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for events the day of Thursday, September 14, 2017.

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Thursday, September 14, 2017

11:00 am in 241 Altgeld Hall,Thursday, September 14, 2017

Partition asymptotics and the polylogarithm

Alexander Dunn (Illinois)

Abstract: n 2015 Vaughn obtained asymptotic formulas for the number of partitions of an integer into squares. Gafni extended this to kth powers. Here we obtain such formulas for the number of partitions into values of an arbitrary integer polynomial $f$ subject to some mild hypotheses. Our methods use an interplay of the circle method, the polylogarithm, and the Matsumoto-Weng zeta function. This is joint work with Nicolas Robles.

12:00 pm in Altgeld Hall,Thursday, September 14, 2017

Self-Similar Interval Exchange Transformations

Kelly Yancey   [email] (Institute for Defense Analyses)

Abstract: During this talk we will discuss the class of self-similar 3-IETs and show that they satisfy Sarnak's conjecture. We will do this by appealing to the theory of joinings. Specifically we will show how to prove the property of minimal self-joinings for substitution systems (self-similar IETs can be thought of in this context).

2:00 pm in 241 Altgeld Hall,Thursday, September 14, 2017

On the percentage of critical zeros of Riemann's zeta function

Kyle Pratt   [email] (UIUC)

Abstract: The Riemann hypothesis (RH) is one of the most important unsolved problems in number theory. RH asserts that all of the important zeros of the Riemann zeta function lie on a specific line, called the critical line. As we lack a solution to RH, it is natural to ask for partial results instead. One way to measure progress towards RH is to prove that some percentage of the zeros are on the critical line. I will sketch a brief history of the results about percentages of zeros on the critical line, and discuss some of the methods of proof. In the latter part of the talk I will discuss the current world record, due to Nicolas Robles and myself, and some of our ideas. The talk should be accessible to any graduate student.

3:00 pm in 345 Altgeld Hall,Thursday, September 14, 2017

Symmetrization vs Constant Term Identities for generalized Macdonald operators: a (directed) walk through Double Affine Hecke Algebras, Toroidal Algebras, and Shuffle algebras. Part II

Philippe Di Francesco (Illinois)

Abstract: We introduce a generalization of the A-type Macdonald difference operators via a symmetrization identity (S) that maps symmetric functions to difference operators. Recall that Macdonald operators have Macdonald polynomials as common eigenfunctions, and play a crucial role in deep combinatorial theorems (factorial n theorem, shuffle theorem, general questions about Schur positivity). We will show how our generalized Macdonald operators can be obtained by implementing the modular group symmetry of the A-type Double Affine Hecke Algebra in its functional representation. The corresponding difference operators obey commutation relations that can be viewed as t-deformations of some particular quantum cluster algebra relations pertaining to the A-type Q-system. We’ll show how these can be canned into a representation of the affine gl1 quantum toroidal algebra with zero central charge. We finally present an alternative definition of our generalized Macdonald difference operators via a constant term identity (CT) that maps symmetric functions to difference operators. This leads to a natural definition of Shuffle product, for which (S) and (CT) are algebra morphisms. We then show how commutation relations for our operators reduce to simple Shuffle identities. This is illustrated in the quantum cluster algebra limit t->infinity.

4:00 pm in 245 Altgeld Hall,Thursday, September 14, 2017

Probabilistic and combinatorial methods in the study of prime gaps

Kevin Ford (Illinois)

Abstract: We will describe how new bounds for the largest gaps between consecutive primes have utilized tools from several different areas, including number theory (efficient prime detecting sieves), probability (randomized congruence system coverings, concentration arguments) and probabilistic combinatorics (hypergraph covering). In particular, we will outline the recent breakthroughs of the speaker together with Ben Green, Sergei Konyain, James Maynard and Terence Tao. We will also describe new work with Konyagin, Maynard, Carl Pomerance and Tao which provides new bounds on consecutive composite values of integers in other sequences, e.g. polynomial sequences.