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.

let $g_{n,p}$ denote the random $n$-vertex graph obtained by including each edge independently and with probability $p$. given a graph $f$, let $\mathrm{ex}(g_{n,p},f)$ denote the size of a largest $f$-free subgraph of $g_{n,p}$. when $f$ is non-bipartite, the asymptotic behavior of $\mathrm{ex}(g_{n,p},f)$ is determined by breakthrough work done independently by conlon-gowers and by schacht, but the behavior for bipartite $f$ remains largely unknown.

we will discuss some recent developments that have been made for bipartite $f$, with a particular emphasis on the case of theta graphs.  based on joint work with gwen mckinley.

as we know, kaehler-ricci flow can be reduced to solve a class of  parabolic   complex monge-amp\`ere equations for kaehler potentials and  the solutions usually depend on the kaehler class of initial metric.   thus there  gives a  degeneration of kaehler metrics arising from the kaehler-ricci flow.  for a class of $g$-spherical manifolds,   we can  use  the local estimate  of  monge-amp\`ere equations as well as  the h-invariant for $c^*$-degeneration  to determine the limit of  kaehler-ricci flow after resales.  in particular,  on such manifolds,  the flow will develop the singularities of  type ii.  

we present a highly effective algorithmic approach, pmm, for generating differentially private synthetic data in a bounded metric space with near-optimal utility guarantees under the 1-wasserstein distance. in particular, for a dataset in the hypercube [0,1]^d, our algorithm generates synthetic dataset such that the expected 1-wasserstein distance between the empirical measure of true and synthetic dataset is o(n^{-1/d}) for d>1. our accuracy guarantee is optimal up to a constant factor for d>1, and up to a logarithmic factor for d=1. also, pmm is time-efficient with a fast running time of o(\epsilon d n). derived from the pmm algorithm, more variations of synthetic data publishing problems can be studied under different settings.

we discuss an analogue of the goldbach conjecture for polynomials with coefficients in semidomains (i.e., subsemirings of an integral domain).