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Abstract: We will discuss a class of ideals in a polynomial ring studied by Mark Haiman in his work on the Hilbert scheme of points and discuss how they are related to homology of affine Springer fibers, Khovanov-Rozansky homology of links, and to the ORS conjecture. We will do some explicit examples of computing KR-homology using combinatorial link recursions developed by Elias and Hogancamp.

An F-decomposition of a graph G is a set of subgraphs of G, each isomorphic to F, whose edge sets partition the edge set of G.  I will speak about a result showing that, for each odd k ≥ 5, any graph G of sufficiently large order n with minimum degree at least (1/2+1/(2k-4)+o(1))n has a C_k-decomposition if and only if k divides |E(G)| and all vertex degrees in G are even.  Our methodology also leads to results on F-decompositions for other 3-partite graphs F. This is joint work with Darryn Bryant, Daniel Horsley, Barbara Maenhaut and Richard Montgomery.

Abstract: The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are quasi-polynomially related.

In this talk, I will present a relaxed but still equivalent version of the conjecture based on a new matrix parameter, $\pm$-rank: the minimum number of all-1 rectangles needed to express the Boolean matrix as a $\pm 1$-sum. $\pm$-rank lies between rank and partition number, and our main result shows that it is in fact equivalent to rank up to a logarithmic factor.

This reframes the log-rank conjecture as: can every signed decomposition of a Boolean matrix be made positive with only quasi-polynomial blowup?

Additionally, I will show that the log-rank conjecture is equivalent to a conjecture on cross-intersecting set systems.

The talk is based on joint work with Shachar Lovett and Morgan Shirley.

Abstract: Given a matching M of a hypercube, does there exist a Hamiltonian cycle that contains every edge of M? This problem was posed by Ruskey and Savage in 1993, and remains open. If M is a perfect matching a Hamiltonian cycle does exist, as shown by Fink in 2007. In this talk, we will explore Fink's proof and examine some subsequent developments in the study of this problem.

Abstract:
A sequence of tournaments is said to be quasirandom if it behaves as a sequence of random tournaments would. In 1991, Chung, Graham, and Wilson provided a list of equivalent properties that any sequence of random tournaments satisfies with high probability. We say that a tournament H forces quasirandomness if every sequence that asymptotically has the expected number of copies of H is quasirandom. Nevertheless, it was shown that almost only transitive tournaments are quasirandom forcing. We then modify the definition of quasirandom forcing by considering only sequences of nearly regular tournaments and characterize all tournaments on at most five vertices that force quasirandomness in this new setting.

Abstract: Algebraic graph theory begins by associating a matrix to a graph and asking how the eigenvalues of the former relate to the structure of the latter. In the past fifteen or so years, horizons have broadened to include variants of graphs (such as signed graphs or directed graphs) which require variants of adjacency matrices such as signed or Hermitian adjacency matrices. I will focus on the topic of characterizing the graphs whose associated matrices have at most three distinct eigenvalues. I will discuss old results in the undirected and signed graph case and then present some new results for oriented graphs.

Based on joint work with S. Akbari, J. Aloni, B. Mohar and S. Xia.