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26 September 

Tomas Gedeon, Montana State University“Combinatorial structure of continuous dynamics in gene regulatory networks.” 
AbstractWe 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. 
17 October 

Bowen Li, Duke University“Variational embedding for quantum groundstate energy problems.” 
AbstractIn this talk, we consider the quantum many body problems and introduce a sumofsquares SDP hierarchy approximating the groundstate 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 byproduct of our investigation, we find that quantum entanglement has the potential to capture the underlying graph of the manybody Hamiltonian. 
*19 October (Thursday) at 2:30 – 3:30 pm in Hill 425Joint with the Hyperbolic & Dispersive PDE Seminar 

André Guerra, ETH Zürich“Homogenization in General Relativity.” 
AbstractGiven their highly nonlinear nature, Einstein’s vacuum equations are not closed under weak convergence and hence sequences of weaklyconvergent solutions may generate a nontrivial 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 Einsteinmassless 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). 
24 October 

Farhan Abedin, Lafayette College“Optimal Transport and TimeDependent MongeAmpere Equations.” 
AbstractI 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 MongeAmpere equation. 
7 November 

Shari Moskow, Drexel University“The Lippmann Schwinger Lanczos algorithm for inverse scattering problems.” 
AbstractWe combine datadriven reduced order models with the LippmannSchwinger 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 datadriven internal solution is produced. This internal solution is then used in the LippmannSchwinger 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 
14 November 

Miroslav Kramar, University of Oklahoma“Predicting Slip Events in Granular Media.” 
AbstractEarthquakes, snow avalanches, and landslides are wellknown phenomena that often lead to widescale destruction. We will concentrate on a smallscale system that exhibits stickslip 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 timedependent shear stress. In the stickslip 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. 
21 November 

Marian Gidea, Yeshiva University“Methods of Geometric Control in Hamiltonian Dynamics.” 
AbstractWe consider an integrable Hamiltonian system subject to a small, timeperiodic 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. 