Events

Title: Quantum graphs, subfactors and tensor categories

Abstract: Graphs and their noncommutative analogues are interesting objects of study from the perspectives of operator algebras, quantum information and category theory. In this talk we will introduce  equivariant graphs with respect to a quantum symmetry along with examples such as classical graphs, Cayley graphs of finite groupoids, and their quantum analogues. We will also see these graphs can be constructed concretely by modeling a quantum vertex set by an inclusion of operator algebras and the quantum edge set by an equivariant endomorphism that is an idempotent with respect to convolution/Schur product. Equipped with this viewpoint and tools from subfactor theory, we will see how to obtain all these idempotents using higher relative commutants and the quantum Fourier transform. Finally, we will state a quantum version of Frucht's Theorem, showing that every quasitriangular finite quantum groupoid arises as certain automorphisms of some categorified graph.

Title:Smooth points on positroid varieties and planar N=4 supersymmetric Yang-Mills theory

 Abstract: Positroid varieties are subvarieties in the Grassmannian defined by cyclic rank conditions and which are related to Schubert varieties. We will provide a criterion for whether positroid varieties are smooth at certain distinguished points, and we will show that this information is sufficient to determine smoothness for the entire positroid variety. This will involve looking at combinatorial diagrams called "affine pipe dreams." We can also form a partial order on positroid varieties given by deletion and contraction, such that there is closure for smooth positroid varieties, and we will characterize the minimal singular elements in this order. Finally, we will discuss a couple of connections between the techniques of this work and planar N=4

SYM: the BCFW bridge decomposition and inverse soft factors.

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

Title: Intersection of Matroids

Abstract: We study simplicial complexes (hypergraphs closed under taking subsets) that are the intersection of a given number k of matroids. We prove bounds on their chromatic numbers (the minimum number of edges required to cover the ground set) and their list chromatic numbers. Settling a conjecture of Kiraly and Berczi--Schwarcz--Yamaguchi, we prove that the list chromatic number is at most k times the chromatic number. The tools used are in part topological. If time permits, I will also discuss a result proving that the list chromatic number of the intersection of two matroids is at most the sum of the chromatic numbers of each matroid, improving a result by Aharoni and Berger from 2006. The talk is based on works joint with Ron Aharoni, Eli Berger, and Daniel Kotlar. In this talk, there is no assumption about background knowledge of matroid theory or algebraic topology.

Title:Descents for Border Strip Tableaux

 Abstract: Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a d-th primitive root of unity yields the number of border strip tableaux with all strips of size d, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for rectangular partitions as cycle type. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new descent statistic for border strip tableaux, extending the classical definition for standard Young tableaux.

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

Title: Problem Decomposition in Optimization:  Algorithmic Advances Beyond ADMM

Abstract:

Decomposition schemes like those coming from ADMM typically start by posing a separable-type problem in the Fenchel duality format.  They then pass to an augmented Lagrangian, which however can interfere with the separability and cause a slow-down.  Progressive decoupling offers a more flexible approach which can utilize augmented Lagrangians while maintaining decomposability.  Based on a variable metric extension of the proximal point algorithm that's applied in a twisted sort of way, progressive decoupling benefits from stopping criteria which can guarantee convergence despite inexact minimization in each iteration.   The convergence is generically at a linear rate, and for convex problems, is global. But the method also works for nonconvex problems when initiated close enough to a point that satisfies a natural extension of the strong sufficient condition for local optimality known from nonlinear programming.

This talk is held as part of the 26^th Annual Midwest Optimization Meeting (“MOM26”).

Title:Vanishing of Schubert coefficients

 Abstract: Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, they are in #P. In this talk we discuss the closely related problem of the vanishing of Schubert coefficients. We prove that this vanishing problem is rather low in the polynomial hierarchy and discuss implications of this result.

This is joint work with Igor Pak.

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

Title: Proximal methods for nonsmooth and nonconvex fractional programs: when sparse optimization meets fractional programs

Abstract:Nonsmooth and nonconvex fractional programs are ubiquitous and also highly challenging. It includes the composite optimization problems studied extensively lately, and encompasses many important modern optimization problems arising from diverse areas such as the recent proposed scale invariant sparse signal reconstruction problem in signal processing, the robust Sharpe ratio optimization problems in finance and the sparse generalized eigenvalue problem in discrimination analysis.

In this talk, we will introduce extrapolated proximal methods for solving nonsmooth and nonconvex fractional programs and analyse their convergence behaviour. Interestingly, we will show that the proposed algorithm exhibits linear convergence for the scale invariant sparse signal reconstruction model,  and the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraints. This is achieved by identifying the explicit desingularization function of the Kurdyka-Ł ojasiewicz inequality for the merit function of the fractional optimization models. Finally, if time permits, we will present some preliminary encouraging numerical results for the proposed methods for sparse signal reconstruction and sparse Fisher discriminant analysis

The talk is based on joint work with R.I. Bo ̧t, M. Dao, T.K. Pong and P. Yu.

Title: Squares of eigenvalues and semi-definite optimization

Abstract: I will share some recent progress on two long-standing conjectures in spectral graph theory, namely the Elphick-Farber-Goldberg-Wocjan conjecture and the Bollob\'{a}s-Nikiforov conjecture. Both conjectures involve bounds on the sum of squares of the eigenvalues of a graph, and a key ingredient in our work is the interpretation of such sums as optimization problems involving semi-definite matrices. Part of the talk is joint work with Gabriel Coutinho and Thomás Jung Spier.

Title: Induced subgraphs of graphs of large $K_{r}$-free chromatic number

Abstract:For an integer $r\geq 2$, the $K_{r}$-free chromatic number of a graph $G$, denoted by $\chi_{r}(G)$, is the minimum size of a partition of the set of vertices of $G$ into parts each of which induces a $K_{r}$-free graph. In this setting, the $K_{2}$-free chromatic number is the usual chromatic number. Which are the unavoidable induced subgraphs of graphs of large $K_{r}$-free chromatic number? Generalizing the notion of $\chi$-boundedness, we say that a hereditary class of graphs is $\chi_{r}$-bounded if there exists a function which provides an upper bound for the $K_{r}$-free chromatic number of each graph of the class in terms of the graph's clique number. With an emphasis on a generalization of the Gy\'arf\'as-Sumner conjecture for $\chi_{r}$-bounded classes of graphs and on polynomial $\chi$-boundedness, I will discuss some recent developments on $\chi_{r}$-boundedness and related open problems. Based on joint work with Mathieu Rundstr\"om and Sophie Spirkl.

Title:Coloring the integers while avoiding monochromatic arithmetic

progressions

 Abstract: Consider coloring the positive integers either red or blue one at a time in order.  Van der Waerden's classical theorem states that no matter how you color the integers, you will eventually have k equally spaced integers all colored the same for any k.  But, how can we minimize the number of times k equally spaced integers are colored the same?  Even for k = 3, this question is unsolved.  We will discuss progress towards proving an existing conjecture by leveraging a connection to coloring the continuous interval [0,1]. Our strategy relies on identifying classes of colorings with permutations and then using mixed integer linear programming.  Joint work with Jonathan Kariv and Noah Williams.

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