In this paper, we consider polynomial optimization with correlative sparsity. We construct correlative sparse Lagrange multiplier expressions (CS-LMEs) and propose CS-LME reformulations for polynomial optimization problems using the Karush-Kuhn-Tucker optimality conditions. Correlative sparse sum-of-squares (CS-SOS) relaxations are applied to solve the CS-LME reformulation. We show that the CS-LME reformulation inherits the original correlative sparsity pattern, and the CS-SOS relaxation provides sharper lower bounds when applied to the CS-LME reformulation, compared with when it is applied to  the original problem. Moreover, the convergence of our approach is guaranteed under mild conditions. In numerical experiments, our new approach usually finds the global optimal value (up to a negligible error) with a low relaxation order, for cases where directly solving the problem fails to get an accurate approximation. Also, by properly exploiting the correlative sparsity, our CS-LME approach requires less computational time than the original LME approach to reach the same accuracy level.

The foundation of equivariant stable homotopy theory is laid by Lewis-May-Steinberger in the 80's, while people's understanding of the computational aspect of the subject is very limited even until today. The reason is that the equivariant homotopy groups are $RO(G)$-graded, and even the coefficient rings of Eilenberg-Maclane spectra involve complicated combinatorics of cell structures. In this talk I'll illustrate the advantages of Tate squares in doing $RO(G)$-graded computations. Several Eilenberg-Maclane spectra of particular interest will be discussed: the Eilenberg-Maclane spectra associated with the constant Mackey functors $\mathbb{Z}$, $\mathbb{F}_2$, and the Burnside ring. Time permitting, I'll also talk about some structures of the homotopy of $HM$, for $M$ a general $C_{2^n}$-Mackey functor.

In 1995, Kuperberg introduced a collection of web bases, which combinatorially encode $SL_2$ and $SL_3$ invariant tensors. By Schur-Weyl duality, these bases are also bases for the Specht modules corresponding to partitions $(k,k) \vdash 2k$ and $(k,k,k) \vdash 3k$ respectively, and have nicer symmetry properties than the standard polytabloid basis. In 2017, Rhoades introduced a basis for the Specht module corresponding to partition $(k,k,1^{n-2k}) \vdash n$ indexed by noncrossing set partitions and with similarly nice symmetry properties. In this talk, we will explore these bases and the connections between them, and discuss how these connections might be used to create a similar basis for the Specht module corresponding to $(k,k,k,1^{n-3k}) \vdash n$.

We'll give a quick introduction to the field of random matrix theory and give partial proof of the famed semicircle law for Wigner matrices. To do so, we will incorporate tools from combinatorics, probability, and topology to understand the genus expansion of a random matrix. No background in probability is required, though it may be helpful.