Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, September 17, 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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Thursday, September 17, 2015

11:00 am in 241 Altgeld Hall,Thursday, September 17, 2015

#### The order of large random permutations with cycle weights

###### Dirk Zeindler   [email] (Lancaster University)

Abstract: The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdős–Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We show that the Erdős–Turán Law can be extended to random permutations chosen according to the so-called generalized Ewens measure and to a generalized weighted measure with polynomially growing cycle weights. Furthermore, we establish for the generalized Ewens measure a local limit theorem as well as, under some extra moment condition, a precise large deviation estimate and also show that the expectation of the logarithm of the order has a remarkable connection with the Riemann hypothesis. In addition, we provide a precise large deviation estimate for random permutations with polynomial growing cycle weights.

12:30 pm in 241 Altgeld Hall,Thursday, September 17, 2015

#### BPS States, Chiral Algebras, and Quivers

###### Clay Cordova (IAS)

Abstract: I will describe a link between BPS states in four-dimensional quantum field theory, and a specific limit of the superconformal index which produces characters of well known chiral algebras.

1:00 pm in 243 Altgeld Hall,Thursday, September 17, 2015

#### Null distance on a spacetime

###### Carlos Vega (Saint Louis University)

Abstract: Given any time function on a spacetime M, we define an induced null distance' function, built from and closely related to the causal structure of M. This null distance' is a conformal pseudometric in general, but is positive-definite under natural conditions. Further, in basic model cases, the causal structure is encoded completely in the resulting metric space. In the cosmological setting, a canonical choice of time function was introduced and studied by Andersson, Galloway, and Howard. We show that under their basic niceness' condition, the induced null distance is definite, and hence provides a uniform way of encoding big bang' and related spacetimes as metric spaces. This is joint work with Christina Sormani.

2:00 pm in 347 Altgeld Hall,Thursday, September 17, 2015

#### Field theories, cohomology theories, and power operations

###### Dan Berwick-Evans   [email] (UIUC Math)

Abstract: I'll give an introduction to some connections between Atiyah-Segal functorial field theories and cohomology theories. A focus will be on a geometric construction of power operations for field theories that recovers previously known power operations in certain cohomology theories. No prior knowledge of field theories or power operations will be assumed.

4:00 pm in 245 Altgeld Hall,Thursday, September 17, 2015

#### Dynamical Systems and the Two-Dimensional Navier-Stokes Equations

###### Gene Wayne (Boston University)

Abstract: Two-dimensional fluid flows exhibit a variety of coherent structures, such as vortices and dipoles, which often serve as organizing centers for the subsequent evolution of the flow. These coherent structures can sometimes be associated with the existence of special geometrical structures in the phase space of the equations and in these cases the evolution of these flows can be studied with the aid of dynamical systems theory.