Seminars and Colloquia by Series

Convex Geometry of the Truncated Moment Problem

Series
High Dimensional Seminar
Time
Wednesday, February 13, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Greg BlekhermanGeorgia Tech

Moment problem is a classical question in real analysis, which asks whether a set of moments can be realized as integration of corresponding monomials with respect to a Borel measure. Truncated moment problem asks the same question given a finite set of moments. I will explain how some of the fundamental results in the truncated moment problem can be proved (in a very general setting) using elementary convex geometry. No familiarity with moment problems will be assumed. This is joint work with Larry Fialkow.

On delocalization of eigenvectors of random non-Hermitian matrices

Series
High Dimensional Seminar
Time
Wednesday, February 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anna LytovaUniversity of Opole

We study delocalization properties of null vectors and eigenvectors of matrices with i.i.d. subgaussian entries. Such properties describe quantitatively how "flat" is a vector and confirm one of the universality conjectures stating that distributions of eigenvectors of many classes of random matrices are close to the uniform distribution on the unit sphere. In particular, we get lower bounds on the smallest coordinates of eigenvectors, which are optimal as the case of Gaussian matrices shows. The talk is based on the joint work with Konstantin Tikhomirov.

Combinatorial methods in frame theory

Series
High Dimensional Seminar
Time
Wednesday, January 30, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles. 006
Speaker
Alex IosevichUniversity of Rochester

We shall survey a variety of results, some recent, some going back a long time, where combinatorial methods are used to prove or disprove the existence of orthogonal exponential bases and Gabor bases. The classical Erdos distance problem and the Erdos Integer Distance Principle play a key role in our discussion.

Few conjectures on intrinsic volumes on Riemannian manifolds and Alexandrov spaces

Series
High Dimensional Seminar
Time
Wednesday, January 23, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Semyon AleskerTel Aviv University

The celebrated Hadwiger's theorem says that linear combinations of intrinsic volumes on convex sets are the only isometry invariant continuous valuations(i.e. finitely additive measures). On the other hand H. Weyl has extended intrinsic volumes beyond convexity, to Riemannian manifolds. We try to understand the continuity properties of this extension under theGromov-Hausdorff convergence (literally, there is no such continuityin general). First, we describe a new conjectural compactification of the set of all closed Riemannian manifolds with given upper bounds on dimensionand diameter and lower bound on sectional curvature. Points of thiscompactification are pairs: an Alexandrov space and a constructible(in the Perelman-Petrunin sense) function on it. Second, conjecturally all intrinsic volumes extend by continuity to this compactification. No preliminary knowledge of Alexandrov spaces will be assumed, though it will be useful.

Convex bodies in high dimensions and algebraic geometry

Series
High Dimensional Seminar
Time
Monday, December 3, 2018 - 12:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yanir RubinshteinUniversity of Maryland

Please Note: Note the special time!

In joint work with J. Martinez-Garcia we study the classification problem of asymptotically log del Pezzo surfaces in algebraic geometry. This turns out to be equivalent to understanding when certain convex bodies in high-dimensions intersect the cube non-trivially. Beyond its intrinsic interest in algebraic geometry this classification is relevant to differential geometery and existence of new canonical metricsin dimension 4.

Lattice points and cube slicing

Series
High Dimensional Seminar
Time
Wednesday, November 28, 2018 - 12:55 for 1 hour (actually 50 minutes)
Location
skiles 006
Speaker
Marcel CelayaGeorgia Institute of technology

In this talk I will describe those linear subspaces of $\mathbf{R}^d$ which can be formed by taking the linear span of lattice points in a half-open parallelepiped. I will draw some connections between this problem and Keith Ball's cube slicing theorem, which states that the volume of any slice of the unit cube $[0,1]^d$ by a codimension-$k$ subspace is at most $2^{k/2}$.

Estimating High-dimensional Gaussian Tails

Series
High Dimensional Seminar
Time
Wednesday, November 14, 2018 - 12:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben CousinsColumbia University

The following is a well-known and difficult problem in rare event simulation: given a set and a Gaussian distribution, estimate the probability that a sample from the Gaussian distribution falls outside the set. Previous approaches to this question are generally inefficient in high dimensions. One key challenge with this problem is that the probability of interest is normally extremely small. I'll discuss a new, provably efficient method to solve this problem for a general polytope and general Gaussian distribution. Moreover, in practice, the algorithm seems to substantially outperform our theoretical guarantees and we conjecture that our analysis is not tight. Proving the desired efficiency relies on a careful analysis of (highly) correlated functions of a Gaussian random vector.Joint work with Ton Dieker.

Analysis and recovery of high-dimensional data with low-dimensional structures

Series
High Dimensional Seminar
Time
Wednesday, November 7, 2018 - 12:52 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Wenjing LiaoGeorgia Tech

High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning and data recovery. Many datasets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for the purpose of compression and inference, and giving performance guarantees depending on the intrinsic dimension of data. I will present two sets of problems: one is related with manifold learning; the other arises from imaging and signal processing where we want to recover a high-dimensional, sparse vector from few linear measurements. In the first problem, we model a data set in $R^D$ as samples from a probability measure concentrated on or near an unknown $d$-dimensional manifold with $d$ much smaller than $D$. We develop a multiscale adaptive scheme to build low-dimensional geometric approximations of the manifold, as well as approximating functions on the manifold. The second problem arises from source localization in signal processing where a uniform array of sensors is set to collect propagating waves from a small number of sources. I will present some theory and algorithms for the recovery of the point sources with high precision.

Smooth valuations and their products

Series
High Dimensional Seminar
Time
Wednesday, October 31, 2018 - 12:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joe FuUGA

Alesker has introduced the notion of a smooth valuation on a smooth manifold M. This is a special kind of set function, defined on sufficiently regular compact subsets A of M, extending the corresponding idea from classical convexity theory. Formally, a smooth valuation is a kind of curvature integral; informally, it is a sum of Euler characteristics of intersections of A with a collection of objects B. Smooth valuations admit a natural multiplication, again due to Alesker. I will aim to explain the rather abstruse formal definition of this multiplication, and its relation to the ridiculously simple informal counterpart given by intersections of the objects B.

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