Department of

Mathematics


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for events the day of Friday, November 4, 2016.

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Friday, November 4, 2016

3:00 pm in 241 Altgeld Hall,Friday, November 4, 2016

KN 1-parameter subgroups for representations of quivers

Itziar Ochoa (UIUC Math)

Abstract: Given a projective variety X with an action of a complex reductive group G, the quotient space $X/ G$ may not exist in the category of algebraic varieties. In order to fix this problem, Geometric Invariant Theory gives a construction of a $G$-invariant open subset $X^{ss}$ of X for which the algebraic quotient exists. The Kirwan-Ness (KN) stratification refines $X$ and its unique open stratum coincides with the set $X^{ss}$. When $X$ is a linear representation $G$, the semistable locus $X^{ss}\subset X$ and a KN stratification of $X \backslash X^{ss}$ are associated to a choice of a homomorphism $G\rightarrow \mathbb{G}_m$ . That is, we can write $X=X^{ss}\sqcup \bigsqcup_{\alpha\in \text{KN}}S_\alpha,$ where $S_{\alpha}$ are locally closed smooth pieces and KN indexes the 1-parameter subgroups that determine the stratification. In this situation, Nevins and McGerty give an algorithm to find the KN 1-parameter subgroups. I will ilustrate the algorithm with some examples and at the end I will focus on the case when $X$ is a representation of the cyclic quiver.

4:00 pm in 241 Altgeld Hall,Friday, November 4, 2016

Continued fractions, modular surfaces, and billiard maps

Claire Merriman (UIUC Math)

Abstract: Continued fractions have been studied in various forms, with Gauss introducing the Gauss or continued fraction map, and itís corresponding invariant measure. In 1985, Series introduced a geometric coding for the regular continued fraction expansion, by looking at geodesics on the modular surface. This coding gave the invariant measure for the natural extension of the Gauss map. In 1997, Bauer and Lopes introduced a geometric coding for the even continued fraction expansion, which uses the billiard map corresponding to the Theta group. I will talk about the geometric constructions in both of these papers.