Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, March 9, 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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30


Thursday, March 9, 2017

11:00 am in 241 Altgeld Hall,Thursday, March 9, 2017

#### New Mock Theta Function Identities

###### Frank Garvan (University of Florida)

Abstract: In his last letter to Hardy, Ramanujan defined ten mock theta functions of order 5 and three of order 7. He stated that the three mock theta functions of order 7 are not related. We give simple proofs of new Hecke double sum identities for two of the order 5 functions and all three of the order 7 functions. We find that the coefficients of Ramanujan's three mock theta functions of order 7 are surprisingly related.

12:00 pm in 243 Altgeld Hall,Thursday, March 9, 2017

#### Polynomial-time curve reduction

###### Mark Bell (Illinois Math)

Abstract: A pair of curves on a surface can appear extremely complicated and so it can be difficult to determine properties such as their intersection number. We will discuss a new argument that, when the curve is given by its intersections with the edges of an ideal triangulation, there is always a "reduction" to a simpler configuration in which such calculations are straightforward. This relies on finding an edge flip or a (power of a) Dehn twist that decreases the complexity of a curve by a definite fraction.

12:30 pm in 464 Loomis Laboratory,Thursday, March 9, 2017

#### Gauged Wess-Zumino actions, equivariant cohomology, and the electromagnetic response of symmetry-protected topological phases

###### Matthew Lapa (Illinois Physics)

Abstract: I will introduce the notion of a symmetry-protected topological (SPT) phase protected by the symmetry of a group G, and then present a calculation of the electromagnetic response of some bosonic SPT phases with G=U(1) in all dimensions. Remarkably, we find that the magnitude of the response of these bosonic SPT phases in spacetime dimensions 2m-1 or 2m differs from that of their more familiar fermionic counterparts by a numerical factor of m!, in agreement with previous results in low dimensions. The calculation uses a description of an SPT phase in terms of a nonlinear sigma model (NLSM) with theta term for the bulk and Wess-Zumino term for the boundary. The target space of the NLSM is a sphere of a particular dimension, and a crucial part of the NLSM description is an action of the group G=U(1) on the target space. I will show that the bulk response of the SPT phase can be deduced from the form of the gauged Wess-Zumino action describing the boundary coupled to the electromagnetic field. The construction of the gauged Wess-Zumino action is related to the U(1)-equivariant cohomology of the sphere, and I will explain this connection in detail. In particular, for even-dimensional spheres our result is equivalent to an equivariant extension of the volume form on the sphere with respect to the U(1) symmetry. On the other hand, for odd-dimensional spheres our result gives a physical interpretation for why such an extension fails. This talk is based on the paper arXiv:1611.03504 written together with Chao-Ming Jian, Peng Ye, and Taylor L. Hughes.

2:00 pm in 241 Altgeld Hall,Thursday, March 9, 2017

#### Playing with partitions and $q$-series

###### Frank Garvan (University of Florida)

Abstract: We start with some open partition problems of Andrews related to Gauss's three triangular numbers theorem. We alter a generating function and find a new Hecke double sum identity. Along the way we need Bailey's Lemma and Zeilberger's algorithm. We finish with some even staircase partitions.

3:00 pm in 347 Altgeld Hall,Thursday, March 9, 2017

#### A Combinatorial approach to Supersymmetry

###### Yan Zhang (San Jose State University)

Abstract: Adinkras are combinatorial tools created to study representations in supersymmetry. Besides having inherent interest for physicists, adinkras offer many easy-to-state and accessible open problems for mathematicians from many diverse subfields including Clifford algebras, posets, coding theory, and algebraic topology. I will discuss some results and problems, but mostly focusing on sharing some very pretty combinatorial objects with you.