Fall 2023

We  first describe the mathematical foundation of DSGRN (Dynamic Signatures Generated by Regulatory Networks), an approach that provides a combinatorial description of global dynamics of a network over its  entire parameter space. Finite description allows comparison of parameterized dynamics between hundreds of networks  to  discard networks that are not compatible with experimental data. We describe a  close connection of DSGRN  to Boolean network models that allows us to view DSGRN as a platform for bifurcation theory of Boolean maps. Finally, we describe current efforts to rigorously connect the combinatorial structures to structure of attractors of continuous ODE models of network dynamics. We discuss  several applications of this methodology  to systems biology.

In this talk, we consider the quantum many body problems and introduce a sum-of-squares SDP hierarchy approximating the ground-state energy from below with a natural quantum embedding interpretation. We establish the connections between our approach and other variational methods for lower bounds, including the RDM method in quantum chemistry and the Anderson bounds. Additionally, inspired by the quantum information theory, we propose efficient strategies for optimizing cluster selection to tighten SDP relaxations while staying within a computational budget. Numerical experiments are presented to demonstrate the effectiveness of our strategy. As a by-product of our investigation, we find that quantum entanglement has the potential to capture the underlying graph of the many-body Hamiltonian.

Given their highly nonlinear nature, Einstein’s vacuum equations are not closed under weak convergence and hence sequences of weakly-convergent solutions may generate a non-trivial energy momentum tensor in the limit. In 1989 Burnett conjectured that, for a sequence of vacuum solutions which oscillates with high frequency, this limit is characterized by the Einstein-massless Vlasov model: in particular, starting from vacuum, matter is generated through homogenization. In this talk we will present a proof of this conjecture, under appropriate gauge and symmetry assumptions. Based on joint work with Rita Teixeira da Costa (Princeton University).

I will present recent work with Jun Kitagawa (Michigan State University) on developing a dynamic approach for solving the optimal transport problem using a parabolic version of the Monge-Ampere equation.

We combine data-driven reduced order models with the Lippmann-Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, in a direct or iterative framework. The approach allows us to process more general transfer functions than the earlier versions of the ROM based inversion algorithms.  We give examples of its use for spectral domain MIMO problems and in the time domain given mono static data, targeting synthetic aperture radar. 

Authors: Vladimir Druskin, Shari Moskow, Mikhail Zaslavsky

Earthquakes, snow avalanches, and landslides are well-known phenomena that often lead to wide-scale destruction. We will concentrate on a small-scale system that exhibits stick-slip dynamics closely connected to the avalanches. In particular, we will consider a system of granular particles confined by two walls. The top wall of the system is pulled by a spring moving with constant velocity. This external forcing exposes the particles to time-dependent shear stress. In the stick-slip regime, the force network, describing the state of the system, evolves slowly and the top wall remains at rest until the spring force becomes sufficiently large. At this point, the wall slips, and an abrupt rearrangement of the force network occurs. In this talk, we will show that the upcoming slip events can be predicted by combining the methods of topological data analysis and Bayesian statistics. We will represent the time evolution of the force network as a time series in the space of persistence diagrams and find relevant descriptors in this space that are amenable to the Bayesian analysis.

We consider an integrable Hamiltonian system subject to a small, time-periodic perturbation. We assume that the perturbed system has a normally hyperbolic invariant manifold (NHIM) whose stable and unstable manifolds intersect transversally. Associated to each transverse intersection one can define a scattering map, which gives the future asymptotic of a homoclinic orbit as a function of its past asymptotic.  We assume that we have a system of such scattering maps. We provide results on the geometric controllability of the system. We show that, under explicit conditions on the scattering maps and on the inner dynamics (restricted to the NHIM),  for any two points on the NHIM, there is an orbit of the Hamiltonian flow that goes from near the first point to near the second point.  Also, for any path on the NHIM, there is an orbit of the Hamiltonian flow that shadows that path.  The upshot is that we use the perturbation as a controller.