Monday, March 22, 2021
03/22/2021 - 1:00pm
Abstract: A major challenge in the study of dynamic systems and boundary value problems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain a linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear systems and boundary value problems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.
email email@example.com for info.
Zoom Meeting ID: 97670014308
03/22/2021 - 4:00pm
Given a sequence of finite volume locally symmetric spaces $\Gamma\backslash X$ we can consider its Benjamini-Schramm limit. This is a probability measure on the space of pointed locally symmetric spaces that captures the geometry aroud a typical point. It is expected that for pairwise non isometric congruence arithmetic orbifolds the only possible B-S limit is the Dirac delta at the universal cover $X$. I proved it in the case of hyperbolic 2 or 3 manifolds. The results in this direction also allow for some control on the complexity of the homotopy type of arithmetic locally symmetric spaces. In my talk I will review the 2 and 3 dimensional hyperbolic cases and discuss recent progress in the case of general locally symmetric spaces (based on j.w. in progress with Sebastian Hurtado and Jean Raimbault)
03/22/2021 - 4:30pm
We explore some surprising symmetries for intersection cohomology of certain moduli of 1-dimensional sheaves and moduli of Higgs bundles, motivated by Gopakumar-Vafa theory concerning enumerative geometry for Calabi-Yau 3-folds. More precisely, we show that, for these moduli spaces, the intersection cohomology is independent of the choice of the Euler characteristic. This confirms a conjecture of Bousseau for P^2, and proves a conjecture of Toda in the case of local toric Calabi-Yau 3-folds. In the proof, a generalized version of Ngô's support theorem refining the decomposition theorem plays a crucial role. Based on joint work with Davesh Maulik.
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)