Department of

Mathematics


Seminar Calendar
for events the day of Thursday, March 12, 2015.

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Thursday, March 12, 2015

11:00 am in 241 Altgeld Hall,Thursday, March 12, 2015

Exact formulas for the smallest parts function

Scott Ahlgren (UIUC Math)

Abstract: Building on ground-breaking work of Hardy and Ramanujan, Rademacher proved an exact formula for the values of the ordinary partition function. More recently, Bruinier and Ono obtained an algebraic formula for these values. We study the smallest parts function. Introduced by Andrews, its generating function is a prototypical example of a mock modular form of weight 3/2. Using automorphic methods, we obtain an exact formula and an algebraic formula for its values. The convergence of the exact formula is not obvious, and requires power savings estimates for sums of Kloosterman sums attached to a multiplier. These are proved with spectral methods following an argument of Goldfeld-Sarnak. (Joint work with Nick Andersen)

1:00 pm in 347 Altgeld Hall,Thursday, March 12, 2015

An first exit time problem inspired by modeling of gene expression

Jay Newby (Ohio State/MBI)

Abstract: A general class of stochastic gene expression models with self regulation is considered. One or more genes randomly switch between regulatory states, each having a different mRNA transcription rate. The gene or genes are self regulating when the proteins they produce affect the rate of switching between regulatory states. Under weak noise conditions, the deterministic forces are much stronger than fluctuations from gene switching and protein synthesis. Metastable transitions, such as bistable switching, can occur under weak noise conditions, causing dramatic shifts in the expression of a gene. A general tool used to describe metastability is the quasi stationary analysis (QSA). A large deviation principle is derived so that the QSA can explicitly account for random gene switching without using an adiabatic limit or diffusion approximation, which are unreliable and inaccurate for metastable events.This allows the existing asymptotic and numerical methods that have been developed for continuous Markov processes to be used to analyze the full model.

1:00 pm in Altgeld Hall 243,Thursday, March 12, 2015

Word Maps and Measure Preservation

Doron Puder (IAS Princeton)

Abstract: We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. More specifically, for every finite group G, a word w in the free group on k generators induces a word map from G^k to G. We say that w is measure preserving with respect to G if given uniform distribution on G^k, the image of this word map distributes uniformly on G. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. In a joint work with O. Parzanchevski, we prove this conjecture.

2:00 pm in 140 Henry Administration Bldg,Thursday, March 12, 2015

Modular Equations in Two Variables

Dan Schultz (Penn State Math)

Abstract: By adding certain equianharmonic elliptic sigma functions to the coefficients of the Borwein cubic theta functions, a set of two-variable theta functions may be derived. These theta functions invert the F1(13;13;13;1|x,y) case of Appell's hypergeometric function and satisfy several identities akin to those satisfied by the Borwein cubic theta functions. I will discuss how these identities arise, several results concerning modular equations satisfied by these functions, and their applicability to a new two-parameter family of solvable nonic equations.

4:00 pm in 245 Altgeld Hall,Thursday, March 12, 2015

Semirandom methods in combinatorics

Jacques Verstraete (Univ of California San Diego)

Abstract: The development of the probabilistic method in combinatorics since it's inception by papers of P. Erdos has led to groundbreaking results across a broad mathematical landscape. In this talk, I will survey a technique which has come to be known as the semirandom method, starting with the ideas of V. Rodl. Some of the highlights include applications to combinatorial and projective geometry, and most notably the recent proof by Keevash of the existence of combinatorial designs. The main ideas will be discussed, without delving too far into the technical details, and a number of open problems will be presented.