Let \(G\) be a countable group acting by isometries on a metric space \((M, d)\), and let \(\mu\) denote a probability measure on \(G\). The \(\mu\)-random walk on \(M\) is the random process defined by 

\[Z_n=X_n \dots X_1 o,\]  
where \(o \in M\) is a fixed base point, and \(X_i\) are independent \(\mu\)-distributed random variables. 

Studying statistical properties of the displacement sequence \(D_n:= d(Z_n, o)\) has been a topic of current research. 

In this talk, which is based on a joint work with Amin Bahmanian, Behrang Forghani, and Ilya Gekhtman, I will discuss a central limit theorem for \(D_n\) in the case that \(M\) is the horospherical product of Gromov hyperbolic spaces. 

I will present some recent joint work with Sorin Popa where we show that, under the continuum hypotheses, any ultraproduct II$_1$ factor contains more than continuum many mutually disjoint singular MASAs. In other words, the singular abelian rank of any ultraproduct II$_1$ factor $M$, $\text{r}(M)$, is larger than $\mathfrak{c}$. Moreover, if the strong continuum hypothesis $2^\mathfrak{c}=\aleph_2$ is assumed, then $\text{r}(M) = 2^\mathfrak{c}$. More generally, these results hold true for any II$_1$ factor $M$ with unitary group of cardinality $\mathfrak{c}$ that satisfies the bicommutant condition $(A_0'\cap M)'\cap M=M$, for all $A_0\subset M$ separable abelian.

In his 1901 thesis, Issai Schur discovered a connection between the representation theory of the symmetric group and general linear group.  One way to understand this connection is through a finite dimensional algebra called the Schur algebra.  I will outline this picture and then describe a new algebra, defined in joint work with Tom Braden, which enhances the Schur algebra and provides a new window into the representation theory of symmetric groups.  Finally, I will explain how we came to discover this algebra by studying the geometry of Hilbert schemes of points in the plane and how this fits into my larger program to uncover representation theory in the geometry of symplectic singularities and their resolutions.

In stochastic modeling we often need to deal with one of two apparently unrelated objects. One is self-similar processes and the other is additive functionals. Self-similar Markov processes are the class of Markovian models that arise as scaling limits of stochastic processes, that are obtained after renormalization of time and space. Additive functionals arise commonly when one considers, for instance, rewards associated to a Markovian model. 

On the one hand, the so-called Lamperti transform ensures that any $R^d$-valued self-similar Markov process admits a polar decomposition, and the argument and the radius of the process are related to a Markov additive process via an explicit time change. On the other hand, any additive functional A of a Markov process X is such that the pair (A, X) is a Markov additive process. A Markov additive process (MAP) is a stochastic process with two components: one that is additive, and real valued, the ordinator, and a general one, the modulator, that rules the behavior of the ordinator. The ordinator has independent and stationary increments, given the modulator. This general structure emulates the structure of processes with independent and stationary increments, Levy processes, as for instance Brownian motion, Cauchy and stable processes, Gamma processes, etc. 

In general, it is too ambitious to try to determine explicitly the whole law of a self-similar Markov process or of an additive functional. But we can aim at understanding properties of the extremes of these processes and to be ready for the best and worst scenarios. In the fluctuation theory of Markov additive processes we aim at developing tools for studying the extremes of the additive part, ordinator, of the process. This has been done in a systematic way during the last four decades under the assumption that the modulator is a constant process, and hence the ordinator is a real valued Levy process. Also, in the 1980-90 period, some foundations were laid to develop a fluctuation theory for MAPs in a general setting.   

In this talk we aim at giving a brief overview of the fluctuation theory of Markov additive processes, to describe some recent results and to provide some applications to the theory of self-similar Markov processes. These applications are mainly related to stable processes, a class of processes that arises often in mathematical physics, potential and harmonic analysis, and in other areas of mathematics. We aim at making this overview accessible to graduate and advanced undergraduate students, with some knowledge of Markov chains and Levy processes, and to point out at some open research questions.

Simply connected smooth 4-manifolds are complicated; understanding anything about them is good progress toward the larger goal of classification. There have been some discoveries in the past few years that distinguish exotic diffeomorphisms (which are topologically isotopic, but not smoothly so) using the families Bauer-Furuta invariant. The goal of this talk is to provide the context of this area, the background for the families Bauer-Furuta invariant, and some ideas for my future research directions. 

Gaussian elimination with partial pivoting (GEPP) remains the most used dense linear solver. For a n x n matrix A, GEPP results in the factorization PA = LU where L and U are lower and upper triangular matrices and P is a permutation matrix. If A is a random matrix, then the associated permutation from the P factor is random. When is this a uniform permutation? How many cycles are in its disjoint cycle decomposition (which equivalently answers how many GEPP pivot movements are needed on A)? What is the length of the longest increasing subsequence of this permutation? We will provide some statistical answers to these questions for select random matrix ensembles and transformations. For particular butterfly permutations, we will present full distributional descriptions for these particular statistics. Moreover, we introduce a random butterfly matrix ensemble that induces the Haar measure on a full 2-Sylow subgroup of the symmetric group on a set of size 2ⁿ.

In this talk, I'll first give a quick introduction to the theory of Drinfeld modules and talk about an equivariant $L$-function associated to Drinfeld modules as defined by Ferrara-Higgins-Green-Popescu in their work on the ETNC. As is usual, these $L$-functions are defined as an infinite product of Euler factors, and the main focus of this talk is a result relating these Euler factors to a certain quotient of Fitting ideals of some algebraically relevant modules. This is joint work with Cristian Popescu.

Human behavior impacts the world around us. From disease control to climate change, understanding human behavior through the lens of evolutionary dynamics provides useful insights and implications for making the world a better place. This multi-disciplinary, data-driven modeling approach combines various introspective processes with interpersonal interactions by accounting for interdependent biological and social network processes across different yet interconnected network layers. In this talk, we will present recent work on modeling complex, multi-faceted human behavior across diverse domains in critical issues of societal importance, ranging from socio-cognitive biases to pandemic compliance. The talk will also discuss the importance of bottom-up behavior and attitude changes, as well as large-scale human cooperation, in addressing urgent challenges facing our common humanity.