Department of

Mathematics


Seminar Calendar
for events the day of Thursday, September 7, 2017.

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Thursday, September 7, 2017

11:00 am in 241 Altgeld Hall,Thursday, September 7, 2017

On the zeros of Riemann's zeta-function

Sieg Baluyot (Illinois)

Abstract: In the first part of this talk, we present a new proof that a positive proportion of the zeros of the Riemann zeta-function lie on the critical line. The proof is an enhancement of a zero-detection method of Atkinson from the 1940’s, and uses the recent estimate of Hughes and Young for the twisted fourth moment of zeta. In the second part, we consider the number of zeros of zeta inside the region with real part larger than $\sigma$ and imaginary part between 0 and T. A bound for this number is called a “zero-density estimate.” We present an improved zero-density estimate for the case when $\sigma$ is larger than 1/2 but close to 1/2. The main theorem confirms an unproved result of Conrey from the 1980’s using his technique of applying Kloosterman sum estimates. Finally, in the third part, we look at hypothetical statements for the vertical distribution of zeros along the critical line and deduce their consequences for the prime numbers and other properties of zeta. The main theorem generalizes results of Goldston, Gonek, and Montgomery that give consequences of the pair correlation conjecture. We apply the theorem to examine implications of the well-known “alternative hypothesis,” which is related to Landau-Siegel zeros.

12:30 pm in 276 Loomis,Thursday, September 7, 2017

Eigenstate thermalization hypothesis in two-dimensional large central charge CFT

Jia-ju Zhang (University of Milano-Bicocca)

Abstract: By the eigenstate thermalization hypothesis (ETH), a highly excited energy eigenstate behaves like the thermal state. We investigate the eigenstate thermalization hypothesis in two-dimensional large central charge CFT and compare the excited state of a primary operator with the canonical ensemble thermal state. To distinguish the excited state and thermal state, we calculate the short interval expansions of the entanglement entropy, relative entropy, Jensen-Shannon divergence, and Schatten 2-norm. We include only contributions from the vacuum conformal family and find that ETH is satisfied in the leading order of central charge and is violated at the next-to-leading order. We also discuss briefly ETH in two-dimensional large central charge CFT for the microcanonical ensemble and generalized Gibbs ensemble thermal states.

1:00 pm in 243 Altgeld Hall,Thursday, September 7, 2017

Dynamics near traveling waves of supercritical KDV equations

Zhiwu Lin (Georgia Institute of Technology)

Abstract: Consider generalized KDV equations with a power non-linearity (u^p)_x. These KDV equations have solitary traveling waves, which are linearly unstable when p>5 (supercritical case). Jointly with Jiayin Jin and Chongchun Zeng, we constructed invariant manifolds (stable, unstable and center) near the orbit of the unstable traveling waves in the energy space. These invariant manifolds are used to give a complete description of the local dynamics near unstable traveling waves. In particular, the global existence with orbital stability is shown on the center manifold of co-dimension two, while the exponential instability is proved for initial data not on the center manifold.

1:00 pm in Altgeld Hall 243,Thursday, September 7, 2017

Character values, combinatorics, and some p-adic representations of finite groups

Michael Geline (Northern Illinois University)

Abstract: There are many questions which remain open from Brauer's modular representation theory of finite groups, and little agreement exists over the extent to which they ought to ultimately depend on the classification of finite simple groups. In studying a classification-free approach to one such question, involving lattices over the p-adics, I was led to an elementary combinatorial problem which is interesting in its own right and somewhat amenable to analysis by means of character values. I will present this problem, the original conjecture of Brauer which gave rise to it, and my own progress in the area. If it is necessary to mention algebraic geometry, there is something of a long-shot analogy between the character values and the Weil conjectures.

2:00 pm in 241 Altgeld Hall,Thursday, September 7, 2017

The Herbrand-Ribet Theorem

Patrick Allen   [email] (UIUC)

Abstract: Kummer's criterion states that a prime number $p \ge 7$ divides the class number of the $p$th cyclotomic field if and only if $p$ divides the numerator of one of the Bernoulli numbers $B_2, B_4, \ldots, B_{p-3}$, or equivalently, one of the values $\zeta(-1), \zeta(-3), \ldots, \zeta(4-p)$ of the Riemann zeta function. It is natural to ask if the individual values $B_k$ correspond to more refined information of the class groups. This is the content of the Herbrand-Ribet Theorem, one direction of which was proved by Herbrand in 1932, and the other by Ribet in 1976. Ribet's converse to Herbrand's Theorem uses the theory of modular forms and their associated Galois representations, and the ideas involved have been highly influential. We'll introduce this theorem, defining all the objects involved, and give some idea of the proof. I will aim to structure this talk so that any graduate student, be it their 1st year or their 45th year, will be able to take away something.

4:00 pm in 245 Altgeld Hall,Thursday, September 7, 2017

Random groups and surfaces

Moon Duchin (Tufts)

Abstract: I'll survey some of the beautiful history of the study of random objects in geometry, topology, and group theory. The focus will be the exchanges between work on random groups and work on random surfaces, including some very recent results and current research topics.

5:15 pm in 245 Altgeld Hall,Thursday, September 7, 2017

IGL Kickoff Meeting

Abstract: Fall 2017 organizational meeting for the Illinois Geometry Lab.