Monday

James Maynard

Maksym Radziwill

Dominique Maldague

Tuesday

Ben Green

Sarah Peluse

Hong Wang

Wednesday

Emmanuel Kowalski

Philippe Michel

Hong Wang

Thursday

Damaris Schindler

Trevor Wooley

Po Lam Yung

Friday

Paul Nelson

James Maynard (University of Oxford): Large values of Dirichlet polynomials and the zeta function

Many important results in analytic number theory concerning primes and the zeta function depend on questions about large values of `Dirichlet polynomials'. Often such questions are almost pure problems in harmonic analysis, and can be studied without much further number-theoretic input. Moreover, despite many different techniques for different regimes, in the many important problems there is a single limiting scenario, and here typically we are unable to improve upon a `trivial bound'. I will highlight a few key questions of this type and the critical limiting scenario in each case.

Maksym Radziwill (California Institute of Technology): Where we get stuck on the 12th moment of the Riemann zeta function

(Pending permission to publish the recording)

Dominique Maldague (University of Cambridge): Difficulty of using decoupling ideas for generalized Dirichlet series and cubic Weyl sums

No recording available.

Ben Green (University of Oxford): Properly understanding why nilpotent groups are important in additive combinatorial problems

Sarah Peluse (Stanford University): Inverse theorems for multidimensional Gowers norms

No recording available.

Hong Wang (NYU Courant): Hausdorff Colloquium

Emmanuel Kowalski (ETH Zürich): Short sums of trace functions: examples, conjectures and applications (part 1)

Philippe Michel (Ecole Polytechnique Federale de Lausanne): Short sums of trace functions: examples, conjectures and applications (part 2)

Hong Wang (NYU Courant): Danzer's problem and quantitative Besikovitch projection theorem

(Pending permission to publish the recording)

Damaris Schindler (Goettingen University): Density of rational points near manifolds

Given a bounded submanifold M in ℝⁿ, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some open problems connected

to these questions and explain arithmetic applications such as in Serre's dimension growth conjecture as well as applications in Diophantine approximation.

(Pending permission to publish the recording)

Trevor Wooley (Purdue University): The paucity of knowledge concerning Weyl sums and their mean values

The past hundred or so years have witnessed tremendous progress in our understanding of exponential sums having polynomial arguments, with recent advances associated with progress on Vinogradov's mean value theorem, the delta-function method, and speculative conjectures on L-functions. Such might be the impressions of an expert in the area. Yet a critic would point out that only for linear and quadratic Weyl sums can knowledge be considered substantial. For cubic Weyl sums the situation remains highly unsatisfactory, with speculative conjectures disguising almost circular arguments. The understanding of exponential sums of high degree, meanwhile, falls far short of what is conjectured to be true. In this talk we seek to identify the common themes underlying present approaches to understanding Weyl sums, and associated obstructions to progress. There are connections with congruences to large moduli, arithmetic stratification, and speculative conjectures on L-functions.

Po Lam Yung (ANU): Discrete restriction in 2+1 vs 1+1 dimensions

Paul Nelson (Aarhus University): Conductor dropping and subconvexity

(Pending permission to publish the recording)