The cohomology of classifying spaces is an important classical topic in algebraic topology. However, much less is known in the equivariant setting, where one wants to know the RO(G)-graded cohomology of classifying G-spaces. The problem is that RO(G)-graded cohomology is notoriously difficult to compute even when G is cyclic.In this talk, I will explain my computations in the case of cyclic 2-groups G while keeping technical details to a minimum. The main goal is to understand rational equivariant characteristic classes, but I will also discuss some mod 2 computations and their relevance to the equivariant dual Steenrod algebra.

The Connes Embedding Problem (CEP) is arguably one of the most famous open problems in operator algebras. Roughly, it asks if every tracial von Neumann algebra can be approximated by matrix algebras. Earlier this year, a group of computer scientists proved a landmark result in complexity theory called MIP*=RE, and, as a corollary, gave a negative solution to the CEP. However, the derivation of the negative solution of the CEP from MIP*=RE involves several very complicated detours through C*-algebra theory and quantum information theory. In this talk, I will present joint work with Bradd Hart where we show how some relatively simple model-theoretic arguments can yield a direct proof of the failure of the CEP from MIP*=RE while simultaneously yielding a stronger, Gödelian-style refutation of CEP as well as the existence of “many” counterexamples to CEP. No prior background in any of these areas will be assumed.

Many of the questions asked during the birth of probability (e.g. what's the probability of getting a certain hand in poker?) are equivalent to basic counting problems, and since then there have been numerous applications of combinatorics to probability (e.g. moment method proofs for the semi-circle law). In this talk, probability strikes back with a vengeance by solving some (non-trivial) counting problems.

Dirac proved that every $n$-vertex graph with minimum degree at least $n/2$ contains a hamiltonian cycle. We prove an analogue for hypergraphs: we give exact bounds for the minimum degree of a uniform hypergraph that implies the existence of hamiltonian Berge cycles. \\ \\ This is joint work with Alexandr Kostochka and Grace McCourt.

A common task in many data-driven applications is to find a low dimensional manifold that describes the data accurately. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there is no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise. In this talk, we will present a method that estimates a manifold and its tangent in the ambient space. Moreover, we establish rigorous convergence rates, which are essentially as good as existing convergence rates for function estimation. This is a joint work with Barak Sober.

Every continuous action of a countable group on a Polish space induces a Borel equivalence relation. We are interested in the problem of realizing (i.e. finding a Borel isomorphic copy of) these equivalence relations as continuous actions on compact spaces. We provide a number of positive results for variants of this problem, and we investigate the connection to subshifts.

A novel approach to simulate simple protein-­ligand systems at large time­ and length­-scales is to couple Markov state models (MSMs) of molecular kinetics with particle-­based reaction­-diffusion (PBRD) simulations; this approach is named MSM/RD. Current formulations of MSM/RD lack an underlying mathematical framework to derive coupling schemes; they are limited to protein-­ligand systems, where the ligand orientation and conformation switching are not taken into account; and they lack multi­particle extensions. In this work, we develop a general MSM/RD framework by coarse­-graining molecular dynamics into hybrid switching diffusion processes, a class of stochastic processes that integrate continuous dynamics and discrete events into the same process. With this MSM/RD framework, it is possi­ble to derive MSM/RD coupling schemes as discretizations of the underlying equations. It also allows conformation switching and the inclusion of all the rotational degrees of freedom. Given enough data to parametrize the model, it is capable of modeling protein­-protein interactions over large time­ and length­-scales, and it can be extended to han­dle multiple molecules. We derive the MSM/RD framework, and we implement and verify it for two protein­-protein benchmark systems and one multiparticle implementation to model the formation of pentameric ring molecules.