Stokes matrices (i.e. unipotent upper triangular matrices) and their nonlinear braid group actions arise naturally in a number of geometric and algebraic contexts. Integral Stokes matrices are often of particular interest, motivating their reduction theory. After reviewing classical work of Markoff treating the case of 3-by-3 matrices, we describe joint work with Yu-Wei Fan for the 4-by-4 case by establishing an exceptional connection to SL2--character varieties of surfaces. This will also serve as an opportunity to present our recent work on effective finite generation of integral points on the latter moduli spaces. Time permitting, we finish by presenting new results (and problems) for Stokes matrices of larger dimension.

Quantum graphs are an operator space generalization of classical graphs that have appeared in different branches of mathematics including operator systems theory, non-commutative topology and quantum information theory. In this talk, I will review the different perspectives to quantum graphs and introduce a chromatic number for quantum graphs using a non-local game with quantum inputs and classical outputs. I will then show that many spectral lower bounds for chromatic numbers in the classical case (such as Hoffman’s bound) also hold in the setting of quantum graphs. This is achieved using an algebraic formulation of quantum graph coloring and tools from linear algebra.

In this talk, we present an algorithm created for Challenge 2 of the Grid Optimization Competition from ARPA-E (Advanced Research Projects Agency--Energy). The competition considered a security constrained optimal power flow (SCOPF) problem whose solution determines optimal dispatch and control settings for power generation and grid control equipment and maximizes the market surplus associated with the operation of the grid, subject to pre- and post-contingency constraints. We will discuss the practical issues associated with the challenge and describe the approach and heuristics established to enhance performance of the algorithm submitted to the competition. Results from the competition will also be presented.

This is joint work with Frank E. Curtis (Lehigh University), Daniel Molzahn (Georgia Tech), Andreas W\"{a}chter and Ermin Wei (Northwestern University)."

We will begin by discussing the multisymplectic structure associated to Hamiltonian PDEs as a generalization of the symplectic structure associated to Hamilton's equations in classical mechanics. Subsequently, we will turn to the question of how to computationally model such Hamiltonian PDEs while preserving the multisymplectic structure at the discrete level. This will lead us to the notion of a multisymplectic integrator. The analogue of these integrators in Hamiltonian mechanics are known as symplectic integrators, which are extremely well-studied and have proven to provide extremely robust and physically faithful simulations of mechanical systems. Multisymplectic integration, on the other hand, is still in its relative infancy.

After establishing the necessary background, we will introduce our construction of variational integrators for Hamiltonian PDEs which automatically yield multisymplectic integrators. This construction gives a systematic framework for constructing such multisymplectic integrators, based on the notion of a Type II generating functional. As an application of our framework, we will derive the class of multisymplectic partitioned Runge--Kutta methods and provide a numerical example with the family of sine--Gordon soliton solutions. This is joint work with Prof. Melvin Leok.

Building on work of Stanley and Bj\"{o}rner--Wachs

In this talk I will introduce Hamilton-Jacobi (HJ) Reachability Analysis for dynamic systems. HJ reachability uses level set methods to compute the set of initial conditions from which a dynamical system is guaranteed to reach its goal and/or avoid unsafe regions despite worst-case conditions. I will discuss challenges, recent advances, and applications.

The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edge-to- edge via translations) has been widely studied due to connections to polygonal billiards and Teichmüller theory. However, much less is known regarding straight-line flows on non-compact infinite translation surfaces. In this talk we will review work on straight line flows on infinite translation surfaces and consider such a flow on the Mucube – an infinite $\mathbb{Z}^3$ periodic half-translation square-tiled surface – first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions for the straight-line flow on the Mucube – in terms of a subgroup of $\mathrm{SL}_2 \mathbb{Z}$. We will use the latter characterization to obtain the group of derivatives of affine diffeomorphisms of the Mucube. This is joint work (in progress) with Andre P. Oliveira, Felipe A. Ramírez and Chandrika Sadanand.

Let $X$ be a perfectoid space with tilt $X^\flat$. We build a natural map $\theta:\mathrm{Pic} X^\flat\to\lim\mathrm{Pic} X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an isomorphism if $R = \Gamma(X,\mathcal{O}_X)$ is a perfectoid ring.

As a consequence we obtain a characterization of when the Picard groups of $X$ and $X^\flat$ agree in terms of the $p$-divisibility of $\mathrm{Pic} X$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence.

In this paper, we study supervised learning tasks on the space of probability measures. We approach this problem by embedding the space of probability measures into $L^2$ spaces using the optimal transport framework. In the embedding spaces, regular machine learning techniques are used to achieve linear separability. This idea has proved successful in applications and when the classes to be separated are generated by shifts and scalings of a fixed measure. This paper extends the class of elementary transformations suitable for the framework to families of shearings, describing conditions under which two classes of sheared distributions can be linearly separated. We furthermore give necessary bounds on the transformations to achieve a pre-specified separation level, and show how multiple embeddings can be used to allow for larger families of transformations. We demonstrate our results on image classification tasks.

Based on joint work with Caroline Moosmueller, Harish Kannan, and Alex Cloninger.

In the Wild West of geometry and groups, some familiar objects feel like home: Euclidean spaces, hyperbolic geometry, and more generally all the geometries described by semisimple Lie groups or similar matrix groups over local fields.

Harmonic analysis and operator algebras have their own wild seas, but again with a small safe haven: the ``Type I", home of the commutative world, compact groups, generalizations of Fourier analysis. In a precise sense, objects that can be ``described".

I will present a connection between these two worlds and show how it leads to previously unexpected classification results.

I will explain recent results on modular, geometrically meaningful compactifications of moduli spaces of K3 surfaces, most of which are joint with Philip Engel. A key notion is that of a recognizable divisor: a canonical choice of a divisor in a multiple of the polarization that can be canonically extended to any Kulikov degeneration. For a moduli of lattice-polarized K3s with a recognizable divisor we construct a canonical stable slc pair (KSBA) compactification and prove that it is semi toroidal. We prove that the rational curve divisor is recognizable, and give many other examples.