
3:00 pm
Prof. Xindong Tang  The Hong Kong Polytechnic University
A correlative sparse Lagrange multiplier expression relaxation for polynomial optimization
Math 278C: Optimization and Data Science
APM 5218
AbstractIn this paper, we consider polynomial optimization with correlative sparsity. We construct correlative sparse Lagrange multiplier expressions (CSLMEs) and propose CSLME reformulations for polynomial optimization problems using the KarushKuhnTucker optimality conditions. Correlative sparse sumofsquares (CSSOS) relaxations are applied to solve the CSLME reformulation. We show that the CSLME reformulation inherits the original correlative sparsity pattern, and the CSSOS relaxation provides sharper lower bounds when applied to the CSLME 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 CSLME approach requires less computational time than the original LME approach to reach the same accuracy level.

12:00 pm
Alexander Schlesinger
Automorphic Forms and RankinSelberg Integrals
Advancement to Candidacy
APM 7321

4:30 pm
Guoqi Yan  University of Notre Dame
$RO(C_{2^n})$graded homotopy of Eilenberg Maclane spectra
Math 292
APM 7321
AbstractThe foundation of equivariant stable homotopy theory is laid by LewisMaySteinberger 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 EilenbergMaclane 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 EilenbergMaclane spectra of particular interest will be discussed: the EilenbergMaclane 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.

4:00 pm
Jesse Kim
Web bases and noncrossing set partitions
Math 269 (Combinatorics Seminar)
APM 6402 (Halkin Room)
AbstractIn 1995, Kuperberg introduced a collection of web bases, which combinatorially encode $SL_2$ and $SL_3$ invariant tensors. By SchurWeyl 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 $(k,k,1^{n2k}) \vdash n$ 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 $(k,k,k,1^{n3k}) \vdash n$.

1:00 pm
Frederick Rajasekaran  UCSD
An Introduction to Random Matrix Theory and the Genus Expansion
Food for Thought
HSS 4025
AbstractWe'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.