Georgia Institute of Technology College of Sciences

The theory of stable polynomials features a key notion called proper position, which generalizes interlacing of real-rooted polynomials to higher dimensions. In a recent paper, I introduced a Lorentzian analog of proper position and used it to give a new characterization of elementary quotients of valuated matroids. This connects the local structure of spaces of Lorentzian polynomials with the incidence geometry of tropical linear spaces. A central object in this connection is the moduli space of codimension-1 tropical linear subspaces of a given tropical linear space. In this talk, I will show some new structural results on this moduli space and their implications for Lorentzian polynomials.

In this talk I will discuss our recent effort to develop structure-preserving machine learning (ML) for time series data, focusing on both dissipative PDEs and singularly perturbed ODEs. The first part presents a data-driven modeling method that accurately captures shocks and chaotic dynamics through a stabilized neural ODE framework. We learn the right-hand-side of an ODE by adding the outputs of two networks together, one learning a linear term and the other a nonlinear term. The architecture is inspired by the inertial manifold theorem. We apply this method to chaotic trajectories of the Kuramoto-Sivashinsky equation, where our model keeps long-term trajectories on the attractor and remains robust to noisy initial conditions. The second part explores structure-preserving ML for singularly perturbed dynamical systems. A powerful tool to address these systems is the Fenichel normal form, which significantly simplifies fast dynamics near slow manifolds. I will discuss a novel realization of this concept using ML. Specifically, a fast-slow neural network (FSNN) is proposed, enforcing the existence of a trainable, attractive invariant slow manifold as a hard constraint. To illustrate the power of FSNN, I will show a fusion-motivated example where traditional numerical integrators all fail.

We consider the mean field Bose gas on the unit torus at temperatures proportional to the critical temperature of the Bose—Einstein condensation phase transition. We discuss trace norm convergence of the Gibbs state to a state given by a convex combination of quasi-free states. Two consequences of this relation are precise asymptotic formulas for the two-point function and the distribution of the number of particles in the condensate. A crucial ingredient of the proof is a novel abstract correlation inequality. This is joint work with Nam Panh Tanh and Marcin Napiorkowski.