MenuEvents for 03/22/2024 from all calendars
Mathematical Physics and Harmonic Analysis Seminar
Time: 11:00AM - 11:50AM
Location: Bloc 302
Speaker: Stephen Shipman, LSU
Title: Localization of Defect States in the Continuum for Bilayer Graphene
Abstract: Algebraic reducibility of the Fermi surface for AB-stacked bilayer graphene provides a mechanism for creating spectrally embedded defect states. Generically, this algebraic structure is not enough to provide sufficient localization in practice. However, it turns out that AB-stacked graphene enjoys additional structure that allows extreme localization of the defect. Also, defect states in the continuum appear numerically to be much more robust to perturbations than expected.
Mathematical Physics and Harmonic Analysis Seminar
Time: 1:50PM - 2:50PM
Location: BLOC 302
Speaker: Daniel Boutros, University of Cambridge
Title: On energy conservation for inviscid hydrodynamic equations: analogues of Onsager's conjecture
Abstract: Onsager's conjecture states that 1/3 is the critical spatial (Hölder) regularity threshold for energy conservation by weak solutions of the incompressible Euler equations. We consider an analogue of Onsager's conjecture for the inviscid primitive equations of oceanic and atmospheric dynamics. The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a 'family' of Onsager conjectures for these equations.
Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. Finally, we present a regularity result for the pressure in the Euler equations, which is of relevance to the Onsager conjecture in the presence of physical boundaries. As an essential part of the proof, we introduce a new weaker notion of pressure boundary condition which we show to be necessary by means of an explicit example. These results are joint works with Claude Bardos, Simon Markfelder and Edriss S. Titi.
Nonlinear Partial Differential Equations
Time: 1:50PM - 2:50PM
Location: BLOC 302
Speaker: Daniel Boutros, University of Cambridge
Title: On energy conservation for inviscid hydrodynamic equations: analogues of Onsager's conjecture
Abstract: Onsager's conjecture states that 1/3 is the critical spatial (Hölder) regularity threshold for energy conservation by weak solutions of the incompressible Euler equations. We consider an analogue of Onsager's conjecture for the inviscid primitive equations of oceanic and atmospheric dynamics. The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a 'family' of Onsager conjectures for these equations. Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. Finally, we present a regularity result for the pressure in the Euler equations, which is of relevance to the Onsager conjecture in the presence of physical boundaries. As an essential part of the proof, we introduce a new weaker notion of pressure boundary condition which we show to be necessary by means of an explicit example. These results are joint works with Claude Bardos, Simon Markfelder and Edriss S. Titi.
Algebra and Combinatorics Seminar
Time: 3:00PM - 4:00PM
Location: BLOC 302
Speaker: Yifan Zhang, UT Austin
Title: Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications
Abstract: In this talk, I will prove a new upper bound on the covering number of real algebraic varieties, images of polynomial maps and semialgebraic sets. The bound remarkably improves the best known general bound by Yomdin and Comte (2004), and its proof is much more straightforward. As a consequence, our result gives new bounds on the volume of the tubular neighborhood of the image of a polynomial map and a semialgebraic set, where results for varieties by Lotz (2015) and Basu, Lerario (2022) are not directly applicable. I will first use this result to derive a near-optimal bound on the covering number of low rank CP tensors. Then I will discuss applications on sketching (general) polynomial optimization problems as well as controlling the generalization error for deep neural networks with rational or ReLU activations.
Free Probability and Operators
Time: 4:00PM - 5:00PM
Location: BLOC 304
Speaker: Ken Dykema, TAMU
Title: On operator-valued R-diagonal and Haar unitary elements (Joint work with John Griffin)
Abstract: R-diagonal elements are naturally defined by conditions on the free cumulants of the pair consisting of the element and its adjoint. In the tracial, scalar-valued context, it is known (due to pioneering work of Nica and Speicher) that being R-diagonal is equivalent to having the same *-distribution as an element with a polar decomposition z=u|z|, where u and |z| are *-free and where u is a Haar unitary. In the operator-valued context (namely, B-valued where B is an operator algebra), this is no longer the case. Freeness need not occur, and even notions of Haar unitary are more complicated in the operator-valued setting. We will (1) examine different notions of operator-valued Haar unitary (2) introduce the notion of a free bipolar decomposition and (3) discuss a specific result about free bipolar decompositions of B-valued circular elements (which are a very special case of B-valued R-diagonal elements) when B is two-dimensional.
Geometry Seminar
Time: 4:00PM - 5:00PM
Location: BLOC 302
Speaker: Paulina Hoyos Restrepo, UT Austin
Title: Manifold Learning in the Presence of Symmetries
Abstract: Graph Laplacian-based algorithms for data lying on a manifold have proven effective for tasks such as dimensionality reduction, clustering, and denoising. Consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group G. In this talk, I will show how to construct a G-invariant graph Laplacian (G-GL) by incorporating the distances between all the pairs of points generated by the action of G on the data set. The G-GL converges to the Laplace-Beltrami operator on the data manifold, with a significantly improved convergence rate compared to the standard graph Laplacian, which uses only the distances between the points in the given data set.
