Ordinary (0-form) symmetries of a QFT act on local operators, while 1-form symmetries naturally act on line operators. These symmetries can mix to form the structure of a 2-group. I'll review this notion and will be drawing a bunch of pretty/useful/pretty useful pictures to elaborate on the structure.

In the first part of this talk we study the existence, regularity and the free boundary of the double obstacle problem in different formulations that involve linear, elliptic, parabolic and fully non linear equations. The second part focuses in the two membranes problem for fully non linear elliptic operators, here we prove existence of solutions and then we prove the optimal regularity when the operators involved are the Pucci extremal operators. Finally we give an example that shows that no regularity for the free boundary is to be expected to hold in general.

The L-space conjecture asserts that for an irreducible, orientable three-manifold M, the following three statements are equivalent: a) M has a left-orderable fundamental group, b) M admits a co-orientable taut foliation and c) M is not a Heegaard Floer L-space. In this talk, we will discuss progress made in the understanding of the implication from statement b) to a) above. We will show that the existence of certain ``nice'' taut foliations on a three-manifold leads to the left-orderability of its fundamental group. It is not clear whether three-manifolds that admit co-orientable taut foliations will admit one with such a ``nice'' property. However, large families of three manifolds that do possess such ``nice'' taut foliations will be presented. This is joint work with Steve Boyer.

To understand a finitely presented group it is natural to explore its finite quotients. For instance, one might try to prove that a presentation does not represent the trivial group by exhibiting a map onto a non-trivial finite group. The potential of such techniques depends on the extent to which the groups being studied are determined by the totality of their finite quotients. If the groups at hand are residually finite, then it is reasonable to expect that one will be able to detect many properties from the totality of its finite quotients. This talk will discuss recent progress on constructing residually finite groups completely determined by their finite quotients.

The constructions in the title originally arose in representation theory (especially modular representation theory). They have since found important applications, as well as new interpretations, in stable homotopy theory. I will give an introduction to these notions, as well as some amusing facts about them. I will then discuss some joint work with Aaron Royer and Saul Glasman exploring these ideas in stable homotopy theory.

Seismic data contains interpretable information about subsurface properties, which are important for exploration geophysics. Full waveform inversion (FWI) is a nonlinear inverse technique that inverts the model parameters by minimizing the difference between the synthetic data from numerical simulations and the observed data at the surface of the earth. The least-squares () norm of this difference is the traditional objective function for FWI, but it is sensitive to the initial model, the data spectrum, the noise in the measurement, and other issues related to optimization. The norm is a point-by-point comparison. Other misfit functions with a global feature have been proposed in the literature to achieve better convexity, but none of them is technically a metric. Here we apply the quadratic Wasserstein () metric of the optimal transport theory to FWI. Both the amplitude differences and the phase mismatches are considered in this new misfit function. Mathematically, we prove the convexity of the metric concerning shift, dilation, and partial amplitude changes of data as well as its insensitivity to noise. Despite these good properties of the metric, solving optimal transport problem in higher dimension is challenging. We first compute the misfit globally by regarding it as a 2D optimal transport problem. Since the Monge-Ampere equation is rigorously related to the metric, we solve the 2D optimal transport problem by solving a fully nonlinear Monge-Ampere equation based on a monotone finite difference solver which has been proved to converge to the viscosity solution. To increase the resolution of the inversion, we further develop another method to compute the Monge-Ampere metric: trace-by-trace comparison based on the 1D optimal transport. The 1D technique can be solved accurately and efficiently and thus is more robust to handle more complicated problems with less computational cost. We also explore the connections between optimal transport and other misfit functions and explain the intrinsic features of the transport-based idea. Since optimal transport problem concerns nonnegative measures, we will also investigate the critical data normalization step which transforms the sign-changing wavefields into probability densities. This is the most important topic to address in applying optimal transport to seismic inversion. With the norm being the misfit function, FWI using the reflection data often results in migration-like features in the model updates. We argue that it is the inherent nonconvexity that prevents it from updating the kinematics with high-frequency data. Through numerical examples and discussions, we demonstrate that the better convexity of the metric can tackle the local minima generated by the high-wavenumber update which appears in addition to the known cycle-skipping issues caused by phase mismatches.

The two membranes problem was first studied by Vergara-Caffarelli in the context of variational inequalities to study the equilibrium position of two elastic membranes that are prescribed to remain one on top of the other and are not allowed to cross. In the linear elliptic case the problem can be reduced to the classical obstacle problem by considering the vertical distance between the two membranes. The nonlinear case in divergence form was studied by Silvestre in '05 where the situation can, again, be reduced to an obstacle type problem for the difference of the two functions. In a recent paper, Caffarelli, De Silva and Savin considered the two membranes problem for two different (possibly nonlocal) operators with not necessarily the same order. In this talk, motivated by a model in mathematical finance, we will consider the two membranes problem for two fully nonlinear operators satisfying a type of compatibility condition.

The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then sketch a new construction of non-compact complete affine manifolds with non-solvable fundamental group whose cohomological dimension is arbitrarily large. At the heart of the issue is understanding the dynamics of discrete groups acting by affine transformations of \mathbb R^n.

Varifolds are a measure-theoretic generalization of differentiable manifolds that are well-suited for problems in the calculus of variations. We will begin by covering the needed background from geometric measure theory, then we will define and discuss varifolds and some of their properties, and we will finish by discussing varifolds in connection to minimal surfaces and Plateau's problem.