Department of

# Mathematics

Seminar Calendar
for events the day of Friday, April 28, 2017.

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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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Friday, April 28, 2017

1:00 pm in 347 Altgeld Hall,Friday, April 28, 2017

#### Estimates of invariant metrics and applications

###### Siqi Fu (Rutgers Univ. -Camden)

Abstract: We will discuss various estimates of invariant metrics (such as the Kobayashi and Bergman metrics) on a smooth bounded pseudoconvex domain, in particular, near a piece of the boundary whose Levi form has constant rank. We will also discuss how these estimates of the invariant metric can be used to characterize geometric properties on the boundary.

3:00 pm in 343 Altgeld Hall,Friday, April 28, 2017

#### On decomposition of the product of Demazure atom and Demazure characters

###### Anna Pun (Drexel University)

Abstract: t is an open problem to prove the Schubert positivity property combinatorially. Recently Haglund, Mason, Remmel, van Willigenburg et al. have studied the skyline fillings (a tableau-combinatorial object giving a combinatorial description to nonsymmetric MacDonald polynomials , proved by Haglund, Haiman and Loehr) specifically for the case of Demazure atoms (atoms) and key polynomials (keys). This suggests a new approach to a combinatorial proof of Schubert positivity property. In this talk, I will introduce Demazure atoms and key polynomials using skyline fillings called semi-standard augmented fillings (SSAFs) and define generalized Demazure atoms by some modifications on SSAF defining atoms and keys. I will illustrate the insertion algorithm on Demazure atoms proved by Mason and describe refinements of Littlewood-Richardson rule proved by Haglund, Mason and Willigenburg. Then I will describe an algorithm to prove the atom positivity property of the product of a monomial and a Demazure atom. The last result gives a positive support to the approach of the combinatorial proof of Schubert positivity property. If time allows, I will show some connection with polytopes and discuss some conjectures.

4:00 pm in 345 Altgeld Hall,Friday, April 28, 2017

#### On definability and interpretability in model theory

###### Ward Henson (UIUC, UC Berkeley)

Abstract: This will be an expository talk. Topics covered/mentioned will include: extension by definition and expansion/extension by interpretation (the eq-construction), Beth's Theorem (characterizing definability), and Makkai's Theorem (characterizing the eq-expansion). The setting will be model theory of classical (discrete) structures and then of metric (real-valued) structures. In the metric setting, definability of sets/relations has some subtleties that do not arise in discrete structures.