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for events the day of Friday, October 27, 2017.

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Friday, October 27, 2017

12:00 pm in 243 Altgeld Hall,Friday, October 27, 2017

Universal Taylor series

Maria Siskaki

Abstract: A series of the form $Σa_n z^n, z \in \mathbb{C}$, with radius of convergence $R=1$, is called a Universal Taylor series if for each compact set $K$ which is disjoint from the unit disc and has connected complement, and each function $h$ in $A(K)$, there exists a subsequence of the sequence of partial sums of the series that approximates $h$ uniformly on $K$. In other words, the series diverges so badly outside the unit disc, that it approximates any reasonably good function defined on any reasonably good set. In this talk I will prove the existence of Universal Taylor series and discuss some of their properties.

3:00 pm in 341 Altgeld Hall,Friday, October 27, 2017

Affine Growth Diagrams

Tair Akhmejanov (Cornell University)

Abstract: We introduce a new type of growth diagram, arising from the geometry of the affine Grassmannian for $GL_m$. These affine growth diagrams are in bijection with the $c_{\vec\lambda}$ many components of the polygon space Poly($\vec\lambda$) for $\vec\lambda$ a sequence of minuscule weights and $c_{\vec\lambda}$ the Littlewood--Richardson coefficient. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of $GL_m$. Letting $m$ go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the $n$-hive of Knutson--Tao--Woodward. The local growth rule satisfied by the diagrams previously appeared in van Leeuwen's work on Littelmann paths, so our results can be viewed as a geometric interpretation of this combinatorial rule.

4:00 pm in 345 Altgeld Hall,Friday, October 27, 2017

On "Universal-homogeneous structures are generic" by Kabluchko and Tent (Part 1)

Erik Walsberg (Illinois Math)

Abstract: I will discuss the recent paper "Universal-homogeneous structures are generic" by Zakhar Kabluchko and Katrin Tent [arXiv]. They show that the set of structures isomorphic to a Fraisse limit is comeager in a certain space of structures. It is rather surprising that this result, which provides a natural connected between model theory and descriptive set theory, was only proven recently.

4:00 pm in 241 Altgeld Hall,Friday, October 27, 2017

Topology from critical points

Nachiketa Adhikari (UIUC)

Abstract: "Every mathematician has a secret weapon. Mine is Morse theory." - Raoul Bott Morse theory is a tool that allows one to study the topology of a manifold by looking at special functions on it. In this introductory talk, we'll first look at some of the fundamental ideas relating critical points of these functions and the homotopy type of the manifold. We will then try to understand how the gradient flows of such functions can yield topological invariants for it. No knowledge beyond the words in this abstract will be assumed.