## Seminars and Colloquia by Series

### Prediction problems and second order equations

Series
Job Candidate Talk
Time
Thursday, December 15, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or https://gatech.zoom.us/j/98373229920
Speaker
Ibrahim EkrenFlorida State University

We study the long-time regime of the prediction with expert advice problem in both full information and adversarial bandit feedback setting. We show that with full information, the problem leads to second order parabolic partial differential equations in the Euclidean space. We exhibit solvable cases for this equation and discuss the optimal behavior of both agents. In the adversarial bandit feedback setting, we show that the problem leads to second order parabolic equations in the Wasserstein space which allows us to obtain novel regret bounds. Based on joint works with Erhan Bayraktar and Xin Zhang.

### Quantum algorithms for Hamiltonian simulation with unbounded operators

Series
Job Candidate Talk
Time
Thursday, December 8, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or https://gatech.zoom.us/j/98355006347
Speaker
Di FangUC Berkeley

Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum machine learning. We will introduce some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. The latter yields a surprising superconvergence result for regular potentials. In the end, I will discuss briefly how Hamiltonian simulation techniques can be applied to a quantum learning task achieving optimal scaling. (The talk does not assume a priori knowledge on quantum computing.)

### Structure for dense graphs: forbidding a vertex-minor

Series
Job Candidate Talk
Time
Tuesday, December 6, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 / hybrid
Speaker
Rose McCartyPrinceton University

Structural graph theory has traditionally focused on graph classes that are closed under both vertex- and edge-deletion (such as, for each surface Σ, the class of all graphs which embed in Σ). A more recent trend, however, is to require only closure under vertex-deletion. This is typically the right approach for graphs with geometric, rather than topological, representations. More generally, it is usually the right approach for graphs that are dense, rather than sparse. I will discuss this paradigm, taking a closer look at classes with a forbidden vertex-minor.

### Mathematical and Statistical Challenges on Large Discrete Structures

Series
Job Candidate Talk
Time
Wednesday, March 16, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/348214744/2450
Speaker
Miklos RaczPrinceton University

From networks to genomics, large amounts of data are abundant and play critical roles in helping us understand complex systems. In many such settings, these data take the form of large discrete structures with important combinatorial properties. The interplay between structure and randomness in these systems presents unique mathematical and statistical challenges. In this talk I will highlight these through two vignettes: (1) inference problems on networks, and (2) DNA data storage.

First, I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph. Furthermore, we obtain the precise threshold for exact community recovery using multiple correlated graphs, which captures the interplay between the community recovery and graph matching tasks.

Next, I will give an overview of DNA data storage. Storing data in synthetic DNA is an exciting emerging technology which has the potential to revolutionize data storage. Realizing this goal requires innovation across a multidisciplinary pipeline. I will explain this pipeline, focusing on our work on statistical error correction algorithms and optimizing DNA synthesis, highlighting the intimate interplay between statistical foundations and practice.

### Matrix Concentration and Synthetic Data

Series
Job Candidate Talk
Time
Thursday, March 10, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/405947238/3475
Speaker
March BoedihardjoUC Irvine

Classical matrix concentration inequalities are sharp up to a logarithmic factor. This logarithmic factor is necessary in the commutative case but unnecessary in many classical noncommutative cases. We will present some matrix concentration results that are sharp in many cases, where we overcome this logarithmic factor by using an easily computable quantity that captures noncommutativity. Joint work with Afonso Bandeira and Ramon van Handel.

Due to privacy, access to real data is often restricted. Data that are not completely real but resemble certain properties of real data become natural substitutes. Data of this type are called synthetic data. I will talk about the extent to which synthetic data may resemble real data under privacy and computational complexity restrictions. Joint work with Thomas Strohmer and Roman Vershynin.

The link to the online talk:  https://bluejeans.com/405947238/3475

### Trees in graphs and hypergraphs

Series
Job Candidate Talk
Time
Tuesday, March 8, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Maya SteinUniversity of Chile

Graphs are central objects of study in Discrete Mathematics. A graph consists of a set of vertices, some of which are connected by edges. Their elementary structure makes graphs widely applicable, but the theoretical understanding of graphs is far from complete. Extremal graph theory aims to find connections between global parameters and substructure. A key topic is how a large average or minimum degree of a graph can force certain subgraphs (where the degree is the number of edges at a vertex). For instance, Erdős and Gallai proved in the 1960's that any graph of average degree at least $k$ contains a path of length $k$. Some of the most intriguing open questions in this area concern trees (connected graphs without cycles) as subgraphs. For instance, can one substitute the path from the previous paragraph with a tree? We will give an overview of open problems and recent results in this area, as well as their possible extensions to hypergraphs.

### Low-rank Structured Data Analysis: Methods, Models and Algorithms

Series
Job Candidate Talk
Time
Tuesday, February 22, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/717545499/6211
Speaker
Longxiu HuangUCLA

In modern data analysis, the datasets are often represented by large-scale matrices or tensors (the generalization of matrices to higher dimensions). To have a better understanding or extract   values effectively from these data, an important step is to construct a low-dimensional/compressed representation of the data that may be better to analyze and interpret in light of a corpus of field-specific information. To implement the goal, a primary tool is the matrix/tensor decomposition. In this talk, I will talk about novel matrix/tensor decompositions, CUR decompositions, which are memory efficient and computationally cheap. Besides, I will also discuss the applications of CUR decompositions on developing efficient algorithms or models to robust decompositions or data completion problems. Additionally, some simulation results will be provided on real and synthetic datasets.

### Zarankiewicz problem, VC-dimension, and incidence geometry

Series
Job Candidate Talk
Time
Thursday, February 17, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/939739653/6882
Speaker
Cosmin PohoataYale University
The Zarankiewicz problem is a central problem in extremal graph theory, which lies at the intersection of several areas of mathematics. It asks for the maximum number of edges in a bipartite graph on $2n$ vertices, where each side of the bipartition contains $n$ vertices, and which does not contain the complete bipartite graph $K_{s,t}$ as a subgraph. One of the many reasons this problem is rather special among Turán-type problems is that the extremal graphs in question, whenever available, always seem to have to be of algebraic nature, in particular witnesses to basic intersection theory phenomena. The most tantalizing case is by far the diagonal problem, for which the answer is unknown for most values of $s=t$, and where it is a complete mystery what the extremal graphs could look like. In this talk, we will discuss a new phenomenon related to an important variant of this problem, which is the analogous question in bipartite graphs with bounded VC-dimension. We will present several new consequences in incidence geometry, which improve upon classical results. Based on joint work with Oliver Janzer.

### On the sum-product problem

Series
Job Candidate Talk
Time
Tuesday, February 15, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
George ShakanCRM

Let A be a subset of the integers of size n. In 1983, Erdos and Szemeredi conjectured that either A+A or A*A must have size nearly n^2. We discuss ideas towards this conjecture, such as an older connection to incidence geometry as well as somewhat newer breakthroughs in additive combinatorics. We further highlight applications of the sum-product phenomenon.

### Recent progress on Hadwiger's conjecture

Series
Job Candidate Talk
Time
Monday, February 14, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Luke PostleUniversity of Waterloo

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t \ge 1$. Hadwiger's Conjecture is a vast generalization of the Four Color Theorem and one of the most important open problems in graph theory. Only the cases when $t$ is at most 6 are known. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t (\log t)^{0.5})$ and hence is $O(t (\log t)^{0.5})$-colorable.  In a recent breakthrough, Norin, Song, and I proved that every graph with no $K_t$ minor is $O(t (\log t)^c)$-colorable for every $c > 0.25$,  Subsequently I showed that every graph with no $K_t$ minor is $O(t (\log \log t)^6)$-colorable.  Delcourt and I improved upon this further by showing that every graph with no $K_t$ minor is $O(t \log \log t)$-colorable. Our main technical result yields this as well as a number of other interesting corollaries.  A natural weakening of Hadwiger's Conjecture is the so-called Linear Hadwiger's Conjecture that every graph with no $K_t$ minor is $O(t)$-colorable.  We prove that Linear Hadwiger's Conjecture reduces to small graphs. In 2005, Kühn and Osthus proved that Hadwiger's Conjecture for the class of $K_{s,s}$-free graphs for any fixed positive integer $s \ge 2$. Along this line, we show that Linear Hadwiger's Conjecture holds for the class of $K_r$-free graphs for every fixed $r$.