Future students

Title:Quasisymmetric Harmonics in Superspace

Abstract: The harmonics of quasisymmetric polynomials in superspace are the  orthogonal complement of the ideal generated by quasisymmetric polynomials without constant term. In this talk, I will discuss the harmonics and present the first basis of this space, which is indexed by a specific family of nested forests.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,

Title:The omega-Condition Number: Applications to Preconditioning and Low Rank Generalized Jacobian Updating

Abstract: Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g.,~in quasi-Newton methods. We study a nonclassic matrix condition number, the omega-condition number}, omega for short. omega is the ratio of: the arithmetic and geometric means of the singular values, rather than the largest and smallest for the classical kappa-condition number. The simple functions in omega allow one to exploit  first order optimality conditions. We use this fact to derive explicit formulae for (i) omega-optimal low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) omega-optimal preconditioners of special structure for  iterative methods for linear systems. In the latter context, we analyze the benefits of omega for (a) improving the clustering of eigenvalues; (b) reducing the number of iterations; and (c) estimating the actual condition of a linear system. Moreover we show strong theoretical connections between the omega-optimal preconditioners and incomplete Cholesky factorizations, and highlight the misleading effects arising from the inverse invariance of kappa. Our results confirm the efficacy of using the omega-condition number compared to the kappa-condition number.

(Joint work with: Woosuk L. Jung, David Torregrosa-Belen.)

Title: Fault Tolerant Routing and High Dimensional Expanders

Abstract: We will go over some recent results about fault tolerant routing from Bafna, Minzer, Vyas. Two parallel series of works from the last few years from Bafna, Lifshitz, Minzer, Vyas as well as Dikstein, Dinur, Lubotzky has led to size efficient PCPs by using high dimensional expanders. We will comment on the context for these works, and briefly go over some high level ideas. Then, we will talk about an application to fault tolerant rounding. The goal in this problem is to design a sparse network supporting efficient fault tolerant interactions between all pairs of nodes. Using the size efficient PCP construction, Bafna and Minzer gave a construction of constant degree networks with efficient protocols that tolerate a constant fraction of adversarial edge faults.

Title:Oriented graded Möbius algebras

Abstract:The graded Möbius algebra B(M) of a matroid M contains much combinatorial information about the flats of M. Its algebraic properties were instrumental in the proof of the Dowling—Wilson top-heavy conjecture. We will introduce a skew-commutative analogue OB(M) associated to every oriented matroid M, and discuss its algebraic structure. This is part of ongoing work joint with Yu Li.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,

Title:Worst-case instances of the stable set problem of graphs for the Lovász–Schrijver SDP hierarchy

Abstract:(Based on joint work with Levent Tunçel.)

In this talk, we discuss semidefinite relaxations of the stable set problem of graphs generated by the lift-and-project operator LS_+ (due to Lovász and Schrijver), and present some of our recent progress on this front. In particular, we show that for every positive integer k, the smallest graph with LS_+-rank k contains exactly 3k vertices. This result is sharp and settles a conjecture posed by Lipták and Tunçel from 2003.

The talk will be accessible to a general audience, and does not assume any prior knowledge of lift-and-project methods.

Title: Almost Tight Additive Guarantees for k-Edge-Connectivity

Abstract: We consider the k-edge-connected spanning subgraph (k-ECSS) problem, where we are given an undirected graph G = (V, E) with nonnegative edge costs, and the goal is to find a minimum-cost subgraph H of G that is k-edge-connected; that is, there exist at least k edge-disjoint paths between every pair of vertices in H.

For even k, we present a polynomial-time algorithm that computes a (k−2)-edge-connected subgraph whose cost is at most that of the natural LP relaxation of k-ECSS. I will try to present an overview of our algorithm and analysis, which is based on the iterative rounding technique.

Title: Two Distinct Eigenvalues from a New Graph Product

Abstract:The parameter q(G) of a graph G is the minimum number of distinct eigenvalues of a symmetric matrix whose pattern is given by G.  We introduce a novel graph product by which we construct new infinite families of graphs that achieve q(G)=2.  Several graph families for which it is already known that q(G)=2 can also be thought of as arising from this new product.

Title:Complete bipartite induced minors (and treewidth)

Abstract:I will present a result that describes the unavoidable induced subgraphs of graphs with a large complete bipartite induced minor, and will discuss the connections and applications to bounding the treewidth in hereditary classes of graphs. If time permits, I will also sketch some proofs.

 Joint work with Maria Chudnovsky and Sophie Spirkl.

Title:The Abelian/non-Abelian correspondence and Littlewood-Richardson

Abstract:The Abelian/non-Abelian correspondence gives rise to a natural basis for the cohomology of flag varieties, which - except for Grassmannians - is distinct from the Schubert basis. I will describe this basis and its multiplication rules, and explain how to relate it to the Schubert basis for two-step flag varieties. I will then explain how this leads to new tableaux Littlewood--Richardson rules for many products of Schubert classes. This is joint work (separately) with Wei Gu and Linda Chen.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,

Title: 2-Y-homogeneous distance-biregular graphs

Abstract: Distance-biregular graphs (DBRGs) generalize distance-regular graphs by admitting a bipartition of the vertex set, where each part satisfies local distance-regularity  under distinct intersection arrays. In recent years, a particular subclass of these graphs, those satisfying the so-called 2-Y-homogeneous condition, has garnered increasing attention due to its rich connections with combinatorial design theory and the representation theory of Terwilliger algebras. In this talk, we will examine the key structural conditions that characterize 2-Y-homogeneous DBRGs. We will survey recent progress in their classification under various combinatorial constraints, highlighting both known results and open problems.