Department of

# Mathematics

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

11:00 am in 241 Altgeld Hall,Thursday, October 19, 2017

#### A survey of tauberian theorems

###### Harold Diamond (UIUC Math)

Abstract: A light look at inversion theorems for Laplace transforms under various hypotheses.

12:00 pm in 243 Altgeld Hall,Thursday, October 19, 2017

#### The Topology of Representation Varieties

###### Maxime Bergeron (University of Chicago)

Abstract: Let H be a finitely generated group, let G be a complex reductive algebraic group (e.g. a special linear group) and let K be a maximal compact subgroup of G (e.g. a special unitary group). I will discuss exceptional classes of groups H for which there is a deformation retraction of Hom(H,G) onto Hom(H,K), thereby allowing us to obtain otherwise inaccessible topological invariants of these representation spaces.

12:30 pm in 222 Loomis,Thursday, October 19, 2017

#### Soft Black Hole Absorption Rates from Large Gauge Symmetry

###### Steven Avery (Michigan State)

Abstract: Recently, a number of exciting connections have been made between large gauge transformations (eg. BMS) and infrared physics (eg. Weinberg's soft graviton theorem). One of the more exciting explorations in this vein was Hawking-Perry-Strominger's (HPS) investigation of the consequences of these new symmetries for black hole physics. I will show very concretely that the Ward identity for the BMS-like large U(1) gauge transformations discussed by HPS fixes the low energy black hole absorption rate for photons. Time permitting, I will discuss broader implications and future extensions.​

2:00 pm in 241 Altgeld Hall,Thursday, October 19, 2017

#### Some classical applications of modular forms in number theory

###### Yifan Yang   [email] (National Chiao Tung University)

Abstract: In this talk, we will give a quick overview of some classical applications of modular forms in number theory, including 1. formulas for the number of representations of an integer as sums of squares, 2. a formula for arithmetic-geometric means, 3. modular forms as solutions of linear ordinary differential equations, 4. modular forms as periods, 5. irrationality of $\zeta(3)$, 6. series representations for $1/\pi$, 7. congruences of the partition function.

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 243 Altgeld Hall,Thursday, October 19, 2017

#### Koszul almost complete intersections

###### Matthew Mastroeni (UIUC Math)

Abstract: Let $R = S/I$ be a quotient of a standard graded polynomial ring $S$ by an ideal $I$ generated by quadrics. If $R$ is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R$ over $S$ can be bounded above by binomial coefficients on the minimal number of generators of $I$. Motivated by previous results for Koszul algebras defined by three quadrics, we give a complete classification of the structure of Koszul almost complete intersections and, in the process, give an affirmative answer to the above question for all such rings.

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

#### Discrete Geometry of polygons and Soliton Equations

###### Gloria Mari Beffa (University of Wisconsin-Madison)

Abstract: The relation between the discrete geometry of surfaces and completely integrable systems has been well stablished in the last few decades, through work of Bobenko, Suris and many others. The recent introduction of discrete moving frames by Mansfield, Mari-Beffa and Wang, and the study of the pentagram map by Richard Schwartz and many others, has produced a flurry of work connecting the discrete geometry of polygons to some completely integrable systems in any dimension, including connections to Combinatorics and the study of the role that the background geometry has in the generation of algebraic structures that often describe integrability. In this talk I will review definitions and background, and will describe recent advances in the proof of the integrability of discretizations of Adler-Gel’fand-Dikii systems (generalized KdV), aided by the use of the geometry of polygons in RPm.

7:00 pm in Live at https://www.storycollider.org,Thursday, October 19, 2017

#### The Story Collider

###### Illinois scientists including Alyssa Loving

Abstract: Be sure to tune in to “The Story Collider” on October 19, 2017, at 7 pm when our graduate student Alyssa Loving will be one of five Illinois scientists (graduates students, faculty, or research scientists), to share their stories. Visit https://www.storycollider.org/