Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman.  The game consists of $3n$ rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors exactly $n$ times during the $3n$ rounds, while Norman is allowed to play normally without any restrictions. We show that a certain greedy strategy is the unique optimal strategy for Rei in this game, and that Norman's expected score is $\Theta(\sqrt{n})$.  We also prove several general theorems about semi-restricted games arising from digraphs.  This is joint work with Erlang Surya, Yuanfan Wang, Ji Zeng.

The 3D incompressible Euler equations in a bounded domain are most often supplemented with impermeable boundary conditions, which constrain the fluid to neither enter nor leave the domain. In this talk, I will explain how we obtain well-posedness of solutions in which the full value of the velocity is specified on inflow and the normal component is specified on outflow. We do this for multiply connected domains, and establish compatibility conditions to obtain arbitrarily high Holder regularity.

This is joint work with Gung-Min Gie and Anna Mazzucato.

Optimal Transport, a theory for optimal allocation of resources, is widely used in various fields such as astrophysics, machine learning, and imaging science. However, many applications impose elementwise constraints on the transport plan which traditional optimal transport cannot enforce. Here we introduce Supervised Optimal Transport (sOT) that formulates a constrained optimal transport problem where couplings between certain elements are prohibited according to specific applications. sOT is proved to be equivalent to an $l^1$ penalized optimization problem, from which efficient algorithms are designed to solve its entropy regularized formulation. We demonstrate the capability of sOT by comparing it to other variants and extensions of traditional OT in color transfer problem. We also study the barycenter problem in sOT formulation, where we discover and prove a unique reverse and portion selection (control) mechanism. Supervised optimal transport is broadly applicable to applications in which constrained transport plan is involved and the original unit should be preserved by avoiding normalization.

Properly proximal groups were introduced recently by Boutonnet, Ioana, and Peterson, where they generalized several rigidity results to the setting of higher-rank groups. In this talk, I will describe how the notion of proper proximality fits naturally in the realm of von Neumann algebras. I will also describe several applications, including that the group von Neumann algebra of a non-amenable inner-amenable group cannot embed into a free group factor, which solves a problem of Popa. This is joint work with Srivatsav Kunnawalkam Elayavalli and Jesse Peterson.

In recent years, our understanding of stable homotopy groups of spheres at $p=2$ increased drastically due to work of Isaksen, Wang, and Xu. A primary method they used is the "cofiber-of-tau philosophy" in the stable infinity category of 2-complete $\mathbb{C}$-motivic spectra. To a sufficiently nice spectrum $E$, Pstragowski produced an infinity-categorical deformation of spectra called "$E$-synthetic spectra," which exhibits and generalizes the cofiber-of-tau phenomena seen in $\mathbb{C}$-motivic spectra. $E$-synthetic spectra are closely related to the $E$-Adams spectral sequence and this relation has had many applications in recent years for Adams spectral sequence calculations. 

In this talk, we discuss some of the basic calculational features of synthetic spectra in the case of $E=H\mathbb{F}_2$, including how to compute bigraded synthetic homotopy groups and their applications to classical Adams spectral sequence calculations for $p=2$. In particular, we discuss our computation of the bigraded synthetic homotopy groups of 2-complete tmf, the connective topological modular forms spectrum.
 

In joint work with Jens Marklof, we prove new upper bounds for theta sums -- finite exponential sums with a quadratic form in the oscillatory phase -- in the case of smooth and box truncations. This generalizes results of Fiedler, Jurkat and Körner (1977) and Fedotov and Klopp (2012) for one-variable theta sums and, in the multi-variable case, improves previous estimates obtained by Cosentino and Flaminio (2015). Key inputs in our approach include the geometry of $\mathrm{Sp}(n, \mathbb{Z}) \backslash \mathrm{Sp}(n, \mathbb{R})$, the automorphic representation of theta functions and their growth in the cusp, and the action of the diagonal subgroup of $\mathrm{Sp}(n, \mathbb{R})$.

We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, fluids, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow and Yamabe flow. I will also present some open questions.

A variety satisfies potential density if it contains a dense subset of rational points after extending its ground field by a finite degree. A collection of varieties satisfies uniform potential density if that degree can be uniformly bounded. I will discuss this property for connected algebraic groups of a fixed dimension and elliptic K3 surfaces. This is joint work with Kuan-Wen Lai.

In this talk, I will give an introduction to subdivision schemes, which are iterative refinement processes for interpolating or approximating discrete data points. Most result on subdivision schemes concern data in vector spaces and rules which are linear. I will present an adaptation of subdivision schemes to operate on manifold-valued data using the intrinsic geometry of the underlying manifold (such as the exponential map). Analysis of convergence and smoothness properties will be presented as well. Subdivision schemes find applications in computer graphics and 3D animated movies.