The Fourier uniformity conjecture seeks to understand what multiplicative functions can have large Fourier coefficients on many short intervals. We will discuss recent progress on this problem and explain its connection with the distribution of prime numbers and with other central problems about the behaviour of multiplicative functions, such as the Chowla and Sarnak conjectures.
An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool not just within representation theory, but also for the construction of an explicit and a categorical local Langlands correspondence, and has applications to the study of automorphic forms, for example. In my talk I will introduce p-adic groups and explain that the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will then provide an overview of what we know about the structure of these Bernstein blocks. In particular, I will sketch how to use a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.
Let π be a cuspidal automorphic representation of Sp2n over ℚ which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is irreducible, that π is ordinary at p, and that p does not divide the conductor of χ. I will describe work in progress which aims to prove that the Bloch-Kato Selmer group attached to the twist of ρ by χ vanishes, under some mild ramification assumptions on π; this is what is predicted by the Bloch-Kato conjectures.
The proof uses "ramified Eisenstein congruences" by constructing p-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of non-classical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of ρ by χ.
In 2014, Fargues realized that one can formulate the local Langlands correspondence over p-adic fields as a geometric Langlands correspondence on the Fargues-Fontaine curve. This raises the question of a similar realization of the local Langlands correspondence over the real numbers. The goal of these lectures is to explain a possible formulation. As part of this, we will give a new perspective on the theory of variations of twistor structure, a generalization of the theory of variations of Hodge structure. This uses the theory of analytic stacks developed in our joint work with Clausen, of which we will give a brief overview.
In the early 2000s Ruijsenaars and Felder-Varchenko have introduced the elliptic gamma function, a remarkable multivariable meromorphic q-series that comes from mathematical physics. It satisfies modular functional equations under the group SL3(ℤ) which make it a higher-dimensional analogue of the Jacobi theta function. In this work, we unveil the place that this function and its avatars play in number theory. Our main thesis is that these functions play the role of modular units in extending the theory of complex multiplication to complex cubic fields. In other words we propose a conjectural solution to Hilbert’s 12th problem for complex cubic fields, following a line of research actually initiated by G. Eisenstein. We give a lot of numerical evidence that support this conjecture, and relate it to the Stark conjecture by proving an analogue of the Kronecker limit formula in this cubic setting.
We discover a surface related to the pair correlation of zeros of the Riemann zeta function. We make a conjecture on the shape of the surface and present partial results and numerical evidence towards the conjecture.
Let G be an infinite discrete group. Finite dimensional unitary representations of G are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps.
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