Number Theory

In the 1980s Beilinson conjectured the existence of motivic cohomology, together with certain precise expected relations to algebraic K-theory, étale cohomology, algebraic cycles, and the category of mixed motives. Apart from the still conjectural category of mixed motives (and its consequences…

In 1972, Borel proved that every holomorphic map from a product of punctured unit disks to a complex Shimura variety extends to a map from a product of disks to its Baily--Borel compactification.  Recently, Oswal--Shankar--Zhu and Patel proved the corresponding p-adic statement over…

Are rational points distributed randomly near curved manifolds? After introducing a simple random model, we focus on this question in hyperbolic regions, which arise naturally in multiplicative Diophantine approximation. We then sketch how tools from homogeneous dynamics and harmonic…

Abstract: The correlations between $d(n)$ and $d(n+h)$, where $d(n)$ is the divisor-counting function and $h$ is a possibly varying non-zero integer are a classical topic in analytic number theory, going back to Ingham. It is intimately related to the fourth moment of the Riemann zeta function.…

Let E(epsilon) be the set of all complex numbers whose real part is within distance epsilon of an integer. The pyjama problem asks if, for every positive epsilon, finitely many rotations of E(epsilon) can cover the entire complex plane. This was answered affirmatively by Manners in…

The circle method is a classical technique in analytic number theory, originally developed to count solutions to polynomial equations. In this talk, I will sketch how this method can be adapted to study moduli spaces of curves on hypersurfaces — objects that have traditionally been approached…

Euler systems -- organized collections of Galois cohomology classes for arithmetically interesting p-adic Galois representations -- have been a useful tool for establishing the conjectured relation between special values of L-functions and the ranks and orders of Selmer groups when they exist.…

Abstract:  The notion of visibility from the origin along straight lines has been generalised by considering lattice points viewed through nonlinear trajectories. Inspired by this, we define polynomial Farey sequences, reducing to the classical Farey sequences for linear polynomials. In…

How do the finite/infinite dichotomy of the Killing–Cartan classification of simple Lie groups & algebras appear in arithmetic geometry? I will explain how this Lie-theoretic dichotomy is realized in the finiteness or infinitude of positive integer solutions to certain Diophantine equations…

We present a new structure on the first Galois cohomology of families of symplectic self-dual p-adic representations of $G_Qp$ of rank two. This is a functorial decomposition into free rank one Lagrangian submodules encoding Bloch-Kato subgroups and epsilon factors, mirroring an underlying…