Events for 11/17/2023 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Enrique Zuazua, Friedrich-Alexander-Universität Erlangen-Nürnberg

Title: Control and Machine Learning

Abstract: In this lecture we shall present some recent results on the interplay between control and Machine Learning, and more precisely, Supervised Learning, Universal Approximation and Normalizing flows. We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets). Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNet, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control. We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies in terms of the structure of the data set. This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. The turnpike property is also analyzed in this context, exploring the interplay between depth and width of the neural network. This lecture is inspired in joint work, among others, withDomènec Ruiz-Balet (Imperial College), Borjan Geshkovski (MIT), Martin Hernandez (FAU) and Antonio Lopez and Rafael Orive (UAM-Madrid).

Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Shixuan Zhang, TAMU (ISEN)

Title: Certification of sums of squares via low-rank optimization

Abstract: To certify a sum of k squares on a real projective variety, one can minimize the distance of the sum of squares of k linear forms from it in the space of quadrics. When k is smaller than the dimension of linear forms, the certification problem can be applied in low-rank semidefinite relaxation of polynomial optimization, similar to the Burer-Monteiro method. We discuss the existence of spurious local minima in this nonconvex certification problems, and show that in some interesting cases, there is no spurious local minima, or any spurious local minimum would lie on the boundary of the sum-of-square cone. These characterizations could potentially lead to efficient algorithms for polynomial and combinatorial optimization.

Geometry Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 302

Speaker: Keller VandeBogert, Notre Dame

Title: Stable Sheaf Cohomology on Flag Varieties

Abstract: The Borel-Weil-Bott (BWB) theorem is a fundamental result that gives a (relatively simple) method of computing the cohomology of line bundles on flag varieties over a field of characteristic 0. The analogue of BWB in positive characteristic is a wide-open problem despite many important results over the decades, and it remains out of reach even from a computational perspective. In this talk, I'll speak on joint work with Claudiu Raicu that shows that, despite the chaos, there is a notion of stability for the cohomology of line bundles on flags in arbitrary characteristic. Moreover, there are many cases where we can compute this stable sheaf cohomology explicitly, and these computations yield sharp, characteristic-free vanishing results for finite-length Koszul modules.

Free Probability and Operators

Time: 4:00PM - 5:00PM

Location: BLOC 306

Speaker: Ryo Toyota, TAMU

Title: Complete Haagerup inequality for Gromov hyperbolic groups

Abstract: In 1978, U Haagerup showed that if f is a function of the free group F_r which is supported on words with length exactly k, then the operator norm of the left regular representation |lambda(f)| is bounded by (k+1) times l^2-norm of f. Now this is called the Haagerup inequality, and its operator valued analogue was proved by Buchholz. In the operator valued case, the above (k+1)-l^2-norms is replaced by different (k+1)-operator norms associated to word decompositions. We will discuss how to generalize it for Gromov hyperbolic groups. This is a joint work with Zhiyuan Yang.