## Seminars and Colloquia Schedule

### Two graph classes with bounded chromatic number

Series
Dissertation Defense
Time
Monday, April 17, 2023 - 09:30 for 1 hour (actually 50 minutes)
Location
Skiles 114 (or Zoom)
Speaker
Joshua SchroederGeorgia Tech

Zoom: https://gatech.zoom.us/j/98256586748?pwd=SkJLZ3ZKcjZsM0JkbGdyZ1Y3Tk9udz0... />
Meeting ID: 982 5658 6748<br />

A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\chi$-bounded for any $t$ with binding function $f(\omega) = O(\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\chi$-binding function $f(\omega) = \omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.

### Invariants of Matrices

Series
Algebra Seminar
Time
Monday, April 17, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Harm DerksenNortheastern University

The group SL(n) x SL(n) acts on m-tuples of n x n matrices by simultaneous left-right multiplication.  Visu Makam and the presenter showed the ring of invariants is generated by invariants of degree at most mn^4. We will also discuss geometric aspects of this action and connections to algebraic complexity and the notion of noncommutative rank.

### Uncovering the Law of Data Separation in Deep Learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 17, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Prof. Weijie SuUniversity of Pennsylvania (Wharton)

The speaker will present in person.

In this talk, we will investigate the emergence of geometric patterns in well-trained deep learning models by making use of a layer-peeled model and the law of equi-separation. The former is a nonconvex optimization program that models the last-layer features and weights. We use the model to shed light on the neural collapse phenomenon of Papyan, Han, and Donoho, and to predict a hitherto-unknown phenomenon that we term minority collapse in imbalanced training.

The law of equi-separation is a pervasive empirical phenomenon that describes how data are separated according to their class membership from the bottom to the top layer in a well-trained neural network. We will show that, through extensive computational experiments, neural networks improve data separation through layers in a simple exponential manner. This law leads to roughly equal ratios of separation that a single layer is able to improve, thereby showing that all layers are created equal. We will conclude the talk by discussing the implications of this law on the interpretation, robustness, and generalization of deep learning, as well as on the inadequacy of some existing approaches toward demystifying deep learning.

### Jones diameter and crossing numbers of satellite knots

Series
Geometry Topology Seminar
Time
Monday, April 17, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Speaker
Effie KalfagianniMichigan State University
It has been long known that the quadratic term in the degree of the colored Jones polynomial of knot provides a lower bound of the crossing number the knot.
I’ll discuss work with Lee where we determine the class of knots for which this bound is sharp and give applications to computing crossing numbers of satellite knots.

### Symplectic trisections and connected sum decompositions

Series
Geometry Topology Seminar
Time
Monday, April 17, 2023 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Lambert-ColeUniversity of Georgia

This talk will have two parts.  The first half will describe how to construct symplectic structures on trisected 4-manifolds. This construction is inspired by projective complex geometry and completely characterizes symplectic 4-manifolds among all smooth 4-manifolds.  The second half will address a curious phenomenon: symplectic 4-manifolds appear to not admit any interesting connected sum decompositions.  One potential explanation is that every embedded 3-sphere can be made contact-type.  I will outline some strategies to prove this from a trisections perspective, describe some of the obstructions, and give evidence that these obstructions may be overcome.

### Global well-posedness for the one-phase Muskat problem

Series
PDE Seminar
Time
Tuesday, April 18, 2023 - 15:00 for
Location
Skiles 006
Speaker
Huy NguyenUniversity of Maryland, College Park

We will discuss the one-phase Muskat problem concerning the free boundary of Darcy fluids in porous media. It is known that there exists a class of non-graph initial boundary leading to self-intersection at a single point in finite time (splash singularity). On the other hand, we prove that the problem has a unique global-in-time solution if the initial boundary is a periodic Lipschitz graph of arbitrary size. This is based on joint work with H. Dong and F. Gancedo.

### Flows (and group-connectivity) in signed graphs

Series
Graph Theory Seminar
Time
Tuesday, April 18, 2023 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jessica McDonaldAuburn University

We discuss flows (and group-connectivity) in signed graphs, and prove a new result about group-connectivity in 3-edge-connected signed graphs. This is joint work with Alejandra Brewer Castano and Kathryn Nurse.

### CANCELED — Multiplier weak type inequalities for maximal operators and singular integrals

Series
Analysis Seminar
Time
Wednesday, April 19, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
This seminar has been cancelled and will be rescheduled next year.
Speaker
Brandon SweetingUniversity of Alabama

This seminar has beeb cancelled and will be rescheduled next year.  We discuss a kind of weak type inequality for the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior. Such inequalities arise in the generalization of weak-type spaces to the matrix weighted setting and find applications in scalar two-weight norm inequalities via interpolation with change of measures. In this talk, I will discuss quantitative estimates obtained for $A_p$ weights, $p > 1$, that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Ríos for $p = 1$. I will also discuss an endpoint result for the Riesz potentials.

### Prethermalization and conservation laws in quasi-periodically driven quantum systems

Series
Math Physics Seminar
Time
Thursday, April 20, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and online https://gatech.zoom.us/j/96817326631
Speaker
Matteo Gallone and Beatrice LangellaSISSA Trieste, Italy

Understanding the route to thermalization of a physical system is a fundamental problem in statistical mechanics. When a system is initialized far from thermodynamical equilibrium, many interesting phenomena may arise. Among them, a lot of interest is attained by systems subjected to periodic driving (Floquet systems), which under certain circumstances can undergo a two-stage long dynamics referred to as "prethermalization", showing nontrivial physical features. In this talk, we present some prethermalization results for a class of lattice systems with quasi-periodic external driving in time. When the quasi-periodic driving frequency is large enough or the strength of the driving is small enough, we show that the system exhibits a prethermal state for exponentially long times in the perturbative parameter. Moreover, we focus on the case when the unperturbed Hamiltonian admits constants of motion and we prove the quasi-conservation of a dressed version of them. We discuss applications to perturbations of the Fermi-Hubbard model and the quantum Ising chain.

Join Zoom Meeting

https://gatech.zoom.us/j/96817326631

### Coalescence, geodesic density, and bigeodesics in first-passage percolation

Series
Stochastics Seminar
Time
Thursday, April 20, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jack HansonCity College, CUNY

Several well-known problems in first-passage percolation relate to the behavior of infinite geodesics: whether they coalesce and how rapidly, and whether doubly infinite "bigeodesics'' exist. In the plane, a version of coalescence of "parallel'' geodesics has previously been shown; we will discuss new results that show infinite geodesics from the origin have zero density in the plane. We will describe related forthcoming work showing that geodesics coalesce in dimensions three and higher, under unproven assumptions believed to hold below the model's upper critical dimension. If time permits, we will also discuss results on the bigeodesic question in dimension three and higher.

### Anderson Localization in dimension two for singular noise, part eight

Series
Mathematical Physics and Analysis Working Seminar
Time
Friday, April 21, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and https://uci.zoom.us/j/93130067385
Speaker

We will finish the proof of the unique continuation theorem, starting with a brief discussion of the growth lemma discussed at our previous talk. After this, we will reduce unique continuation for untitled squares to unique continuation for tilted squares, and using the tilted square growth lemma prove such unique continuation result.

### Beyond Moments: Robustly Learning Affine Transformations with Asymptotically Optimal Error

Series
ACO Student Seminar
Time
Friday, April 21, 2023 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
He JiaGeorgia Tech CS

We present a polynomial-time algorithm for robustly learning an unknown affine transformation of the standard hypercube from samples, an important and well-studied setting for independent component analysis (ICA). Total variation distance is the information-theoretically strongest possible notion of distance in our setting and our recovery guarantees in this distance are optimal up to the absolute constant factor multiplying the fraction of corruption. Our key innovation is a new approach to ICA (even to outlier-free ICA) that circumvents the difficulties in the classical method of moments and instead relies on a new geometric certificate of correctness of an affine transformation. Our algorithm is based on a new method that iteratively improves an estimate of the unknown affine transformation whenever the requirements of the certificate are not met.

### Constructing Exotic 4-manifolds

Series
Geometry Topology Working Seminar
Time
Friday, April 21, 2023 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Jon SimoneGeorgia Tech

This week, we'll continue discussing the rational blowdown and use it to construct small exotic 4-manifolds. We will see how we can view the rational blowdown as a "monodromy substitution." Finally, if time allows, we will discuss knot surgery on 4-manifolds.

### Shortest closed curve to inspect a sphere

Series
Combinatorics Seminar
Time
Friday, April 21, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker

We show that in Euclidean 3-space any closed curve which contains the unit sphere in its convex hull has length at least $4\pi$, and characterize the case of equality, which settles a conjecture of Zalgaller. Furthermore, we establish an estimate for the higher dimensional version of this problem by Nazarov, which is sharp up to a multiplicative constant, and is based on Gaussian correlation inequality. Finally we discuss connections with sphere packing problems, and other optimization questions for convex hull of space curves. This is joint work with James Wenk.

### Bifurcations in patterns of human sleep under variation in homeostatic dynamics

Series
CDSNS Colloquium
Time
Friday, April 21, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and online
Speaker
Christina AthanasouliGeorgia Tech