# Past Events

The Muskat problem on the half-plane models motion of an interface between two fluids of distinct densities in a porous medium that sits atop an impermeable layer, such as oil and water in an aquifer above bedrock. We develop a local well-posedness theory for this model in the stable…

Abstract: We introduce a method of constructing (generalized) capillary surfaces via Gromov's "$\mu$-bubble" method. Using this, we study low-dimensional manifolds with nonnegative scalar curvature and strictly mean convex boundary. We prove a fill-in question of Gromov, a band-width estimate,…

Abstract: I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range. Central to the framework is a new formulation of nonlinear gravitational perturbations of Kerr in a geometric gauge tailored to the…

Morrison, Walker, and Wedrich’s skein lasagna modules are 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining…

The two-dimensional one-component plasma (OCP), also known as the Coulomb gas, is a system that consists of identical electrically charged particles embedded in a uniform background of the opposite charge, interacting through a logarithmic potential and kept at a fixed temperature. In the 1990s…

Given a smooth closed n-manifold M and a k-tuple of basepoints in M, we define a Morse-type A∞-algebra called the based multiloop A∞-algebra and show the equivalence with the higher-dimensional Heegaard Floer A∞-algebra of k disjoint cotangent fibers of T*M.

Symplectic cohomology is a fundamental invariant of a symplectic manifold M with contact type boundary that is defined in terms of dynamical information and counts of pseudoholomorphic genus zero curves, and carries algebraic structures that parallel the algebraic structures on the Hochschild (…

We introduce a mod p analogue of the Mumford—Tate conjecture, which governs the p-adic monodromy of families of mod p abelian varieties. It turns out that the conjecture is closely related to a notion of formal linearity of mod p Shimura varieties. Surprisingly, the conjecture…

I will introduce the book "Superconcentration and Related Topics" by Sourav Chatterjee, which will be the topic of this quarter's seminar. Superconcentration occurs when classical concentration of measure gives suboptimal bounds on the order of fluctuations. I will go over several examples…

Derivation of the Boltzmann equation, introduction of the collision operator, the hard sphere case, conservation laws and entropy.