourier approach to geometric tomography - the relation between the derivatives of the parallel section function of a body and the Fourier transform (in the sense of distributions) of powers of the norm generated by this body, and the Fourier characterization of intersection bodies. 2. The Busemann-Petty problem asks whether symmetric convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. We will prove an isomorphic version of the problem with a constant depending on the distance from the class of intersection bodies. This will include a generalization to arbitrary measures in place of volume. 3. The slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. We will consider a version of this problem for arbitrary measures in place of volume. We will show that the answer is affirmative for many classes of bodies, but in general the constant must be of the order 1/√n. 4. Optimal estimates for the maximal distance from a convex body to the classes of intersection bodies and the unit balls of subspaces of Lp. 5. We will use the Fourier approach to prove that the only polynomially integrable convex bodies, i.e. bodies whose parallel section function in every direction is a polynomial of the distance from the origin, are ellipsoids in odd dimensions.
Floating Bodies and Approximation - Part 3

(Georgia Institute of Technology, 2019-12-12) Werner, Elisabeth

Two important closely related notions in affine convex geometry are the floating body and the affine surface area of a convex body. The floating body of a convex body is obtained by cutting off caps of volume less or equal to a fixed positive constant. Taking the right-derivative of the volume of the floating body gives rise to an affine invariant, the affine surface area. This was established for all convex bodies in all dimensions by Schuett and Werner. There is a natural inequality associated with affine surface area, the affine isoperimetric inequality, which states that among all convex bodies, with fixed volume, affine surface area is maximized for ellipsoids. Due to its important properties, which make them effective and powerful tools, affine surface area and floating body are omnipresent in geometry and have applications in many other areas of mathematics, e.g., in problems of approximation of convex bodies by polytopes and for the notion of halfspace depth for multivariate data from statistics.
Fourier Analysis in Geometric Tomography - Part 2

(Georgia Institute of Technology, 2019-12-11) Koldobsky, Alexander

Geometric tomography is the study of geometric properties of solids based on data about sections and projections of these solids. The lectures will include: 1. An outline of proofs of two of the main features of the Fourier approach to geometric tomography - the relation between the derivatives of the parallel section function of a body and the Fourier transform (in the sense of distributions) of powers of the norm generated by this body, and the Fourier characterization of intersection bodies. 2. The Busemann-Petty problem asks whether symmetric convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. We will prove an isomorphic version of the problem with a constant depending on the distance from the class of intersection bodies. This will include a generalization to arbitrary measures in place of volume. 3. The slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. We will consider a version of this problem for arbitrary measures in place of volume. We will show that the answer is affirmative for many classes of bodies, but in general the constant must be of the order 1/√n. 4. Optimal estimates for the maximal distance from a convex body to the classes of intersection bodies and the unit balls of subspaces of Lp. 5. We will use the Fourier approach to prove that the only polynomially integrable convex bodies, i.e. bodies whose parallel section function in every direction is a polynomial of the distance from the origin, are ellipsoids in odd dimensions.
Floating Bodies and Approximation - Part 2

(Georgia Institute of Technology, 2019-12-11) Werner, Elisabeth

Two important closely related notions in affine convex geometry are the floating body and the affine surface area of a convex body. The floating body of a convex body is obtained by cutting off caps of volume less or equal to a fixed positive constant. Taking the right-derivative of the volume of the floating body gives rise to an affine invariant, the affine surface area. This was established for all convex bodies in all dimensions by Schuett and Werner. There is a natural inequality associated with affine surface area, the affine isoperimetric inequality, which states that among all convex bodies, with fixed volume, affine surface area is maximized for ellipsoids. Due to its important properties, which make them effective and powerful tools, affine surface area and floating body are omnipresent in geometry and have applications in many other areas of mathematics, e.g., in problems of approximation of convex bodies by polytopes and for the notion of halfspace depth for multivariate data from statistics.
Concentration and Convexity - Part 1

(Georgia Institute of Technology, 2019-12-10) Paouris, Grigoris

The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles.
Floating Bodies and Approximation - Part 1

(Georgia Institute of Technology, 2019-12-09) Werner, Elisabeth

Two important closely related notions in affine convex geometry are the floating body and the affine surface area of a convex body. The floating body of a convex body is obtained by cutting off caps of volume less or equal to a fixed positive constant. Taking the right-derivative of the volume of the floating body gives rise to an affine invariant, the affine surface area. This was established for all convex bodies in all dimensions by Schuett and Werner. There is a natural inequality associated with affine surface area, the affine isoperimetric inequality, which states that among all convex bodies, with fixed volume, affine surface area is maximized for ellipsoids. Due to its important properties, which make them effective and powerful tools, affine surface area and floating body are omnipresent in geometry and have applications in many other areas of mathematics, e.g., in problems of approximation of convex bodies by polytopes and for the notion of halfspace depth for multivariate data from statistics.
Fourier Analysis in Geometric Tomography - Part 1

(Georgia Institute of Technology, 2019-12-09) Koldobsky, Alexander

Geometric tomography is the study of geometric properties of solids based on data about sections and projections of these solids. The lectures will include: 1. An outline of proofs of two of the main features of the Fourier approach to geometric tomography - the relation between the derivatives of the parallel section function of a body and the Fourier transform (in the sense of distributions) of powers of the norm generated by this body, and the Fourier characterization of intersection bodies. 2. The Busemann-Petty problem asks whether symmetric convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. We will prove an isomorphic version of the problem with a constant depending on the distance from the class of intersection bodies. This will include a generalization to arbitrary measures in place of volume. 3. The slicing problem of Bourgain asks whether every symmetric convex body of volume one has a hyperplane section with area greater than an absolute constant. We will consider a version of this problem for arbitrary measures in place of volume. We will show that the answer is affirmative for many classes of bodies, but in general the constant must be of the order 1/√n. 4. Optimal estimates for the maximal distance from a convex body to the classes of intersection bodies and the unit balls of subspaces of Lp. 5. We will use the Fourier approach to prove that the only polynomially integrable convex bodies, i.e. bodies whose parallel section function in every direction is a polynomial of the distance from the origin, are ellipsoids in odd dimensions.
Concentration and Convexity - Part 2

(Georgia Institute of Technology, 2019-12) Paouris, Grigoris

The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. In these talks I will introduce the phenomenon and I will show how High dimensional Geometry affects the concentration. In particular I will explain how "convexity" can be used to establish strong concentration inequalities in the Gauss space and how the "convexity" of the underline measure is responsible for deviation principles.
6-connected graphs are two-three linked

(Georgia Institute of Technology, 2019-11-11) Xie, Shijie

Let $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on Four Color Theorem, Hadwiger's conjecture for $K_6$, and J\o rgensen's conjecture, Robertson and Seymour asked when does $G$ contain disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We prove that if $G$ is 6-connected then such $G_1,G_2$ exist. Joint work with Robin Thomas and Xingxing Yu.
The proxy point method for rank-structured matrices

(Georgia Institute of Technology, 2019-11-06) Xing, Xin

Rank-structured matrix representations, e.g., $\mathcal{H}^2$ and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their application is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form. This dissertation is mainly about the study and application of a hybrid analytic-algebraic compression method, called \textit{the proxy point method}. This work uncovers the full strength of this presently underutilized method that could potentially resolve the above bottleneck for all rank-structured matrix techniques. As a result, this work could extend the applicability and improve the performance of rank-structured matrix techniques and thus facilitate dense matrix computations in a wider range of scientific computing problems, such as particle simulations, numerical solution of integral equations, and Gaussian processes. Application of the proxy point method in practice is presently very limited. Only two special instances of the method have been used heuristically to compress interaction blocks defined by specific kernel functions over points. We address several critical problems of the proxy point method which limit its applicability. A general form of the method is then proposed, paving the way for its wider application in the construction of different rank-structured matrix representations with kernel functions that are more general than those usually used. In addition to kernel-defined interactions between points, we further extend the applicability of the proxy point method to compress the interactions between charge distributions in quantum chemistry calculations. Specifically, we propose a variant of the proxy point method to efficiently construct an $\mathcal{H}^2$ matrix representation of the four-dimensional electron repulsion integral tensor. The linear-scaling matrix-vector multiplication algorithm for the constructed $\mathcal{H}^2$ matrix is then used for fast Coulomb matrix construction which is an important step in many quantum chemical methods. Two additional contributions related to $\mathcal{H}^2$ and HSS matrices are also presented. First, we explain the exact equivalence between $\mathcal{H}^2$ matrices and the fast multipole method (FMM). This equivalence has not been rigorously studied in the literature. Numerical comparisons between FMM and $\mathcal{H}^2$ matrices based on the proxy point method are also provided, showing the relative advantages and disadvantages of the two methods. Second, we consider the application of HSS approximations as preconditioners for symmetric positive definite (SPD) matrices. Preserving positive definiteness is essential for rank-structured matrix approximations to be used efficiently in various algorithms and applications, e.g., the preconditioned conjugate gradient method. We propose two methods for constructing HSS approximations to an SPD matrix that preserve positive definiteness.