Last week, Yingjia gave an excellent overview of results in the theory of Stochastic Processing networks. I will be taking this opportunity to
demonstrate some of the concepts she mentioned in depth using a specific example. In particular, I will be discussing the fluid limit of a
processor sharing queue, how its properties can be used to prove a dimension reduction (state space collapse), and how this can be used to ultimately prove the diffusion approximation.

We study tuples $(X_1,\dots,X_d)$ of self-adjoint operators in a
tracial $W^*$-algebra whose non-commutative distribution is the free Gibbs
law for a (sufficiently regular) convex potential $V$. Such tuples model
the large $N$ behavior of random matrices $(X_1^{(N)}, \dots, X_d^{(N)})$
chosen according to the measure $e^{-N^2 V(x)}\,dx$ on
$M_N(\mathbb{C})_{sa}^d$. Previous work showed that
$W^*(X_1,\dots,X_d)$ is isomorphic to the free group factor
$L(\mathbb{F}_d)$. In a recent preprint, we showed that an isomorphism
$\phi: W^*(X_1,\dots,X_d)$ can be chosen so that $W^*(X_1,\dots,X_k)$ is
mapped to the canonical copy of $L(\mathbb{F}_k)$ inside $L(\mathbb{F}_d)$
for each $k$. The idea behind the proof is to apply PDE methods for
constructing transport to Gaussian to the conditional density of
$X_j^{(N)}$ given $X_1^{(N)}, \dots, X_{j-1}^{(N)}$. Then we analyze the
asymptotic behavior of these transport maps as $N \to \infty$ using a new
type of functional calculus, which applies certain
$\|\cdot\|_2$-continuous functions to tuples of self-adjoint operators
to self-adjoint tuples in (Connes-embeddable) tracial $W^*$-algebras.

Line search methods for unconstrained optimization based on satisfying the Wolfe conditions impose a restriction on the value of the directional derivative of the objective function at the new iterate. Projected search methods for bound-constrained optimization involve a line search along a continuous piecewise-linear path, which makes it impossible to apply the conventional Wolfe conditions. We propose a new quasi-Wolfe line search for piecewise differentiable functions. The behavior of the line search is similar to that of a conventional Wolfe line search, except that a step is accepted under a wider range of conditions. These conditions take into consideration steps at which the line search function is not differentiable. Some basic results associated with a conventional Wolfe line search are established for the quasi-Wolfe case. After identifying the practical considerations needed for converting a Wolfe line search into a quasi-Wolfe line search, details of the imp
lementation along with some numerical results will be presented.

We show that any set of $n$ points in general position in the plane
determines $n^{1-o(1)}$ pairwise crossing segments. The best
previously known lower bound, $\Omega(\sqrt{n})$ was proved more than
25 years ago by Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach,
and Schulman. Our proof is fully constructive, and extends to dense
geometric graphs. This is joint work with J\'anos Pach and G\'abor
Tardos.

I will present the recent tools I have developed to prove existence and
regularity properties of the critical points of anisotropic functionals.
In particular, I will provide the anisotropic extension of Allard's
celebrated rectifiability theorem and its applications to the
anisotropic Plateau problem. Three corollaries are the solutions to the
formulations of the Plateau problem introduced by Reifenberg, by
Harrison-Pugh and by Almgren-David. Furthermore, I will present the
anisotropic counterpart of Allard's compactness theorem for integral
varifolds. To conclude, I will focus on the anisotropic isoperimetric
problem: I will provide the anisotropic counterpart of Alexandrov's
characterization of volume-constrained critical points among finite
perimeter sets. Moreover I will derive stability inequalities associated
to this rigidity theorem.
Some of the presented results are joint works with De Lellis, De
Philippis, Ghiraldin, Gioffr\'e, Kolasinski and Santilli.

Finding roots of a function is one of the most fundamental tasks in mathematics. What if
the function is random ?

We are going to survey some of the main developments in the theory of random functions in the last 80 years or so, from the works of Polya, Erdos, Littlewood, Offord in the early 1900s to this very sunny day.

Gerrymandering is the (currently very relevant) issue in
representative democracies of drawing electoral districts to give a
political advantage to one party or group. Political events within the
last few years have sparked a lot of research activity from
mathematicians, computer scientists, and statisticians related to
detecting and quantifying gerrymandered electoral plans. In this talk I
will give an introduction to the problem of gerrymandering, discuss some
historical attempted methods of quantifying gerrymanders and their
shortcomings, and then talk about a promising new method called ``metagraph
Markov Chain Monte Carlo'' currently being researched and implemented.

We examine the relationship between having an algebraic Hecke character attached to the cohomology of a smooth projective variety $X$ equipped with a finite-order automorphism and the algebraicity of some Hodge/Tate classes on the product $X^n$. As a consequence, we deduce the Hodge and Tate conjectures for some self-products of varieties, including some self-products of $K3$ surfaces.

I will report some recent progress on p-adic Simpson
correspondence.