Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, October 3, 2013.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2013          October 2013          November 2013
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1  2  3  4  5  6  7          1  2  3  4  5                   1  2
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Thursday, October 3, 2013

8:00 am in Second Floor, Levis Faculty Center,Thursday, October 3, 2013

#### Second International Meeting of the Association for the Philosophy of Mathematical Practice

Abstract: The Second International Meeting of the APMP will be held October 3-4, 2013. All talks will take place on the second floor of the Levis Faculty Center. See the conference website http://institucional.us.es/apmp/index_APMP2013.htm for more information.

11:00 am in 241 Altgeld Hall,Thursday, October 3, 2013

#### Colored Partition Identities Arising from Ramanujan's Formulas for Multipliers

###### Bruce Berndt (UIUC Math)

Abstract: Beginning with work of Farkas and Kra, a large number of partition identities have been established from theta function identities and modular equations in the past dozen years. In this lecture we examine a particular type of modular equation, namely, formulas for multipliers, and derive several interesting identities for colored partitions. This is joint work with Rui Zhou, who was a visiting graduate student from Dalian University during the past year.

12:30 pm in 243 Altgeld Hall,Thursday, October 3, 2013

#### Ribbon categories and anyons

###### Abishek Roy (Illinois Physics)

Abstract: An expository talk on ribbon categories and their application to physics. Using the graphical method of Reshetikhin-Turaev diagrams, we will prove modular invariance. I shall try to include several models of physical anyons - abelian, Ising, Fibonacci and (if time permits) Kitaev's quantum double model.

1:00 pm in 241 Altgeld Hall ,Thursday, October 3, 2013

#### Quasiconformal Mappings on Planar Surfaces

###### Colleen Ackermann (UIUC Math)

Abstract: The geometric definition of quasiconformality is a condition involving all quadrilaterals (Jordan domains with four boundary points identified). Past research has shown that one need only look at rectangles to determine quasiconformality. We will investigate whether it suffices to just consider squares. Next we will look at a method of studying quasiconformal mappings on the Grushin plane. I will give a quasisymmetry from the Grushin plane to the complex plane and use it to develop an analytic definition of quasisymmetry on the Grushin plane. We will also discuss several characterizations of comformal mappings on the Grushin plane.

1:00 pm in Altgeld Hall 347,Thursday, October 3, 2013

#### Factors of IID on Trees

###### Russel Lyons (Indiana University)

Abstract: Classical ergodic theory for integer group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We discuss a few known results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. No background will be assumed.

2:00 pm in 149 Henry Administration Building,Thursday, October 3, 2013

#### Revisiting Lehmers' Picturesque Exponential Sums (joint with Dan Schultz)

###### Michael DiPasquale (UIUC Math)

Abstract: Let $\zeta$ be a $k$th root of unity and $b_k(i)$ be the sum of the digits of $i$ when $i$ is written in base $k$. We consider a graphical representation $G(n,k)$ of the partial sums $S(n,k)=\sum_{i=0}^n \zeta^{b_k(i)}x^i$, where $x$ is a complex root of unity. Graphical representations of sums of this type were considered by D.H. and Emma Lehmer for the case $x=\zeta$. If $x$ is replaced by an arbitrary $m$th root of unity, the resulting graph exhibits fractal-like properties which can be explained by a formula for the partial sums $S(n,k)$ which extends a previously known formula discovered by Apostol. There will be pictures.

3:00 pm in 243 Altgeld Hall,Thursday, October 3, 2013

#### Local cohomology modules over polynomial rings of prime characteristic - Part IV

###### Yi Zhang (UIUC)

Abstract: Let $R=k[x_1,\cdots, x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0.$ If $I$ is an ideal of $R,$ we denote $H^i_I(R)$ the $i$-th local cohomology module of $R$ with support in $I.$ We discuss an adjointness theorem of Frobenius map. Then we use this property to study the dimension of the associated primes of $H^i_I(R),$ the grading on

4:00 pm in 245 Altgeld Hall,Thursday, October 3, 2013

#### Random orderings and unique ergodicity of automorphism groups

###### Russell Lyons (Indiana University)

Abstract: Is there a natural way to put a random total ordering on the vertices of a finite graph? Natural here means that all finite graphs get an isomorphism-invariant random ordering and induced subgraphs get the random ordering that is inherited from the larger graph. Thus, the uniformly random ordering is natural; are there any others? What if we restrict to certain kinds of graphs? What about finite hypergraphs or finite metric spaces? We discuss these questions and sketch how their answers give unique ergodicity of corresponding automorphism groups; for example, in the case of graphs, the group is the automorphism group of the infinite random graph. This is joint work with Omer Angel and Alexander Kechris.