CR geometry studies boundaries of domains in $\mathbb{C}^n$ and their generalizations. In characterizing CR structures, a central role is played by the Levi form $L$ of a CR manifold $M$, which measures the failure of the CR bundle to be integrable, so that when $L$ has a nontrivial kernel of constant rank, $M$ is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold $N$, $M$ is called straightenable, and the Tanaka-Chern-Moser classification of CR hypersurfaces with nondegenerate Levi form can be applied to $N$. It remains to classify those $M$ for which $L$ is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, Isaev-Zaitzev, Medori-Spiro, and Pocchiola. I will discuss their results, my progress on the problem in dimension 7, and my work (joint with Igor Zelenko) modifying Tanaka's prolongation procedure to treat the equivalence problem in arbitrary dimension.

Biological data sets, such as gene expressions or protein levels, are often high-dimensional, and thus difficult to interpret. Finding important structural features and identifying clusters in an unbiased fashion is a core issue for understanding biological phenomena. In this talk, we describe the dynamical behavior of the important tumor suppressor gene p53 in a data-driven manner. By using simulations from nonlinear models that describe the experimentally observed oscillatory behavior of p53, we first identify parameters which qualitatively change the behavior of the system. Focusing on these parameters, we then show that the effective dimension of the parameter and state space can be recovered from time-series data, providing a minimal realization of the underlying nonlinear system. To this end, we apply the methods of bifurcation analysis and diffusion maps. This is joint work with M. Kooshkbaghi, D. Sroczynski, Z. Belkhatir, M. Pouryahya, A. Tannenbaum, I. Kevrekid is.

A unital $C^\ast$-algebra $A$ is tracially stable if maps on $A$ that are approximately (in trace) unital $\ast$-homomorphisms can be
approximated (in trace) by honest unital $\ast$-homomorphisms on $A$. Tracial stability is closed under free products and tensor products
with abelian $C^\ast$-algebras. In this talk we expand these results to show that for a graph from a certain class, the corresponding
graph product (a simultaneous generalization of free and tensor products) of abelian $C^\ast$-algebras is tracially stable. We will
then discuss two applications of this result: a selective version of Lin’s Theorem and a characterization of the amenable traces
on certain right-angled Artin groups.

Ergodic theory deals with a particular type of measure-preserving transformation on a probability space, and are mostly used in the study of dynamical systems. On the other hand, continued fractions are just a way to represent any real number as an infinte/finite sequence of integers such that this sequence when written as a 'fraction' gives this real number. In this talk, I shall set up the preliminaries of ergodic theory and state the ergodic theorem that we shall need, and then move on to define some notions in continued fractions, and then bridge these two seemingly unrelated subjects. The main application would be to find out the rate of convergence of these 'fractions' to the real number, and thereby coming across some interesting constants like the Khinchin's constant and the Lévy's constant. The talk will not require any prerequisites except for being familiar with the notion of a measure space, which again, is not too necessary.
If time permits, I shall just mention some interesting patterns observed in the continued fraction expansion of e and its $1/n$ and $2/n$ th power, or give an idea how working with the continued fraction of $\sqrt{n}$ is helpful in finding all the solutions of the Pell's equations, i.e. $x^2 - n y^2 = 1$ (or -1).

In this talk, I will present a general upper bound for the number of incidences with k-dimensional varieties in R$^d$ such that their incidence graph does not contain K$_{s,t}$ for fixed positive integers s,t,k,d (where s,t$>$1 and k$<$d). The leading term of this new bound generalizes previous bounds for the special cases of k=1, k=d-1, and k=d/2. Moreover, we find lower bounds showing that this leading term is tight (up to sub-polynomial factors) in various cases. To prove our incidence bounds, we define k/d as the dimension ratio of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties in each proof. If time permits, I will mention other incidence bounds with traversal varieties and hyperplanes in complex spaces. This is joint work with Adam Sheffer.

Cyclic sieving, identified in work with Dennis Stanton and Dennis White, is a happy situation, where counting how many among some objects enjoy cyclic symmetry is as easy as $q$-counting all of the objects. We will illustrate this with two kinds of examples: old ones that still plague us with only uninsightful proofs, and new ones that have joined our list of favorites.

A bounded strictly pseudoconvex domain in C$^n$, n$>1$, supports a unique complete Kahler-Einstein metric determined by the Cheng-Yau solution of Fefferman's Monge-Ampere equation. The smoothness of the solution of Fefferman's equation up to the boundary is obstructed by a local CR invariant of the boundary called the obstruction density. In the case n=2 the obstruction density is especially important, in particular in describing the logarithmic singularity of the Bergman kernel. For domains in C$^2$ diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. (This is a strong form of the Ramadanov Conjecture.)

For two fixed graphs $T$ and $H$, a positive integer $n$ and a real number $p$ in $[0, 1]$ let $ex(G(n, p), T, H)$
be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n, p)$. In this talk we discuss this variable, its phase transition as a function of $p$
and its connection to the deterministic function counting the maximum number of copies of $T$ in an $H$-free graph on $n$ vertices.

Based on joint works with N. Alon, A. Kostochka and W. Samotij.

Every tensor is a sum of rank-1 tensors and the decomposition in a minimal number is called canonical polyadic decompositions. In this talk, we will introduce some decomposition methods for third-order tensors based on standard linear algebra. They all reduce tensor decomposition problems to matrix decomposition problems. Generalized Schur decomposition, simultaneous matrix diagonalization, and generalized eigenvalue decomposition will be used respectively.

We prove a pointwise ergodic theorem for locally countable ergodic quasi-pmp (nonsingular) graphs, which gives an increasing sequence
of Borel subgraphs with finite connected components, averages over which converge a.e. to the expectations of $L^1$-functions.
This can be viewed as a random analogue of pointwise ergodic theorems for group actions: instead of taking a (deterministic) sequence
of subsets of the group and using it at every point to compute the averages, we allow every point to coherently choose such a sequence
at random with a strong condition that the sets in the sequence determine aconnected subgraph of the Schreier graph of the action.

In 1999, David Aldous conjectured that a certain natural 'random walk' on the space of binary combinatorial trees should have a continuum analogue, which would be a diffusion on the Gromov-Hausdorff space of continuum trees. This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently verified this conjecture with a path-wise construction of the diffusion. This construction combines our work on dynamics of certain projections of the combinatorial tree-valued random walk with our previous construction of interval-partition-valued diffusions.

For each function field multiple zeta value (defined by Thakur), we construct a t-module with an attached logarithmic vector such that a specific coordinate of the logarithmic vector is a rational multiple of that multiple zeta value. We then show that the other coordinates of this logarithmic vector contain hyperderivatives of a deformation of these multiple zeta values, which we call t-motivic multiple zeta values. This allows us to give a logarithmic expression for monomials of multiple zeta values. Joint work with Chieh-Yu Chang and Yoshinori Mishiba.

A C$^*$-algebra consists of an algebra of bounded linear operators acting on a Hilbert space which is closed the adjoint operation (roughly, the transpose) and is complete in a certain metric. Typical examples include the ring of $n \times n$ complex matrices and the ring $C(X)$ of representation of continuous functions from a compact space $X$ to the complex numbers. Many more interesting examples arise from various dynamical objects (e.g. group and group actions) and from various geometric/topological constructions.

The structure of finite dimensional C*-algebras is well understood: they are finite direct sums of complex matrix algebras. The class of approximately finite-dimensional (AF) C*-algebras, ones which may be written as (the closure of) an increasing union of f1inite-dimensional subalgebras, are also well understood: they are determined up to isomorphism by their module structure. However, the class of subalgebras of AF-algebras is still rather mysterious; it includes, for instance, all commutative C*-algebras and all C*-algebras generated by amenable groups. It is a long-standing problem to find an abstract characterization of subalgebras of AF-algebras.

I will discuss the AF-embedding problem for C$^*$-algebras and a recent partial solution to this problem which gives a nearly complete characterization of C$^*$-subalgebras of simple AF-algebras.

One method of building extremal objects is a random construction subject to constraints. For instance, one can build a tree by randomly adding edges as long as they do not form a cycle. However, analyzing these constructions is rarely so simple and even finding good asymptotic bounds can be difficult. The Differential Equations Method provides a powerful tool for analyzing random constructions subject to constraints by building a tractable system of differential equations out of a combinatorial construction, solving the system, and then proving that the random process is 'close' to the system solution with high probability. We present the differential equations method and give an application in finding lower bounds for graph Ramsey number asymptotics. Following the treatment in Bohman and Keevash (2013), we sketch the proof that R(3,t) $>$ ((1/4) - o(1))t$^2$/log(t).

A log Del Pezzo surface is a normal log terminal surface with anti-ample canonical bundle. Over the complex numbers, Keel and McKernan have classified all but a bounded family of the simply connected log Del Pezzo surfaces of rank one. In this talk we extend their classification in positive characteristic, and in particular we prove that for $p>5$ every log Del Pezzo surface of rank one lifts to characteristic zero with smooth base. As a consequence, we see that Kawamata-Viehweg vanishing holds in this setting. Finally, we exhibit some counter-examples in characteristic two, three and five.