In this talk, I will discuss some new results on intersection graphs of pseudo-segments in the plane and their applications in graph drawing.  These results are joint work with Jacob Fox and Janos Pach.
 

For $(R, R^{+})$ an analytic perfectoid ring in char $p$, let $\mathbf{A}_{\textrm{inf}}(R^{+})$ be the ring of Witt vectors with the induced topology from $(R, R^{+})$. We prove that $\textrm{Spa}(\mathbf{A}_{\textrm{inf}}(R^{+}),\mathbf{A}_{\textrm{inf}}(R^{+}))$ is a sheafy adic space and its structure sheaf is acyclic. We first show $\mathbf{A}_{\textrm{{inf}}}(R^{+})$ is a stably uniform Banach ring. The 'stably uniform implies sheafy' argument is applied to Tate Huber rings in Buzzard-Verberkmoes(2015) and is generalized to analytic Huber rings in Kedlaya(2019). Here we show that the 'stably uniform implies sheafy' argument in Kedlaya(2019) can be applied to general stably uniform Banach rings whose underlying topological ring is a Huber ring. Finally we show the equivalence of categories of vector bundles over $\textrm{Spa}(\mathbf{A}_{\textrm{inf}}(R^{+}),\mathbf{A}_{\textrm{inf}}(R^{+}))$ and finite projective modules over $\mathbf{A}_{\textrm{inf}}(R^{+})$.

Many classical locally compact groups $G$ admit a very large compact subgroup $K$, where "very large" has been formalized by Gelfand in 1950. Examples include $G=\mathrm{SL}_n(\mathbb{R})$ with $K=\mathrm{SO}(n)$, or $G=\mathrm{SL}_n(\mathbb{Q}_p)$ with $K=\mathrm{SL}_n(\mathbb{Z}_p)$. More generally, all semi-simple algebraic groups and some tree automorphism groups.

In these explicit examples, there is also an "Iwasawa decomposition" which formalizes the fact that $G$ has a homogeneous Frustenberg boundary, even homogeneous under $K$. This is a very strong restriction for general groups.

Using no structure theory whatsoever, we prove that this homogeneity (and Iwasawa decomposition) holds for all Gelfand Pairs. This implies, in some geometric cases, a classification of Gelfand pairs. (This is related to a small part of my 2021 zoom colloquium at UCSD).
 

We consider the range of Gaussian analytic functions (GAF) with finite radius of convergence. We show that any unbounded GAF has dense image in the plane. We moreover show that if in addition the coefficients have sufficiently regular variances, then the image is the whole complex plane. We do this by exploiting an approximate connection between the coverage problem and spatial branching processes, analogous to the branching structure that appears in the log-correlated GAF and circular beta ensembles. This answers a long-standing open question of J.-P. Kahane, with sufficient regularity.  

Joint work with Alon Nishry.

Micrometer-sized particles made from responsive polymer networks (that is, responsive microgel colloids) are of high potential for the design of functional soft materials due to their adaptive compressibility and stimuli-triggered volume transition. In this talk, I will discuss models and theoretical approaches, such as Langevin simulations and classical (dynamic) density functional theory (DFT), to describe the structural and dynamical behavior of dispersions of these responsive colloids in and out of equilibrium. Moreover, I will argue that chemical fueling and the inclusion of chemomechanical feedback loops may lead to excitable and oscillatory dynamics of the active colloids, establishing first steps to a well-controlled design of artificial cells and their emergent behavior.

Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.

By a pseudo-representation, I mean an umbrella term for several abstractions/generalizations of finite dimensional representations. In my talk, I will discuss two types of pseudo-representations, explain their relevance to number theory, and highlight the relationship between the two. I will conclude by mentioning a few open questions. 

Different constructions of cohomology allow us to measure different properties of manifolds. To construct either the singular cohomology or the de Rham cohomology groups, the representatives of each are distinct things: cochains and closed differential forms. Yet the de Rham theorem states there is a correspondence between them. And we can see this directly via examples displaying how integration on a manifold is greatly influenced by the manifold’s topology. In this talk, I will cover highlights of the proof of the De Rham theorem and show how Stokes’ theorem lends itself to defining an explicit homomorphism between the two cohomology groups: de Rham and singular. I will also discuss a further direction that this correspondence allows which leads us into Hodge theory. 

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 current plan on using these tools for computing the Chow ring.

Aiming to understand the solvability of semilinear elliptic boundary values problems in bounded domains, we will review the best known techniques for establishing the existence of critical points of functionals whose critical points are solutions to such problems. The mountain pass lemma, the Nehari manifold,
the Morse index, and bifurcation analysis will be discussed to conclude the existence of seven solutions for an asymptotically linear elliptic problem.