This Week's Seminars and Colloquia

A simplified approach to interacting Bose gases

Series
Time
Tuesday, January 21, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ian JauslinPrinceton University

In 1963, Lieb introduced an effective theory to approximate the ground state energy of a system of Bosons interacting with each other via a repulsive pair potential, in the thermodynamic limit. Lieb showed that in one dimension, this effective theory predicts a ground state energy that differs at most by 20% from its exact value, for any density. The main idea is that instead of considering marginals of the square of the wave function, as in Hartree theory, we consider marginals of the wave function itself, which is positive in the ground state. The effective theory Lieb obtained is a non-linear integro-differential equation, whose non-linearity is an auto-convolution. In this talk, I will discuss some recent work about this effective equation. In particular, we proved the existence of a solution. We also proved that the ground state energy obtained from this simplified equation agrees exactly with that of the full N-body system at asymptotically low and at high densities. In fact, preliminary numerical work has shown that, for some potentials, the ground state energy can be computed in this way with an error of at most 5% over the entire range of densities. This is joint work with E. Carlen and E.H. Lieb.

Convection in a coupled free-flow porous media flow system

Series
PDE Seminar
Time
Tuesday, January 21, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Xiaomin WangSouthern University of Science and Technology

We show that the Principle of Exchange of Stability holds for convection in a layer of fluids overlaying a porous media with proper interface boundary conditions and suitable assumption on the parameters. The physically relevant small Darcy number regime as well as the dependence of the convection on various parameters will be discussed. A theory on the dependence of the depth ratio of the onset of deep convection will be put forth together with supporting numerical evidence. A decoupled uniquely solvable, unconditionally stable numerical scheme for solving the system will be presented as well.

Topology of knot complements

Series
Geometry Topology Student Seminar
Time
Wednesday, January 22, 2020 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyun Ki MinGeorgia Tech

Note the unusual time

Gordon and Luecke showed that the knot complements determine the isotopy classes of knots in S^3. In this talk, we will study the topology of various knot complements in S^3: torus knots, cable knots, satellite knots, etc. As an application, we will see some knot invariants using knot complements.

On the Log-Brunn-Minkowski conjecture and other questions

Series
Analysis Seminar
Time
Wednesday, January 22, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna LivshytsGeorgia Tech

We will discuss certain isoperimetric-type problems for convex sets, such as the Log-Brunn-Minkowski conjecture for Lebesgue measure, and will explain the approach to this type of problems via local versions of inequalities and why it arises naturally. We consider a weaker form of the conjecture and prove it in several cases, with elementary geometric methods.  We shall also consider several illustrative ``hands on’’ examples. If time permits, we will discuss Bochner’s method approach to the question and formulate some new results in this regard. The second (optional!) part of this talk will be at the High-dimensional seminar right after, and will involve a discussion of more involved methods. Partially based on a joint work with Hosle and Kolesnikov.

On the L_p-Brunn-Minkowski inequality for measures

Series
High Dimensional Seminar
Time
Wednesday, January 22, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Galyna LivshytsGeorgia Tech

The first part of this pair of talks will be given at the Analysis seminar right before; attending it is not necessary, as all the background will be given in this lecture as well, and the talks will be sufficiently independent of each other.

I will discuss the L_p-Brunn-Minkowski inequality for log-concave measures, explain ‘’Bochner’s method’’ approach to this problem and state and prove several new results. This falls into a general framework of isoperimetric type inequalities in high-dimensional euclidean spaces. Joint with Hosle and Kolesnikov.

Matroids, log-concavity, and expanders

Series
Job Candidate Talk
Time
Thursday, January 23, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cynthia VinzantNorth Carolina State University

Abstract:  Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties.  I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

Geometric statistics for shape analysis of bioimaging data

Series
Job Candidate Talk
Time
Thursday, January 23, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nina MiolaneStanford University

The advances in bioimaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, statistical analyses in biomedical research are poised to incorporate more shape data. This leads to the question: how do we define quantitative descriptions of shape variability from images?

Mathematically, landmarks’ shapes, curve shapes, or surface shapes can be seen as the remainder after we have filtered out the corresponding object’s position and orientation. As such, shape data belong to quotient spaces, which are non-Euclidean spaces.

In this talk, I introduce “Geometric statistics”, a statistical theory for data belonging to non-Euclidean spaces. In the context of shape data analysis, I use geometric statistics to prove mathematically and experimentally that the “template shape estimation” algorithm, used for more than 15 years in biomedical imaging and signal processing, has an asymptotic bias. As an alternative, I present variational autoencoders (VAEs) and discuss the accuracy-speed trade-off of these procedures. I show how to use VAEs to estimate biomolecular shapes from cryo-electron microscopy (cryo-EM) images. This study opens the door to unsupervised fast (cryo-EM) biological shape estimation and analysis.

Fast uniform generation of random graphs with given degree sequences

Series
Combinatorics Seminar
Time
Friday, January 24, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
tBA
Speaker
Andrii ArmanEmory University

In this talk we discuss an algorithm that generates a graph with given degree sequence uniformly at random. Provided that $\Delta^4=O(m)$, where $\Delta$ is the maximal degree and $m$ is the number of edges,  the algorithm runs in expected time $O(m)$. Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time $O(m^2\Delta^2)$ for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also  obtain algorithms with expected run time which is (i) linear for power-law degree sequences in cases where the previous best was $O(n^{4.081})$, and  (ii) $O(nd+d^4)$ for $d$-regular graphs when $d=o(\sqrt n)$, where the previous best was $O(nd^3)$.