The Ramsey number of graphs G1 and G2, the smallest N so that any red/blue coloring of the N-clique contains either a red G1 or a blue G2, is one of the most studied notions in combinatorics. We study a related process called the online ordered Ramsey game, played between two players, Builder and Painter. Builder has two graphs, G1 and G2, each of which has a specific ordering on its vertices. Builder starts with an edgeless graph on an ordered vertex set (the integers) and attempts to build either an ordered red copy of G1 or an ordered blue copy of G2 by adding one edge at a time. When Builder adds an edge, Painter is required to decide, at the time of creation, whether an edge is red or blue. Ramsey’s Theorem tells us that Builder can eventually win; their objective is to do so using the minimum number of turns, and Painter’s objective is to delay them as long as possible. The *online ordered Ramsey number* of G1 and G2 is the number of turns taken when both players play optimally.

Online ordered Ramsey numbers were introduced by Perez-Gimenez, P. Pralat, and West in 2021. In this talk we will discuss their relation to other types of Ramsey numbers and present some results on the case when at least one of G1,G2 is sparse.

(Joint work with Emily Heath, Grace McCourt, Hannah Sheats, and Justin Wisby)

I will introduce some recent progress towards understanding the scalability of Markov chain Monte Carlo (MCMC) methods and their comparative advantage with respect to variational inference. I will fact-check the folklore that "variational inference is fast but biased, MCMC is unbiased but slow". I will then discuss a combination of the two via reverse diffusion, which holds promise of solving some of the multi-modal problems. This talk will be motivated by the need for Bayesian computation in reinforcement learning problems as well as the differential privacy requirements that we face.

The extreme eigenvalues of a random matrix have been important objects of study since the inception of random matrix theory and also have a variety of applications.  The Laplacian matrix is the workhorse of spectral graph theory and is the key player in many practical algorithms for graph clustering, network control theory and combinatorial optimization.  In this talk, we discuss the fluctuations of the extreme eigenvalues of a random Laplacian matrix with gaussian entries.  The proof relies on a broad set of techniques from random matrix theory and free probability.  We will also describe some recent progress on a broader class of random Laplacian matrices.

This is joint work with Andrew Campbell and Sean O'Rourke.

Large-scale linear systems, Ax=b, frequently arise in data science and scientific computing at massive scales, thus demanding effective iterative methods to solve them. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized Kaczmarz algorithm (RK) was studied extensively as an efficient iterative solver for such systems. However, the convergence study of RK in the noisy regime is limited and considers measurement noise in the right-hand side vector, b. Unfortunately, that is not always the case, and the coefficient matrix A can also be noisy. In this talk, we motivate and discuss doubly noise linear systems and the performance of the Kaczmarz algorithm applied to such systems.