Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, October 19, 2017.

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    September 2017          October 2017          November 2017
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Thursday, October 19, 2017

2:00 pm in 243 Altgeld Hall,Thursday, October 19, 2017

#### The Schwarz range domain for harmonic mappings of the unit disc with boundary normalization

###### Józef Zajac (State School of Higher Education, Chelm, Poland)

Abstract: The family ${\mathcal F}$ of boundary normalized harmonic mappings of the unit disc into itself is considered by the author. An expression of the Schwarz type inequalities concerning the mappings in question will be presented. In particular, introduced by Partyka and the author, the Schwarz range domain $$\bigcup_{F\in {\mathcal F}} \{ F(z) \colon |z|\leq r\},$$ where $0 \leq r < 1$, is well described within the class ${\mathcal F}$. Previously obtained result says that $|F(0)| \leq \frac{2}{3}$ for $F \in {\mathcal F}$, whereas the present one describes precisely the Schwarz range domain of $F(0)$ for $F \in {\mathcal F}$. This research has been strongly motivated by certain problems of gas flow mechanics. Boundary normalization appears to be naturally applicable to aerodynamics when studying mechanics of the jet outlet stream described by harmonic mappings. Some conjectures coming out from this research, concerning jet engine construction will be also presented.

3:00 pm in 345 Altgeld Hall,Thursday, October 19, 2017

#### Hall algebras and the Fukaya category

###### Peter Samuleson (University of Edinburgh)

Abstract: The Hall algebra is an invariant of an abelian (or triangulated) category C whose multiplication comes from "counting extensions in C." Recently, Burban and Schiffmann defined the "elliptic Hall algebra" using coherent sheaves over an elliptic curve, and this algebra has found applications in knot theory, mathematical physics, combinatorics, and more. In this talk we discuss some background and then give a conjectural description of the Hall algebra of the Fukaya category of a topological surface. This is partially motivated by an isomorphism between the elliptic Hall algebra and the skein algebra of the torus, which we also discuss. (Joint works with H. Morton and with B. Cooper.)

4:00 pm in 245 Altgeld Hall,Thursday, October 19, 2017