## Seminars and Colloquia by Series

### Symmetric Tropical Rank 2 Matrix Completion

Series
Other Talks
Time
Monday, May 23, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
May Cai

An important recent topic is matrix completion, which is trying to recover a matrix from a small set of observed entries, subject to particular requirements. In this talk, we discuss results on symmetric tropical and symmetric Kapranov rank 2 matrices, and establish a technique of examining the phylogenetic tree structure obtained from the tropical convex hulls of their columns to construct the algebraic matroid of symmetric tropical rank 2 $n \times n$ matrices. This matroid directly answers the question of what entries of a symmetric $n \times n$ matrix needs to be specified generically to be completable to a symmetric tropical rank 2 matrix, as well as to a symmetric classical rank 2 matrix.

This is based on joint work with Cvetelina Hill and Kisun Lee.

### Concentration of the Chromatic Number of Random Graphs

Series
Graph Theory Seminar
Time
Tuesday, May 17, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lutz WarnkeUCSD
What can we say about the chromatic number \chi(G_{n,p}) of an n-vertex binomial random graph G_{n,p}? From a combinatorial perspective, it is natural to ask about the typical value of \chi(G_{n,p}), i.e., upper and lower bounds that are close to each other. From a probabilistic combinatorics perspective, it is also natural to ask about the concentration of \chi(G_{n,p}), i.e., how much this random variable varies. Among these two fundamental questions, significantly less is known about the concentration question that we shall discuss in this talk. In terms of previous work, in the 1980s Shamir and Spencer proved that the chromatic number of the binomial random graph G_{n,p} is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p\in (0,1). In this talk, we prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and also discuss several intriguing questions about the chromatic number \chi(G_{n,p}) that remain open. Based on joint work with Erlang Surya; see https://arxiv.org/abs/2201.00906

### Approximation of invariant manifolds for Parabolic PDEs over irregular domains

Series
CDSNS Colloquium
Time
Friday, May 13, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Jorge GonzalezGeorgia Tech

The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.

### Thesis defense: Invariance of random matrix

Series
Time
Thursday, May 12, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
JunTao DuanGeorgia institute of technology

Random matrix has been found useful in many real world applications. The celebrated Johnson-Lindenstrauss lemma states that certain geometric structure of deterministic vectors is preserved when projecting high dimensional space $R^n$ to a lower dimensional space $R^m$. However, when random vectors are concerned, it is still unclear how the distribution of the geometry is affected by random matrices. Since random projection or embedding introduces dependence to independent random vectors, does it imply random matrices are inferior for transforming random vectors?

We will start with establishing a new  central limit theorem  for random variables with certain product dependence structure. At the same time, we obtain its Berry-Esseen type rate of convergence. Then we apply this general central limit theorem to random projections and embeddings of two independent random vectors $X, Z$. In particular, we show the distribution of inner product structure is preserved by random matrices. Roughly speaking, two independent random vectors remain "independent" in the randomly projected lower dimensional space or randomly embedded high dimensional space. More importantly, we also quantitatively characterize the distortion of distribution introduced by random matrices. The error term has a bound at most $O(\frac{1}{\sqrt{m}} + \frac{1}{\sqrt{n}})$.

Then we also establish the fact that random matrices have low distortion on the norm of a random vector. It is first justified by establishing concentration of the projected or embedded norm under sub-Gaussian assumptions. A central limit theorem for the randomly projected norm is established as well similar to the CLT for inner product.

### An army of one: stable solitary states in the second-order Kuramoto model

Series
CDSNS Colloquium
Time
Friday, May 6, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005; streaming via Zoom available
Speaker
Igor BelykhGeorgia State University

Symmetries are  fundamental concepts in modern physics and biology. Spontaneous symmetry breaking often leads to fascinating  dynamical patterns such as  chimera states in which structurally and dynamically identical oscillators  split into coherent and incoherent clusters.  Solitary states in which one oscillator separates from the coherent cluster and oscillates with a different frequency represent  “weak” chimeras. While a rigorous stability analysis of a “strong” chimera with a multi-oscillator incoherent cluster  is typically out of reach for finite-size networks, solitary states offer a unique test bed for the development of stability approaches to large chimeras. In this talk, we will present such an approach and study the stability of solitary states in Kuramoto networks of identical 2D phase oscillators with inertia and a phase-lagged coupling.   We will derive asymptotic stability conditions for such solitary states as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our analysis for the emergence of rotatory chimeras and splay states. This is a joint work with V. Munyaev, M. Bolotov, L. Smirnov, and G. Osipov.

### Two conjectures on the spread of graphs

Series
Combinatorics Seminar
Time
Friday, April 29, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael TaitVillanova University

Given a graph $G$ let $\lambda_1$ and $\lambda_n$ be the maximum and minimum eigenvalues of its adjacency matrix and define the spread of $G$ to be $\lambda_1 - \lambda_n$. In this talk we discuss solutions to a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs.

The first, referred to as the spread conjecture, states that over all graphs on $n$ vertices the join of a clique of order $\lfloor 2n/3 \rfloor$ and an independent set of order $\lceil n/3 \rceil$ is the unique graph with maximum spread. The second, referred to as the bipartite spread conjecture, says that for any fixed $e\leq n^2/4$, if $G$ has maximum spread over all $n$-vertex graphs with $e$ edges, then $G$ must be bipartite.

We show that the spread conjecture is true for all sufficiently large $n$, and we prove an asymptotic version of the bipartite spread conjecture. Furthermore, we exhibit an infinite family of counterexamples to the bipartite spread conjecture which shows that our asymptotic solution is tight up to a multiplicative factor in the error term. This is joint work with Jane Breen, Alex Riasanovsky, and John Urschel.

### Back to boundaries in billiards

Series
CDSNS Colloquium
Time
Friday, April 29, 2022 - 13:00 for
Location
Speaker
Yaofeng SuSoM, GT

Abstract: This talk has 4 or 5 parts

1. I will start with a physical toy model to introduce billiards/open billiards, which describe the dynamics of a particle in a compact manifold/in a particular open area of this manifold.

2. One of the main questions of open billiards is Poisson approximations. It describes the asymptotic behavior of the dynamics in statistical distributions.  I will define it for billiards systems.

3. The main result is that such approximations hold for a billiard system that has arbitrarily slow chaos.

4. I will sketch the idea of the proof.

5. If time permits, I will talk about the connection between this work and riemann hypothesis.

This is a joint work with Prof. Leonid Bunimovich.

### Log-concavity of coefficients of characteristic polynomials of matroids.

Series
Algebra Student Seminar
Time
Friday, April 29, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Tong JinGeorgia Tech

This is an expanded version of a 10-minute presentation in MATH 6422. I'll explain what matroids and their characteristic polynomials as well as log-concavity mean, and then sketch a proof due to Petter Brändén and Jonathan Leake (arXiv:2110.00487). If time permits, I'll describe several consequences of this and/or other existing yet different proofs.

### Reconstructing ancestral sequences in large trees

Series
Mathematical Biology Seminar
Time
Thursday, April 28, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and ONLINE
Speaker
Brandon LegriedSoutheast Center for Mathematics and Biology

Statistical consistency in phylogenetics has traditionally referred to the accuracy of estimating mutation rates and phylogenies for a fixed number of species as we increase the amount of data within their signatures, such as DNA and protein sequences. Analyzing sequences undergoing indel mutations (insertions and deletions of sites) has provided a venue for understanding what power can be provided by a lot of data. In this talk, we discuss some of the failings of this approach. For instance, it will be shown that phylogeny estimation is impossible for infinitely long sequences, even with infinite data. This motivates a dual type of statistical consistency, where the number of species is taken to infinity rather than the size of each signature. Here, we give polynomial-time algorithms for ancestral sequence estimation and sequence alignment for reference phylogenies with so many species that they are sufficiently dense. Based on joint work with Louis Fan and Sebastien Roch.

### On Difference Graph Covers and the Local Dimension of the Boolean Lattice

Series
Time
Thursday, April 28, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Ariana Hall

Please Note: MS Thesis Defense