The Transformer from "Attention Is All You Need" is now the ubiquitous architecture in state-of-the-art models such as GPT and AlphaFold. At Transformer's heart and soul is the "attention mechanism". We propose an attention-based deep direct sampling method for solving Electrical Impedance Tomography, a class of boundary value inverse problems. We try to provide an example of whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming neural operators. Then, inspired by the idea of employing partial differential equation (PDE) solvers as preconditioners for this inverse problem, we investigate a "solver-in-the-loop" training mechanism for Transformer-based neural operators. It allows the interaction between learnable parameters in a PDE solver module and those in neural networks for image reconstruction. The new method achieves superior reconstruction quality compared with its predecessors and is robust to noise.

I will present two complementary small-data stability theorems for the 3D irrotational and isentropic compressible Euler equations in spherical symmetry. We consider the Cauchy problem with initial data that are perturbations of a non-vacuum constant state.

In the first theorem, we treat all equations of state except that of the Chaplygin gas. For an open set of initial data with tails at infinity, we construct a maximal globally hyperbolic development (MGHD), both to the future and to the past, provide a complete description of its global structure, and prove its uniqueness. The uniqueness is a substantive part of the result, since Eperon, Reall, and Sbierski proved that MGHDs need not be unique for general quasilinear hyperbolic equations. The boundary of this MGHD consists of three types of components: creases where singularities first occur in positive and negative time, singular boundaries where gradient blowup occurs, and Cauchy horizons emanating from the creases. This is the first MGHD existence-uniqueness-stability result for shock-forming solutions for any multi-dimensional quasilinear hyperbolic wave-like system.

In the second theorem, we consider another open set of initial data whose tails have the opposite sign from those in the first theorem. We prove global existence both to the future and to the past. This is the first small-data global existence result in the entire spacetime for any 3D quasilinear hyperbolic wave-like system that fails to satisfy the null condition and the weak null condition.

This is joint work with Leonardo Abbrescia and Jared Speck.

A long-standing problem in metric geometry is to recognize those metric spaces that admit a “good” parametrization by the Euclidean space or the unit sphere. In this talk, we will discuss the quasisymmetric uniformization problem and some recent advancements in terms of the Menger curvature, a curvature suited for Euclidean sets with no a priori smooth structure. The talk is based on a joint work with G. C. David (Ball State University).

In this talk, we study the problem of minimizing the distance from a data point to a subvariety of the Grassmannian from the perspective of metric algebraic geometry. An important invariant in this line of work is the number of critical points of this optimization problem for general data. When the data is a general point in the ambient space, this number is called the Euclidean distance (ED) degree of the variety. However, we focus mainly on the case when the data is a general point in the Grassmannian, and call the number of critical points the Grassmann distance (GD) degree. As we will see, the GD degree is more subtle than the ED degree. We give formulas for the ED and GD degrees of prominent subvarieties of the Grassmannian, including Schubert and matroid varieties.

David and Semmes proved that, for an Ahlfors regular measure, (L^2)-boundedness of a sufficiently rich class of singular integral operators implies uniform rectifiability. I will discuss analogues in parabolic space and the Heisenberg group. This is joint work with John Hoffman (EHU, University of the Basque Country).