I’ll demonstrate how careful analysis of group relations yields unexpected constructions, addressing several central questions in group theory. These include a question by Elliott, Jonusas, Mesyan, Mitchell, Morayne, and Peresse regarding Zariski topologies on groups and semigroups, a series of questions by Amir, Blachar, Gerasimova, and Kozma concerning algebraic group laws, and a longstanding question by Wiegold on invariably generated groups.

In 1970, Frisch and Bourret introduced a q-deformation of independent Gaussian random variables (say "q-Gaussian system"). In one-variable case, q-Gaussian is the distribution whose orthogonal polynomials are q-Hermite polynomials, and this distribution interpolates between Rademacher (q=-1), semicircle (q=0), Gaussian (q=1) distribution. In multivariable case, q-Gaussian system is represented as a tuple of operators (which are non-commutative in general) on the q-deformed Fock space introduced by Bożejko and Speicher. 

In this talk, I will explain related combinatorics (pair partitions and number of crossings) and analysis for q-Gaussian system. 

Initially studied by Markov around 1880, the Lagrange spectrum, $L$, and the Markov spectrum, $M$, are complicated subsets of the real line that play a crucial role in the study of Diophantine approximation and binary quadratic forms. Perron's 1920s description of the spectra in terms of continued fractions allowed powerful dynamical machinery to come to bear on many problems. In this talk, I will discuss recent work with Harold Erazo and Carlos Gustavo Moreira investigating the function $d_\textrm{loc}(t)$ that determines the local Hausdorff dimension at a point $t$ in $L'$.

Biholomorphic mapping problems for domains in complex Euclidean space and in complex manifolds will be discussed.

Placing a two-dimensional lattice on another with a small rotation gives rise to periodic “moiré” patterns on a superlattice scale much larger than the original lattice.  The Bistritzer-MacDonald (BM) model attempts to capture the electronic properties of twisted bilayer graphene (TBG) by an effective periodic continuum model over the bilayer moiré pattern. We use the mathematical techniques developed to study waves in inhomogeneous media to identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at the monolayer Dirac points of linear dispersion, up to error that we rigorously estimate. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first “magic" angle where the group velocity of the wave packet is zero and strongly correlated electronic phases (superconductivity, Mott insulators, etc.) are observed. 

We are working to develop models of TBG which account for the effects of mechanical relaxation and to couple our relaxed BM model with interacting TBG models.  We are also extending our approach to essentially arbitrary moirématerials such as twisted multilayer transition metal dichalcogenides (TMDs) or even twisted heterostructures consisting of layers of distinct 2D materials.

Ultrafilters show up in many places, including logic, topology, and analysis. Despite this, the concept does not seem to be well-known among mathematicians (indeed, the speaker completed several years of graduate school without learning about them). The goal of this talk is to present a friendly introduction to ultrafilters and to highlight a few of their various manifestations. If all goes well, the talk will include a topological proof of Arrow’s Impossibility Theorem, a classic result from political science.

We will discuss some ongoing work on the subject, specifically in the rank $2$, genus $2$ case. The talk will start with a quick review of existing literature on $M_2$ and some of its étale covers, along with results and constructions involving moduli of rank $2$ bundles. We will go over their generalizations to the universal setting and outline the usage of these tools for computing the Chow ring. Lastly, we shall go over some ideas to relate the generators to tautological classes.