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A number theorist's adventures in the land of spectral theory

Abstract

I will discuss some of the interactions between number theory and the spectral theory of the Laplacian. Some have very classical background, such as the connection with lattice point problems. Others are newer, including connections between random matrix theory, the zeros of the Riemann zeta function, and spectral statistics on the moduli space of hyperbolic surfaces.​​​

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Professor Zeev Rudnick (Tel Aviv University) was awarded a PhD from Yale University in 1990, followed by positions at Stanford University and Princeton University, before joining Tel-Aviv in 1995. His main research interests lie broadly in Number Theory (especially analytic Number Theory) and Mathematical Physics (especially quantum chaos). He is widely recognised for his outstanding contributions to these fields including, but not limited to, uniform distribution theory, the study of the Riemann zeta function, developing the arithmetic  of function fields over finite fields by studying analogues of various well-known problems over integers, developing connections of number theory with mathematical physics and random matrix theory, and the formulation of various conjectures (most notably the Quantum Unique Ergodicity conjecture with Peter Sarnak). Zeev has been widely recognised by various awards and prizes, including the Erdos prize in 2001 and two highly prestigious ERC grants, he was elected a Fellow of the AMS 2012 and was an invited speaker at the ICM in 2014.

Counting linear relations between zeros of Dirichlet L-functions

Abstract

Not much is known about Diophantine properties of the imaginary parts of zeros of Dirichlet L-functions. A well known conjecture states that such zeros are linearly independent over the rationals. Progress in this direction would make it possible to say something new about the oscillations of error terms in classic number theoretic problems. In this talk I will:

1. Give an overview of how oscillations of error terms are connected to Diophantine problems involving the zeros
2. Show how one may formulate probabilistic versions of the conjecture mentioned above which may be approached by combining the Guinand-Weil explicit formula with techniques from the geometry of numbers.
3. Discuss the extent to which progress on 2 may be used to say something new about oscillations of error terms.

Bio

Dr. Bryce Kerr completed his PhD in 2017 in the area of analytic number theory under the supervision of Prof. Igor Shparlinski. He continues to work in this area and the range of his interests  includes bounds of exponential sums, additive combinatorics and the distribution of values of  the Riemann zeta-function. He has held postdoctoral positions with A/Prof. Timothy Trudgian at UNSW Canberra, Prof. Kaisa Matomäki at the University of Turku and a prestigious 12 months fellowship at the Max Planck Institute for Mathematics in Bonn. He is currently an ARC DECRA fellow at UNSW Canberra, and he was recently awarded yet another ARC Discovery Project.

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