This talk will introduce the space of $id{e}les$ for a global field $K$. For convenience, the construction given in the talk will use the rational numbers. In the process, completions, additive/multiplicative valuations, and connections to ideals will be discussed. If time permits, we will generalize to an arbitrary Galois number field, ultimately ending with some Galois-theoretic properties of the $id{e}les$.

We will discuss rigidity results for holomorphic mappings in CR and complex geometry, emphasizing the connections between the two types of rigidity. We discuss in more detail rigidity of volume-preserving maps between Hermitian symmetric spaces, based on the work of Mok-Ng and my recent joint work with Fang and Huang.

Mean curvature flow is the negative gradient flow of volume, so any closed hypersurface flows in the direction of steepest descent for volume and eventually becomes extinct in finite time. In most cases, the flow develops singularities before its extinction time. It is known that the asymptotic behaviors of the flow near a singularity are modeled on a special class of solutions to mean curvature flow, which are called self-shrinkers. In this talk, we will outline a program on the classification of noncompact two-dimensional self-shrinkers, and report some recent progress with an emphasis on the geometry at infinity of these self-shrinkers.

In this talk, I will start with an introduction to the Einstein 4-manifold. Then I will discuss some earlier result on classification of the positive case. Finally I will mention some recent development in this area.

A common problem in modern statistical applications is to select, from a large set of candidates, a subset of variables which are important for determining an outcome of interest. For instance, the outcome may be disease status and the variables may be hundreds of thousands of single nucleotide polymorphisms on the genome. For data coming from low-dimensional ($n \ge p$) linear homoscedastic models, the knockoff procedure recently introduced by Barber and Cand\'es solves the problem by performing variable selection while controlling the false discovery rate (FDR). In this talk I will discuss an extension of the knockoff framework to arbitrary (and unknown) conditional models and any dimensions, including $n < p$, allowing it to solve a much broader array of problems. This extension requires the design matrix be random (independent and identically distributed rows) with a covariate distribution that is known, although the procedure appears to be robust to unknown/estimated distributions. No other procedure solves the variable selection problem in such generality, but in the restricted settings where competitors exist, I will demonstrate the superior power of knockoffs through simulations. Finally, applying the new procedure to data from a case-control study of Crohn’s disease in the United Kingdom resulted in twice as many discoveries as the original analysis of the same data.

The classic 1935 paper of Erdos and Szekeres entitled ``A combinatorial problem in geometry" was a starting point of a very rich discipline within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres studied the following geometric problem. For every integer $n \geq 3$, determine the smallest integer $ES(n)$ such that any set of $ES(n)$ points in the plane in general position contains n members in convex position, that is, n points that form the vertex set of a convex polygon. Their main result showed that $ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}$. In 1960, they showed that $ES(n) \geq 2^{n-2} + 1$ and conjectured this to be optimal. Despite the efforts of many researchers, no improvement in the order of magnitude has been made on the upper bound over the last 81 years. In this talk, we will sketch a proof showing that $ES(n) =2^{n +o(n)}$.

In joint work with Daniel Hast and Joseph we count degree 3 and 4 covers of the projective line over finite fields. This is a geometric analogue of the number field side of counting cubic and quartic fields. We take a geometric approach, by using a vector bundle parametrization of these curves which is different from the recent work of Manjul Bhargava, Arul Shankar, Xiaoheng Wang "Geometry of numbers methods over global fields: Prehomogeneous vector spaces" in which the authors extend the geometry of numbers methods to global fields. Our count is just for $S_3$ and $S_4$ covers, and we put the rest of the curves in our error term.

We start by discussing Bochner's canonical decomposition of negative definite functions on the space of infinite sequences of real numbers and then look at extensions of this theorem within the framework of more general infinite-dimensional groups like the infinite symmetric group. The chosen approach relies on the theory of spherical functions developed by G. Olshanski.

In classical Hodge theory, variations of Hodge structure and their associated period mappings play a crucial role. In the $p$-adic world, it turns out there are *two* natural kinds of period maps associated with variations of $p$-adic Hodge structure: the ``Grothendieck-Messing" period maps, which roughly come from comparing crystalline and de Rham cohomology, and the ``Hodge-Tate" period maps, which come from comparing de Rham and $p$-adic etale cohomology. I'll discuss these period maps, their applications, and some new results on their construction and geometry. This is partially joint work with Jared Weinstein.

I will describe a construction which associates a canonical $p$-adic L-function with a refined cohomological Hilbert modular form $(\pi, \alpha)$ under some mild and natural assumptions. The main novelty is that we do not impose any hypothesis of “small slope” or “noncriticality” on the allowable refinements. Over $\mathbb{Q}$, this result is due to Bellaiche. Our strategy for dealing with critical refinements is roughly parallel to his, and in particular relies on a careful study of the local geometry of eigenvarieties at classical (but possibly critical) points. This is joint work with John Bergdall.

Data assimilation or filtering of nonlinear dynamical systems combines numerical models and observational data to provide the best statistical estimates of the systems. Ensemble-based methods have proved to be indispensable filtering tools in atmosphere and ocean systems that are typically high dimensional turbulent systems. In operational applications, due to the limited computing power in solving the high dimensional systems, it is desirable to use cheap and robust reduced-order forecast models to increase the number of ensemble for accuracy and reliability. This talk describes a multiscale data assimilation framework to incorporate reduced-order multiscale forecast methods for filtering high dimensional complex systems. A reduced-order model for two-layer quasi-geostrophic equations, which uses stochastic modeling for unresolved scales, will be discussed and applied for filtering to capture important features of geophysical flows such as zonal jets. If time permits, a generalization of the ensemble-based methods, multiscale clustered particle filters, will be discussed, which can capture strongly non-Gaussian statistics using relatively few particles.