University of Oxford: 24th April 2026

Directions: The location of the Mathematical Institute and some information about transport can be found here: https://www.maths.ox.ac.uk/about-us/travel-maps

Lecture room: We will be in room L4, in the basement level of the Andrew Wiles Building. The room will be signposted from the central staircase leading down from the entrance lobby.

Registration is now closed.

Schedule:

10.15 - 11.00: Arrival and coffee

11.00 - 12.00: Cédric Pilatte

12.15 - 13.15: Chris Birkbeck

13.15 - 14.30: Lunch

14.30 - 15.30: Chengyang Bao

15.30 - 16.00: Tea/coffee break

16.00 - 17.00: Samir Siksek

Titles & Abstracts

Cédric Pilatte (Oxford)

Title: Short exponential sums of the Liouville function

Abstract: The Liouville function is the completely multiplicative function λ defined by λ(p) = -1 for every prime p. In this talk, we consider the pseudo-random behaviour of this function in short intervals [x, x+h] where the length h is much smaller than x. 

The Fourier uniformity conjecture predicts that the Liouville function has negligible correlations with linear phases n ↦ e2πinɑ over almost all short intervals, in a precise sense. Using improved combinatorial arguments and deep number-theoretic inputs, we extend the range of interval lengths h for which this conjecture is known to hold. 

Chris Birkbeck (UEA)

Title: Formalising modular forms and sphere packings using AI and Lean.

Abstract: I will discuss some joint work on formalising modular forms in Lean and how it was used in the formalisation of Maryna Viazovska’s Fields Medal-winning paper proving that no packing of unit balls in Euclidean space R^8 has density greater than that of the E8-lattice packing. I will discuss both the human and AI contributions to the project. This is joint work with Sidharth Hariharan, Gareth Ma, Bhavik Mehta, Seewoo Lee, Maryna Viazovska and Math.inc. 

Chengyang Bao (Imperial)

Title: Equidistribution of modular points on local deformation rings.

Abstract: In his proof of the Fontaine–Mazur conjecture, Kisin proved that every irreducible component of the patched ring R_\infty contains at least one automorphic Galois representation. In this work, we investigate a quantitative question in the GL(2,Q_p) case: how many automorphic Galois representations lie on each component. We show that the answer depends on the singularity structure of certain decorated local deformation rings, together with the number of automorphic representations in the Fontaine–Laffaille range, up to twist. This is joint work with John Bergdall and Brandon Levin

Samir Siksek (Warwick)

Title: Galois groups of low degree points on curves

Abstract: Low degree points on curves have been subject of intense study for several decades, but little attention has been paid to the Galois groups of those points. In this talk we recall primitive group actions, and focus on low degree points whose Galois group is primitive. We shall see that such points are relatively rare, and that they interfere with each other. This talk is based on joint work with Maleeha Khawaja.