Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 13, 2020.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 13, 2020

9:00 am in Zoom,Tuesday, October 13, 2020

#### Gröbner geometry for skew-symmetric matrix Schubert varieties

###### Eric Marberg   [email] (The Hong Kong University of Science and Technology)

Abstract: Matrix Schubert varieties are the orbits of $B\times B$ acting on all n-by-n matrices, where $B$ is the group of invertible lower triangular matrices. Knutson and Miller identified a Gröbner basis for the prime ideals of these varieties. They also showed that the corresponding initial ideals are Stanley-Reisner ideals of shellable simplicial complexes. Skew-symmetric matrix Schubert varieties are the nonempty intersections of matrix Schubert varieties with the subspace of skew-symmetric matrices, and in this talk I will discuss some new extensions of Knutson and Miller's results to this setting. In particular, I will describe a natural Gröbner basis for the prime ideals of skew-symmetric matrix Schubert varieties, along with a primary decomposition for the corresponding initial ideals involving certain involution pipe dreams. As one application, this will lead to a geometric explanation for some recent formulas for involution Schubert polynomials. This is joint work with Zachary Hamaker and Brendan Pawlowski. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, October 13, 2020

#### Stochastically modeled reaction network

###### Jinsu Kim (University of California Irvine)

Abstract: A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system are small, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of the challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature. In this talk, I will provide a class of reaction networks whose associated stochastic process is stable. I will also provide how this stability can be used in system biology and control theory.

2:00 pm in Zoom,Tuesday, October 13, 2020

#### Asymptotic Structure for the Clique Density Theorem

###### Jaehoon Kim (KAIST)

Abstract: The famous Erdos-Rademacher problem asks for the smallest number of $r$-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all $r$ was determined only recently by Reiher. Here we describe the asymptotic structure of all almost extremal graphs. This task for $r=3$ was previously accomplished by Pikhurko and Razborov. This is joint work with Hong Liu, Oleg Pikhurko and Maryam Sharifzadeh.