Seminars and Colloquia Schedule

The Fractional Laplacian: Approximation and Applications

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 22, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Hans-Werner van Wyk – Auburn University
The fractional Laplacian is a non-local spatial operator describing anomalous diffusion processes, which have been observed abundantly in nature. Despite its many similarities with the classical Laplacian in unbounded domains, its definition on bounded regions is more problematic. So is its numerical discretization. Difficulties arise as a result of the integral kernel's singularity at the origin as well as its unbounded support. In this talk, we discuss a novel finite difference method to discretize the fractional Laplacian in hypersingular integral form. By introducing a splitting parameter, we first formulate the fractional Laplacian as the weighted integral of a function with a weaker singularity, and then approximate it by a weighted trapezoidal rule. Our method generalizes the standard finite difference approximation of the classical Laplacian and exhibits the same quadratic convergence rate, for any fractional power in (0, 2), under sufficient regularity conditions. We present theoretical error bounds and demonstrate our method by applying it to the fractional Poisson equation. The accompanying numerical examples verify our results, as well as give additional insight into the convergence behavior of our method.

Genuine Equivariant Operads

Series
Geometry Topology Seminar
Time
Monday, October 22, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Luis Alexandre Pereira – Georgia Tech
A fundamental result in equivariant homotopy theory due to Elmendorf states that the homotopy theory of G-spaces, with w.e.s measured on all fixed points, is equivalent to the homotopy theory of G-coefficient systems in spaces, with w.e.s measured at each level of the system. Furthermore, Elmendorf’s result is rather robust: analogue results can be shown to hold for, among others, the categories of (topological) categories and operads. However, it has been known for some time that in the G-operad case such a result does not capture the ”correct” notion of weak equivalence, a fact made particularly clear in work of Blumberg and Hill discussing a whole lattice of ”commutative operads with only some norms” that are not distinguished at all by the notion of w.e. suggested above. In this talk I will talk about part of a joint project which aims at providing a more diagrammatic understanding of Blumberg and Hill’s work using a notion of G-trees, which are a generalization of the trees of Cisinski-Moerdijk-Weiss. More specifically, I will describe a new algebraic structure, which we dub a ”genuine equivariant operad”, which naturally arises from the study of G-trees and which allows us to state the ”correct” analogue of Elmendorf’s theorem for G-operads.

Knot invariants and algebraic structures based on knots

Series
Research Horizons Seminar
Time
Wednesday, October 24, 2018 - 12:20 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Thang Le – Georgia Tech
A knot is a simple closed curve in the 3-space. Knots appeared as one of the first objects of study in topology. At first knot theory was rather isolated in mathematics. Lately due to newly discovered invariants and newly established connections to other branches of mathematics, knot theory has become an attractive and fertile area where many interesting, intriguing ideas collide. In this talk we discuss a new class of knot invariants coming out of the Jones polynomial and an algebra of surfaces based on knots (skein algebra) which has connections to many important objects including hyperbolic structures of surfaces and quantum groups. The talk is elementary.

Introduction to geometric tomography

Series
High Dimensional Seminar
Time
Wednesday, October 24, 2018 - 12:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dmitry Ryabogin – Kent State University –

We will discuss several open problems concerning unique determination of convex bodies in the n-dimensional Euclidean space given some information about their projections or sectionson all sub-spaces of dimension n-1. We will also present some related results.

On the fifth Busemann-Petty problem

Series
Analysis Seminar
Time
Wednesday, October 24, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmirty Ryabogin – Kent State University –
In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.

Boothby Wang Fibrations, K-Contact Structures and Regularity

Series
Geometry Topology Student Seminar
Time
Wednesday, October 24, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Surena Hozoori – Georgia Institute of Technology –
Boothby Wang fibrations are historically important examples of contact manifolds and it turns out that we can equip these contact manifolds with extra structures, namely K-contact structures. Based on the study of the relation of these examples and the regularity properties of the corresponding Reeb vector fields, works of Boothby, Wang, Thomas and Rukimbira gives a classification of K-contact structures.

Strongly dissipative systems with a quasi-periodic forcing term

Series
Math Physics Seminar
Time
Wednesday, October 24, 2018 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guido Gentile – Universita di Roma 3 –
We consider a class of singular ordinary differential equations, describing systems subject to a quasi-periodic forcing term and in the presence of large dissipation, and study the existence of quasi-periodic solutions with the same frequency vector as the forcing term. Let A be the inverse of the dissipation coefficient. More or less strong non-resonance conditions on the frequency assure different regularity in the dependence on the parameter A: by requiring a non-degeneracy condition on the forcing term, smoothness and analyticity, and even Borel-summability, follow if suitable Diophantine conditions are assumed, while, without assuming any condition, in general no more than a continuous dependence on A is obtained. We investigate the possibility of weakening the non-degeneracy condition and still obtaining a solution for arbitrary frequencies.

Strong chromatic index for (3, Δ)-bipartite graphs

Series
Graph Theory Working Seminar
Time
Wednesday, October 24, 2018 - 16:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Chi-Nuo Lee – Georgia Tech
Erdős and Nešetřil conjectured in 1985 that every graph with maximum degree Δ can be strong edge colored using at most (5/4)Δ^2 colors. A (Δ_a, Δ_ b)-bipartite graphs is an bipartite graph such that its components A,B has maximum degree Δ_a, Δ_ b respectively. R.A. Brualdi and J.J. Quinn Massey (1993) conjectured that the strong chromatic index of (Δ_a, Δ_ b)-bipartite graphs is bounded by Δ_a*Δ_ b. In this talk, we focus on a recent result affirming the conjecture for (3, Δ)-bipartite graphs.

Gabor Lugosi lectures on combinatorial statistics (3 of 3)

Series
Other Talks
Time
Thursday, October 25, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lectures on Combinatorial Statistics – Pompeu Fabra University, Barcelona –

Thanks are due to our colleague, Vladimir Koltchinskii, for arranging this visit. Please write to Vladimir if you would like to meet with Professor Gabor Lugosi during his visit, or for additional information.

In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable.

TRIAD Distinguished Lecture Series: Lectures on Combinatorial Statistics

Series
School of Mathematics Colloquium
Time
Thursday, October 25, 2018 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gábor Lugosi – Pompeu Fabra University, Barcelona
In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable.

Matroids over pastures

Series
Graph Theory Seminar
Time
Thursday, October 25, 2018 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Matthew Baker – Math, GT
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects which we call pastures; they generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over pastures; in fact, there are at least two natural notions of matroid in this general context, which we call weak and strong matroids. We present ``cryptomorphic'’ descriptions of each kind of matroid. To a (classical) rank-$r$ matroid $M$ on $E$, we can associate a universal pasture (resp. weak universal pasture) $k_M$ (resp. $k_M^w$). We show that morphisms from the universal pasture (resp. weak universal pasture) of $M$ to a pasture $F$ are canonically in bijection with strong (resp. weak) representations of $M$ over $F$. Similarly, the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios'', which we call the foundation of $M$, parametrizes rescaling classes of weak $F$-matroid structures on $M$. As a sample application of these considerations, we give a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.

An Introduction to Étale Cohomology

Series
Student Algebraic Geometry Seminar
Time
Thursday, October 25, 2018 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel Minahan – Georgia Tech
We will discuss some basic concepts in étale cohomology and compare them to the more explicit constructions in both algebraic geometry and algebraic topology.

Stabilization of Diffusion Limited Aggregation in a Wedge

Series
Stochastics Seminar
Time
Thursday, October 25, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eviatar Procaccia – Texas A&M –
We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA as a finite time growth process and questions about the number of arms, growth and dimension. I will present some conjectures and open problems. This is joint work with Ron Rosenthal (Technion) and Yuan Zhang (Pekin University).

High-Dimensional Robust Mean Estimation in Nearly-Linear Time

Series
ACO Student Seminar
Time
Friday, October 26, 2018 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yu Cheng – CS, Duke University –
We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent error guarantees for several families of structured distributions. In this work, we give the first nearly-linear time algorithms for high-dimensional robust mean estimation. Specifically, we focus on distributions with (i) known covariance and sub-gaussian tails, and (ii) unknown bounded covariance. Given $N$ samples on $R^d$, an $\epsilon$-fraction of which may be arbitrarily corrupted, our algorithms run in time $\tilde{O}(Nd)/poly(\epsilon)$ and approximate the true mean within the information-theoretically optimal error, up to constant factors. Previous robust algorithms with comparable error guarantees have running times $\tilde{\Omega}(N d^2)$. Our algorithms rely on a natural family of SDPs parameterized by our current guess $\nu$ for the unknown mean $\mu^\star$. We give a win-win analysis establishing the following: either a near-optimal solution to the primal SDP yields a good candidate for $\mu^\star$ -- independent of our current guess $\nu$ -- or the dual SDP yields a new guess $\nu'$ whose distance from $\mu^\star$ is smaller by a constant factor. We exploit the special structure of the corresponding SDPs to show that they are approximately solvable in nearly-linear time. Our approach is quite general, and we believe it can also be applied to obtain nearly-linear time algorithms for other high-dimensional robust learning problems. This is a joint work with Ilias Diakonikolas and Rong Ge.

Chi-y genera of generic intersections in algebraic tori and refined tropicalizations

Series
Algebra Seminar
Time
Friday, October 26, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andreas Gross – Colorado State University
An algorithm to compute chi-y genera of generic complete intersections in algebraic tori has already been known since the work of Danilov and Khovanskii in 1978, yet a closed formula has been given only very recently by Di Rocco, Haase, and Nill. In my talk, I will show how this formula simplifies considerably after an extension of scalars. I will give an algebraic explanation for this phenomenon using the Grothendieck rings of vector bundles on toric varieties. We will then see how the tropical Chern character gives rise to a refined tropicalization, which retains the good properties of the usual, unrefined tropicalization.

Critical behavior for percolation on graphs with given degrees

Series
Combinatorics Seminar
Time
Friday, October 26, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Souvik Dhara – Microsoft Research New England
We discuss some recent developments on the critical behavior of percolation on finite random networks. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdos-Renyi random graph (ERRG). Subsequently, there has been a surge in the literature, revealing several interesting scaling limits of these critical components, namely, the component size, diameter, or the component itself when viewed as a metric space. Fascinatingly, when the third moment of the asymptotic degree distribution is finite, many random graph models has been shown to exhibit a universality phenomenon in the sense that their scaling exponents and limit laws are the same as the ERRG. In contrast, when the asymptotic degree distribution is heavy-tailed (having an infinite third moment), the limit law turns out to be fundamentally different from the ERRG case and in particular, becomes sensitive to the precise asymptotics of the highest degree vertices. In this talk, we will focus on random graphs with a prescribed degree sequence. We start by discussing recent scaling limit results, and explore the universality classes that arise from heavy-tailed networks. Of particular interest is a new universality class that arises when the asymptotic degree distribution has an infinite second moment. Not only it gives rise to a completely new universality class, it also exhibits several surprising features that have never been observed in any other universality class so far. This is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden and Sanchayan Sen.

Invariant Manifolds Associated to Non-resonant Spectral Subspaces II

Series
Dynamical Systems Working Seminar
Time
Friday, October 26, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 156
Speaker
Jiaqi Yang – GT Math
We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace, and some non-resonance conditions are satisfied. Then the map leaves invariant a smooth (as smooth as the map) manifold, which is unique among C^L invariant manifolds. Here, L only depends on the spectrum of the linearization. This is based on a work of Prof. Rafael de la Llave.