Department of

Mathematics


Seminar Calendar
for events the week of Tuesday, August 11, 2020.

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Tuesday, August 11, 2020

2:00 pm in Zoom,Tuesday, August 11, 2020

New Upper Bounds on Generalized Ramsey Numbers

Emily Heath (University of Illinois, Urbana-Champaign)

Abstract: A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ in which each $p$-clique contains edges of at least $q$ distinct colors. We are interested in the function $f(n,p,q)$, first introduced by Erdos and Shelah, which is the minimum number of colors needed for a $(p,q)$-coloring of the complete graph $K_n$. The best-known general upper bound on $f(n,p,q)$ was given by Erdos and Gyarfas in 1997 using a probabilistic argument. Since then, improved bounds in the case where $p=q$ have been obtained only for $p = 4$ and $p = 5$. In this talk, I will introduce a general strategy for finding new constructive upper bounds and explain how to apply this technique to obtain improved bounds for $p=6$ and $p=8$.

This is joint work with Alex Cameron.

Please email Sean English at SEnglish (at) illinois (dot) edu for Zoom details.

Thursday, August 13, 2020

3:00 pm in Zoom,Thursday, August 13, 2020

Gröbner geometry of Schubert polynomials through ice

Anna Weigandt   [email] (University of Michigan)

Abstract: The geometric naturality of Schubert polynomials and the related combinatorics of pipe dreams was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of vexillary matrix Schubert varieties. We will discuss general diagonal degenerations, relating them to an older formula of Lascoux (2002) in terms of the 6-vertex ice model. Lascoux's formula was recently rediscovered by Lam, Lee, and Shimozono (2018), as "bumpless pipe dreams." We will explain this connection and discuss conjectures and progress towards understanding diagonal Gröbner degenerations of matrix Schubert varieties. This is joint work with Zachary Hamaker and Oliver Pechenik. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.