Seminars and Colloquia by Series

Coloring graphs with no K5-subdivision: disjoint paths in graphs

Series
Dissertation Defense
Time
Tuesday, March 12, 2019 - 10:00 for 1.5 hours (actually 80 minutes)
Location
203 Classroom D.M. Smith
Speaker
Qiqin Xie – Georgia Institute of Technology
The Four Color Theorem states that every planar graph is 4-colorable. Hajos conjectured that for any positive integer k, every graph containing no K_{k+1}-subdivision is k-colorable. However, Catlin disproved Hajos' conjecture for k >= 6. It is not hard to prove that the conjecture is true for k <= 3. Hajos' conjecture remains open for k = 4 and k = 5. We consider a minimal counterexample to Hajos' conjecture for k = 4: a graph G, such that G contains no K_5-subdivision, G is not 4-colorable, and |V (G)| is minimum. We use Hajos graph to denote such counterexample. One important step to understand graphs containing K_5-subdivisions is to solve the following problem: let H represent the tree on six vertices, two of which are adjacent and of degree 3. Let G be a graph and u1, u2, a1, a2, a3, a4 be distinct vertices of G. When does G contain a topological H (i.e. an H-subdivision) in which u1, u2 are of degree 3 and a1, a2, a3, a4 are of degree 1? We characterize graphs with no topological H. This characterization is used by He, Wang, and Yu to show that graph containing no K_5-subdivision is planar or has a 4-cut, establishing conjecture of Kelmans and Seymour. Besides the topological H problem, we also obtained some further structural information of Hajos graphs.

Using 2-torsion to obstruct topological isotopy

Series
Geometry Topology Seminar
Time
Monday, March 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Hannah Schwartz – Bryn Mawr
It is well known that two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other. In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but are not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres; this shows that Gabai's recent ``4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis.

Sectional monodromy groups of projective curves

Series
Algebra Seminar
Time
Monday, March 11, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Borys Kadets – MIT –

Let X be a degree d curve in the projective space P^r.

A general hyperplane H intersects X at d distinct points; varying H defines a monodromy action on X∩H. The resulting permutation group G is the sectional monodromy group of X. When the ground field has characteristic zero the group G is known to be the full symmetric group.

By work of Harris, if G contains the alternating group, then X satisfies a strengthened Castelnuovo's inequality (relating the degree and the genus of X).

The talk is concerned with sectional monodromy groups in positive characteristic. I will describe all non-strange non-degenerate curves in projective spaces of dimension r>2 for which G is not symmetric or alternating. For a particular family of plane curves, I will compute the sectional monodromy groups and thus answer an old question on Galois groups of generic trinomials.

Spheres in 4-manifolds

Series
Geometry Topology Seminar Pre-talk
Time
Monday, March 11, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 257
Speaker
Hannah Schwartz – Bryn Mawr
In this talk, we will examine the relationship between homotopy, topological isotopy, and smooth isotopy of surfaces in 4-manifolds. In particular, we will discuss how to produce (1) examples of topologically but not smoothly isotopic spheres, and (2) a smooth isotopy from a homotopy, under special circumstances (i.e. Gabai's recent work on the ``4D Lightbulb Theorem").

Mathapalooza!

Series
Other Talks
Time
Saturday, March 9, 2019 - 13:00 for 4 hours (half day)
Location
Ebster Recreation Center, Decatur
Speaker
Evans Harrell, Matt Baker, and GT Club Math, among others – Georgia Tech, Emory, and others –

Mathapalooza! is simultaneously a Julia Robinson Mathematics Festival and an event of the Atlanta Science Festival. There will be puzzles and games, a magic show by Matt Baker, mathematically themed courtroom skits by GT Club Math, a presentation about math and dance by Manuela Manetta, a presentation about math and music by David Borthwick, and a gallery of mathematical art curated by Elisabetta Matsumoto. It is free, and we anticipate engaging hundreds of members of the public in the wonders of mathematics. More info at https://mathematics-in-motion.org/about/Be there or B^2 !

Measure-valued splines and matrix optimal transport

Series
GT-MAP Seminar
Time
Friday, March 8, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Yongxin Chen – GT AE

Two recent extensions of optimal mass transport theory will be covered. In the first part of the talk, we will discuss measure-valued spline, which generalizes the notion of cubic spline to the space of distributions. It addresses the problem to smoothly interpolate (empirical) probability measures. Potential applications include time sequence interpolation or regression of images, histograms or aggregated datas. In the second part of the talk, we will introduce matrix-valued optimal transport. It extends the optimal transport theory to handle matrix-valued densities. Several instances are quantum states, color images, diffusion tensor images and multi-variate power spectra. The new tool is expected to have applications in these domains. We will focus on theoretical side of the stories in both parts of the talk.

1-d parabolic Anderson model with rough spatial noise

Series
Stochastics Seminar
Time
Thursday, March 7, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Samy Tindel – Purdue University
In this talk I will first recall some general facts about the parabolic Anderson model (PAM), which can be briefly described as a simple heat equation in a random environment. The key phenomenon which has to be observed in this context is called localization. I will review some ways to express this phenomenon, and then single out the so called eigenvectors localization for the Anderson operator. This particular instance of localization motivates our study of large time asymptotics for the stochastic heat equation. In the second part of the talk I will describe the Gaussian environment we consider, which is rougher than white noise, then I will give an account on the asymptotic exponents we obtain as time goes to infinity. If time allows it, I will also give some elements of proof.

Packing and covering triangles

Series
Graph Theory Working Seminar
Time
Wednesday, March 6, 2019 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Youngho Yoo – Georgia Tech

Let $\nu$ denote the maximum size of a packing of edge-disjoint triangles in a graph $G$. We can clearly make $G$ triangle-free by deleting $3\nu$ edges. Tuza conjectured in 1981 that $2\nu$ edges suffice, and proved it for planar graphs. The best known general bound is $(3-\frac{3}{23})\nu$ proven by Haxell in 1997. We will discuss this proof and some related results.

Global Convergence of Neuron Birth-Death Dynamics

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, March 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joan Bruna Estrach – New York University –
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).

A restriction estimate in $\mathbb{R}^3$

Series
Analysis Seminar
Time
Wednesday, March 6, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hong Wang – MIT

If $f$ is a function supported on a truncated paraboloid, what can we say about $Ef$, the Fourier transform of f? Stein conjectured in the 1960s that for any $p>3$, $\|Ef\|_{L^p(R^3)} \lesssim \|f\|_{L^{\infty}}$.

We make a small progress toward this conjecture and show that it holds for $p> 3+3/13\approx 3.23$. In the proof, we combine polynomial partitioning techniques introduced by Guth and the two ends argument introduced by Wolff and Tao.

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