Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, May 5, 2015.

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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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Tuesday, May 5, 2015

11:00 am in 243 Altgeld Hall,Tuesday, May 5, 2015

#### Calculations of Higher Topological Hochschild Homology

###### Ayelet Lindenstrauss   [email] (IU Bloomington)

Abstract: T. Pirashvili defined the higher Hochschild homology groups of a commutative ring (with coefficients in the ring itself or more generally a bimodule) as the homotopy groups of the Loday construction of that ring (and module) evaluated on spheres. Having strictly associative and commutative ring spectra allows us to do the same for spectra, obtaining higher topological Hochschild homology. I would like to discuss some basic calculations (joint with I. Bobkova, K. Poirier, B. Richter, and I. Zakharevich) of higher Hochschild homology, namely of Z/p[x] and Z/p[x]/x^a when p divides a. I would then like to explain how these lead to a calculation (joint with B. Dundas and B. Richter) of higher topological Hochschild homology of number rings with coefficients in their residue fields, and to a re-calculation of higher topological Hochschild homology of finite fields (done originally by other methods by M. Basterra and M. Mandell).

1:00 pm in 345 Altgeld Hall,Tuesday, May 5, 2015

#### Polish groups with ample generics

###### Maciej Malicki (Warsaw School of Economics and Caltech)

Abstract: A Polish group G has ample generics if every diagonal conjugation action of G on its finite power has a comeager orbit. In recent years, this notion has drawn attention of many researchers, mainly because of its very interesting and strong consequences such as the automatic continuity property or the small index property. In this talk, I will give a brief overview of results on Polish groups with ample generics, and discuss my own contributions to this subject. For example, I will characterize Polish ultrametric spaces whose isometry groups have a neighborhood basis at the identity consisting of open subgroups with ample generics. I will formulate a condition that implies that the automorphism group of a Polish metric structure shares all the main consequences of the existence of ample generics. I will also present an example of a Polish group with ample generics that fails to be non-archimedean.

2:00 pm in 347 Altgeld Hall,Tuesday, May 5, 2015

#### Diffusion Processes and Invariant Gibbs Measures

###### Samantha Xu (UIUC Math)

Abstract: In this talk, we discuss the connection between various diffusion processes and invariant Gibbs measures for Hamiltonian PDEs. We analyze various examples of this connection, and discuss some recent results.

3:00 pm in 243 Altgeld Hall,Tuesday, May 5, 2015

#### On the varying lengths of binary forms

###### Bruce Reznick (UIUC Math)

Abstract: If the coefficients of a binary form p(x,y) of degree d are in a field F and $F \subset K \subset C$, one can ask for the minimal r such that $p = \sum_{i=1}^r c_i (a_i x + b_i y)^d$, with $a_i, b_i, c_i \in K$. I call this the K-length of p. Using two theorems of Sylvester, it is easy to show that K-length can vary with K. A few years ago, I gave an example of a real quintic which has lengths 3, 4 and 5 over various fields. In this talk, I'll show that such examples exist for all degrees $\ge 5$, and they are not particularly exotic: for even degree, $(xy)^k$ and for odd degree, $(xy)^k(x-y)$.

3:00 pm in 241 Altgeld Hall,Tuesday, May 5, 2015

#### Spectral characterization of graphs with a given matching number k

###### Keivan Hassani Monfared   [email] (Western Illinois University Department of Mathematics)

Abstract: We provide a characterization of graphs with a given matching number k, in terms of the spectra they can realize. Here, we allow the entries of the matrix corresponding to the edges of the graph to be any nonzero real number. This is a joint work with Sudipta Mallik.