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Title:Quasisymmetric Harmonics in Superspace

Abstract:ÌýThe harmonics of quasisymmetric polynomials in superspace are theÌýÌýorthogonal 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: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.

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Title:ÌýTwo 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.Ìý We introduce a novel graph product by which we construct new infinite families of graphs that achieve q(G)=2.Ìý 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.

ÌýJoint work with Maria Chudnovsky and Sophie Spirkl.

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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:Ìý2-Y-homogeneousÌýdistance-biregularÌýgraphs

Abstract:ÌýDistance-biregular graphs (DBRGs) generalize distance-regular graphs by admitting a bipartition of the vertex set, where each part satisfies local distance-regularity Ìýunder distinct intersection arrays. In recent years, a particular subclass of these graphs, those satisfying the so-calledÌý2-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 characterizeÌý2-Y-homogeneous DBRGs. We will survey recent progress in their classification under various combinatorial constraints, highlighting both known results and open problems.