The Israel–Stewart equations are a model for relativistic viscous fluids in which the viscous fluxes are not given by constitutive laws, as in the Navier–Stokes equations, but are independent unknowns governed by additional equations. Away from vacuum, the Israel–Stewart equations can be written as a first-order symmetric hyperbolic system, and local existence and uniqueness follow from the classical theory. In this talk, we consider the Israel–Stewart equations with bulk viscosity as the only source of dissipation, for a gas surrounded by vacuum: the density vanishes on the fluid–vacuum interface, which is advected by the fluid velocity. In this setting, the system degenerates at the vacuum boundary and the classical theory no longer applies.

A natural regime is the one in which the boundary moves with finite, nonzero acceleration. In this case, the density and the bulk viscous pressure are required to decay at specific rates near the boundary. These decay rates are referred to as the physical vacuum boundary condition. We show that solutions satisfying this condition exist locally in time in weighted Sobolev spaces. The main difficulties are the degeneracy at the boundary and a loss of derivatives in the estimates for the linearized equations. They are overcome by a Nash–Moser iteration scheme combining both Lagrangian and Eulerian coordinates.

In this talk I’ll give a brief but gentle introduction to some guiding problems in geometry, motivating the mean curvature flow. An important limiting factor in applying the mean curvature flow is that it often develops singularities, but there have been many recent advances towards understanding these well. With these in hand, I’ll then discuss some recent joint work with Ao Sun on using the mean curvature flow to produce minimal surfaces. 

(Joint Algebra-Data Seminar)

Invariant polynomials have been recently shown to be useful for finding maximally entangled quantum systems for small numbers of particles.  Invariant rings for spaces of tensors are particularly nice: they are finitely generated, and Cohen-Macaulay, and in the tame cases (the set of orbits depending on finitely many parameters) they are even polynomial rings. We can compute their Hilbert series via computations with finite groups. But what if we want to really get our hands on computations with invariants? Schur-Weyl duality and Young symmetrizers provide a way to access and evaluate these functions at least in low degrees. The first case of qubits with wild orbit type is with 5 qubits, and in this case it might be too difficult to compute all generators of the invariant ring. Instead, we could be happy with a set of primary invariants. I will discuss our implementations of these invariant theory computations in Macaulay2 and attempts to find sets of primary invariants in this case.

The incompressible Euler equations are the classical low Mach number limit of the compressible Euler system, and the usual justifications proceed through compactness or relative entropy, which effectively rule out highly oscillatory behavior. I will take the opposite route and ask whether the incompressible system is a universal attractor: can every incompressible weak solution be realized as a low Mach limit of compressible ones? The answer is yes. Given any \(L^2\) weak solution of incompressible Euler, we construct, for each small Mach number, infinitely many weak solutions of the rescaled compressible system converging strongly to it as the Mach number vanishes. The proof builds a compressible subsolution with small Reynolds stress and then runs a convex integration scheme. The main new ingredient is an \(L^2\)-type constraint inserted into the non-constructive \(L^\infty\) (Baire category) scheme, which lets one control the turbulent perturbation by the trace of the Reynolds stress, which is a bound that is automatic in the constructive \(C^\alpha\) theory but was not previously available here. This is joint work with Alexis Vasseur, Dehua Wang, and Cheng Yu.

We study the use of feedforward neural networks for solving parametrized systems of polynomial equations over the real numbers. Our model involves a classifier network to predict the number of real solutions for a given parameter and a regressor network for each realizable number of solutions. We highlight the interaction between algebraic geometry and machine learning throughout. In particular, the classifier training is sensitive to the singularities of the real discriminant locus, and the regressor training is sensitive to the presence of nontrivial real monodromy loops. We demonstrate the model on toy problems as well as systems of equations arising in dynamical systems and robotics.  This is based on joint work with Julianne Barnhart, Jon Hauenstein, Ikenna Nometa, Maggie Regan, Matthew Trang, and Charles Wampler.

Judith Sally proved in 1980 that the associated graded ring of a one-dimensional Gorenstein local ring of multiplicity and embedding dimension \(e-2\) is Cohen-Macaulay. Numerical semigroup rings provide a large and important class of one-dimensional Cohen-Macaulay rings. In 2014, Herzog and Stamate showed that the numerical semigroup \(\langle e,e+1,e+4,e+5,\dots,2e+1\rangle\)  defines a Gorenstein semigroup ring satisfying Sally’s conditions. We call a numerical semigroup Sally type if its multiplicity is one more than its width; equivalently, it can be obtained from the interval semigroup \(\langle e,e+1,\dots, 2e-1\rangle\) by removing some generators.

In this talk, we first focus on the case where two generators are removed: \(S= \{e, e+1,.., 2e-1\} \backslash \{e+m,e+n\}\), and study the following properties: Frobenius numbers, Cohen-Macaulay type, and computing minimal generating sets for their defining ideals using Hochster’s formula. If time permits, we will use Apéry sets to understand when “similar” numerical semigroups have the same Betti numbers. 

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).