In this talk, I will present recent results on game theoretical formulation of optimal debt management problems in an infinite time horizon with exponential discount, modeled as a noncooperative interaction between a borrower and a pool of risk-neutral lenders. Here, the yearly income of the borrower is governed by a stochastic process and bankruptcy instantly occurs…

Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc. This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques…

 Models for networks that evolve and change over time are ubiquitous in a host of domains including modeling social networks, understanding the evolution of systems in proteomics, the study of the growth and spread of epidemics etc. This talk will give a brief summary of three recent findings in this area where stochastic approximation techniques…

The goal of this talk is to price American-type financial contracts in the presence of Knightian uncertainty. In particular, instead of choosing a particular probabilistic model to represent the price process of some underlying asset (on which an American option is written), we first restrict our attention to the whole class of models that are…

Consider a dense Erdös-Rényi random graph with parameters n and p, with p fixed in (0,1). Let H_n(p) be the hitting time between two distinct vertices: run simple random walk from a vertex w, wait until it hits another vertex v, and average over the random walk. What does the distribution of H_n(p) look like? Heuristic…

We investigate the long-time behavior and stationary fluctuations of an infinite-dimensional rank-based diffusion process, called the Atlas model, where particles move as independent Brownian motions, with the lowest ranked particle at any time getting a unit upward drift. The associated process of gaps between successive ranked particles possesses an uncountable collection of invariant measures. We…

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and…

We consider a class of disordered mean-field combinatorial optimization problems, focusing on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the edge weights are sampled from a general distribution. We prove Central Limit Theorems for the log-partition function, the weight of a typical configuration, and the…

The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained via a connection with branching Brownian motion, beginning with works of McKean and Bramson in the 70s. I will discuss an extension of this probabilistic…

Conventional superconductivity emerges for weakly interacting Fermi gases, and its emergence has been studied in mathematical physics. Such conventional superconductors, however, have a very low critical temperature, making them very expensive in applications. Unconventional superconductors, such as cuperates, on the other hand exhibit a very high critical temperature, but we have very little understanding of…

In this talk we will discuss a collection of convolution inequalities for real valued functions on the hypercube, motivated by combinatorial applications. Speaker's website: https://sites.google.com/site/joseramonmadridpadilla/home

Multiple financial assets’ time-series data is stored in a matrix upon which we perform principal component analysis to find predominant factors in the market. Random matrix theory helps us to identify the number of factors present in the data, with the top eigenvalue-eigenvector pair bearing a strong resemblance to the market’s capitalization-weighted portfolio. This resemblance…

We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes G = G(n), whose ranks and nilpotency classes are uniformly bounded.  For some k = k(n) such that 1 << log k << log |G|, we pick a random set of generators S = S(n) by sampling k elements…

A number of continuous interacting particle systems can be described as collections of Brownian particles on the real line whose collision dynamics are mediated by the local times associated with the gaps between adjacent particles.  Examples of systems in this class include the ordered particle dynamics of rank-based diffusions and certain eigenvalue processes arising in…

The trace reconstruction problem asks to reconstruct an unknown n-bit string x given independent random "traces" of x, where a random trace is obtained by first deleting each bit of x independently with some probability (say 0.5), and then outputting the concatenation of the remaining bits of x. A basic question is to determine the…

Given a set K in R^2 with Hausdorff dimension t \in , what can we say about a typical orthogonal projection of K? Marstrand (1954) proved that for Lebesgue almost all unit vectors \theta \in S^1, the dimension of the projection \pi_\theta (K) to \theta is min{t, 1}. To refine the question, we can replace…

Speaker's website: https://www.stat.berkeley.edu/~vadicgor/

The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles underlying the success of deep learning, the core technology behind these breakthroughs, is still far from well-understood. In this presentation, I will share some interesting…

Speaker's website: https://home.olemiss.edu/~leth/