We will discuss a historical theorem of Gelfand and Neumark
characterizing abelian C*-algebras. The main indication for the importance of
this result is the fact that it relates two areas in math, namely topology and
operator algebras. We will take the time to discuss the ideas behind the proof
of this beautiful theorem as well as some other applications and related ideas.

Interior methods provide an effective approach for the treatment of inequality constraints in nonlinearly constrained optimization. A new primal-dual interior method is proposed that has favorable global convergence properties, yet, under suitable assumptions, is equivalent to the conventional path-following interior method in the neighborhood of a solution. The method may be combined with a primal-dual shifted penalty function for the treatment of equality constraints to provide a method for general optimization problems with a mixture of equality and inequality constraints.

Error-correcting codes are often used when data is transmitted over
a channel in which noise can occur, thereby damaging some of the data. There
are several types of error-correcting codes. In this talk, we will discuss an errorcorrecting
code that is defined in terms of a particular finite geometry. This finite
geometry comes from the incidence matrix of the so-called Wenger graphs. These
graphs are well-known to those working in extremal graph theory. The talk will
begin with a brief introduction to error-correcting codes, followed by linear codes.
We will then define the finite geometry, and discuss some progress on an open
problem of Cioab\u{a}, Lazebnik, and Li.

\'{S}niady constructed a random matrix model whose empirical eigenvalue distribution converges to the $q$-Gaussian random variable. In this talk, we prove that the Segal-Bargmann transform defined on the \'{S}niady random matrix model converges to the $q$-Segal-Bargmann transform.

Ancient solution is a type of Ricci flow that plays a fundamental role in singularity analysis. We introduce some results for ancient solutions, especially the classification of three-dimensional Type I ancient solutions, and a rigidity theorem for the four-dimensional shrinking cylinder.

This talk introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between n data points and a set of $n_R$ reference points, where $n_R$ can be drastically smaller than n. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as $\Vert p-q \Vert \sim$ O($n^{-1/2+\delta})$ for any $\delta>$ 0 based on a result of convergence in distribution of the test statistic. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

Many problems in stochastic modeling come down to study the crossing time of a certain stochastic process through a given boundary, lower or upper. Typical fields of application are in risk theory, epidemic modeling, queueing, reliability and sequential analysis. The purpose of this talk is to present a method to determine boundary crossing probabilities linked to stochastic point processes having the order statistic property. A very well-known boundary crossing result is revisited, a detailed proof is given. the same arguments may be used to derive results in trickier situations. We further discuss the practical implications of this classical.

I will explain the arithmetic analogue of the Gan-Gross-Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it.

Fogarty showed in the 1970s that the Hilbert scheme of n points on a smooth surface is itself smooth. Interest in these Hilbert schemes has grown since it has been shown they arise in hyperkahler geometry, geometric representation theory, and algebraic combinatorics. In this talk we will explore the geometry of certain tautological bundles on the Hilbert scheme of points. In particular we will show that these tautological bundles are (almost always) stable vector bundles. We will also show that each sufficiently positive vector bundle on a curve C is the pull back of a tautological bundle from an embedding of C into the Hilbert scheme of the projective plane.