For any $C_2$-equivariant spectrum, we can functorially assign two non-equivariant spectra - the underlying spectrum and the geometric fixed point spectrum. They both induce maps from $RO(C_2)$-graded cohomology to classical cohomology. In this talk, I will compare the $RO(C_2)$-graded squaring operations with the classical squaring operations along the induced maps. This is joint work with Prasit Bhattacharya and Bertrand Guillou.

I will talk about two recent projects on solving inverse problems for kinetic models, where a change of perspective between Lagrangian and Eulerian is highly beneficial. In the first project, we are interested in recovering the initial temperature of the nonlinear Boltzmann equation given macroscopic quantities observed at a later time. With the problem formulated as constrained optimization, our proposed adjoint DSMC method, together with the well-known (forward) DSMC method, makes it possible to evaluate Boltzmann-constrained gradient within seconds, independent of the size of the parameter. In the second project, we are interested in calibrating the parameter in the chaotic dynamic system. Transforming the long-time trajectories to an invariant measure significantly improves the inverse problem's ill-posedness. It also turns the original ODE model into a PDE model (continuity equation or Fokker-Planck equation), allowing efficient gradient calculation for the resulting PDE-constrained optimization problem.

Abstract: Strong 1-boundedness is a notion introduced by Kenley Jung which captures (among Connes-embeddable von Neumann algebras) the property of having "a small amount of matrix approximations". Some examples are diffuse hyperfinite von Neumann algebras, vNa's with Property Gamma, vNa's that are non prime, vNa's that have a diffuse hyperfinite regular subalgebra and so on. This is a von Neumann algebra invariant, and was used to prove in a unified approach remarkable rigidity theorems. One such is the following: A non trivial free product of von Neumann algebras can never be generated by two strongly 1-bounded von Neumann subalgebras with diffuse intersection. This result cannot be recovered by any other methods for even hyperfinite subalgebras. In this talk, I will present joint work with David Jekel and Ben Hayes, settling an open question from 2005, whether Property (T) II$_1$ factors are strongly 1-bounded, in the affirmative. Time permitting, I will discuss some insights and future directions.

In an $n$-resident town, Oddtown, all of their clubs must satisfy the following properties: all clubs must have an odd number of members and amongst any two distinct clubs, there must be an even number of residents in common. The classical oddtown theorem states that any such town can have at most $n$ clubs. In this talk, we explore how the residents can have $n+1$ clubs of odd size and minimize the chance of the town catching them. That is, we'd like to minimize the number of pairs of clubs with an odd number of members in common. We will also explore a similar problem with Eventown.

The letters D and O are topologically indistinguishable (both are circles). However, after superimposing each symbol with their reflections across several axes, one \emph{can} distinguish between them. The curvature in the O results in a different evolution in first homology when compared to the angled D. In this talk we will expand on this idea by explaining a white box classification algorithm which classifies an image as one of the 26 letters in the (capitalized) English alphabet. The driving force is the theory of persistent homology, as implemented in the Ripser package. This technique is less powerful than traditional techniques of machine learning (such as a neural net), but it is much more explainable.

An independent set $I$ of a graph $G$ is said to be a maximal independent set (MIS) if it is maximal with respect to set inclusion. Nielsen proved that the maximum number of MIS's of size $k$ in an $n$-vertex graph is asymptotic to $(n/k)^k$, with the extremal construction being a disjoint union of $k$ cliques with sizes as close to $n/k$ as possible. In this talk we study how many MIS's of size $k$ an $n$-vertex graph $G$ can have if $G$ does not contain a clique $K_t$. We prove for all fixed $k$ and $t$ that there exist such graphs with $n^{\lfloor\frac{(t-2)k}{t-1}\rfloor-o(1)}$ MIS's of size $k$ by utilizing recent work of Gowers and B. Janzer on a generalization of the Ruzsa-Szemer\'edi problem. We prove that this bound is essentially best possible for triangle-free graphs when $k\le 4$. \medskip This is joint work with Xiaoyu He and Jiaxi Nie.

The Monge-Ampere equation det$(D^2u) = 1$ arises in prescribed curvature problems and in optimal transport. An interesting feature of the equation is that it admits singular solutions. We will discuss new examples of convex functions on $R^n$ that solve the Monge-Ampere equation away from finitely many points, but contain polyhedral and Y-shaped singular structures. Along the way we will discuss geometric motivations for constructing such examples, as well as their connection to a certain obstacle problem.

The method of moments is a statistical method for density estimation that equates sample moments to moment equations for a given family of densities. When the underlying distribution is assumed to be a convex combination of Gaussian densities, the resulting moment equations are polynomial in the density parameters. We examine the asymptotic behavior of the variety stemming from these equations as the number of components and the dimension of each component increases. This is joint work with Jose Israel Rodriguez and Carlos Amendola.

A Cayley diagram is a labeling of a graph $G$ that encodes an action of a group which induces $G$. For instance, a $d$-edge coloring of a $d$-regular tree is a Cayley diagram for the group $(\mathbb{Z}/2\mathbb{Z})^{*d}$. In this talk, we will investigate when a Cayley graph $G=(\Gamma, E)$ admits an $\operatorname{Aut}(G)$-f.i.i.d. Cayley diagram and show that $\Gamma$-f.i.i.d. solutions to local labeling problems for such graphs lift to $\operatorname{Aut}(G)$-f.i.i.d. solutions.

We describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.

The moduli space of stable curves parameterizes tuples $(C,p_1,...,p_n)$ of a compact, complex curve $C$ together with distinct marked points $p_1,\dots, p_n$. Inside this moduli space, there are natural subsets, called the strata of $k$-differentials, defined by the condition that there exists a meromorphic $k$-differential on $C$ with zeros and poles of some fixed multiplicities at the points $p_i$. I will discuss basic properties of these strata and explain a conjecture relating their fundamental class to the so-called double ramification cycles on the moduli space. I explain the idea of the proof of this conjecture and some ongoing work with Costantini and Sauvaget on how to use this relation to compute intersection numbers of the strata with $\psi$-classes on the moduli of curves.