Events

Title: Asymptotic count of edge-bicolored graphs

 Abstract: I will talk about recent joint work with Chiara Meroni and Max Wiesmann, where we showed that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex structures.

The asymptotic behavior is governed by the critical points of a polynomial. An interesting application of this purely combinatorial work to mathematical physics is the Ising model on a random graph. I will explain how its phase transitions arise from our formula.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title:The Inexact Augmented Lagrangian Method: Optimal Complexity Bounds and Applications to Solving Huge SDPs

Abstract:In the first part of this talk, we present optimal and nearly-optimal parameter-free augmented Lagrangian (AL) methods for convex and strongly optimization with linear constraints. Our AL methods employ tractable inexact criteria for solving their inner subproblems, which accelerated methods can be shown to achieve in a finite number of iterations that depends on the target accuracy.

In the second part of this talk, we present a new inexact augmented Lagrangian method, namely, HALLaR, that employs a Burer-Monteiro style low-rank factorization for solving large-scale semidefinite programs (SDPs). The AL subproblems are solved by a hybrid low-rank method, which is based on a combination of a low-rank Frank-Wolfe method and a nonconvex accelerated inexact proximal point method. In contrast to the classical low-rank method by Burer and Monteiro, HALLaR finds a near-optimal solution (with provable complexity bounds) of SDP instances satisfying strong duality. Computational results comparing HALLaR to state-of-the-art solvers on several large SDP instances show that the former finds higher accurate solutions in substantially less CPU time than the latter ones. For example, in less than 20 minutes, HALLaR can solve (on a personal laptop) a maximum stable set SDP with 1 million vertices and 10 million edges within 1e-5 relative accuracy.

This talk is based on joint work with Saeed Ghadimi and Henry Wolkowicz from University of Waterloo and Diego Cifuentes and Renato Monteiro from Georgia Tech.

Title: Finite bivariate Tratnik functions

Abstract: In the context of algebraic combinatorics, P- and Q-polynomial association schemes are important objects and are closely related to distance-regular graphs. The polynomials appearing in these structures are classified by Leonard's theorem, and they belong to the discrete part of the (q-)Askey scheme. Relatively recently, the notions of P- and Q-polynomial association schemes as well as of distance-regular graphs have been generalized to the multivariate case. There is however no multivariate analog of Leonard's theorem. With the purpose of progressing in that direction, I will discuss ongoing work concerning certain finite families of bivariate functions, said of Tratnik type, which are expressed as an intricate product of univariate polynomials of the (q-)Askey scheme. The goal is to classify such functions which satisfy some generalized bispectral properties, that is, two recurrence relations and two (q-)difference equations of certain types.

Title: The quasisymmetric Macdonald polynomials are quasi-Schur positive at t = 0

 Abstract: The quasisymmetric Macdonald polynomials G_\gamma (X; q, t) are a quasisymmetric refinement of the symmetric Macdonald polynomials that specialize to the quasisymmetric Schur functions QS_\alpha (X). We study the t = 0 specialization G_\gamma (X; q,0), which can be described as a sum over weighted multiline queues. We show that G_\gamma (X; q, 0) expands positively in the quasisymmetric Schur basis and give a charge formula for the quasisymmetric Kostka-Foulkes polynomials K_{\gamma,\alpha}(q) in the expansion G_\gamma (X; q, 0) = \sum K_{\gamma,\alpha}(q) QS_\alpha(X). The proof relies heavily on crystal operators, and if you do not know what that means, come find out! This is joint work with Olya Mandelshtam.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,