Bodleian Libraries have a range of engaging wellbeing activities for staff and students. Taking a break is important for your mental health and overall wellbeing, so take some time for yourself. You can see a list of activities on the Bodleian Libraries Wellbeing page.
Mon, 25 Nov 2024
16:00
16:00
C3
Gap distributions and the Metric Poissonian Property
Sophie Maclean
(King's College London)
Abstract
When studying dilated arithmetic sequences, it is natural to wonder about their distribution. Whilst it is relatively achievable to ascertain whether the resulting sequence is equidistributed, is it much more difficult to say much about gap size between consecutive elements of the new set? In this talk I will explore the gap distributions in dilated arithmetic sequences modulo 1, including what it means for a sequence to have the metric poissonian property. I will also give an overview of the current progress and what I am aiming to discover in my own work.
Mon, 21 Oct 2024
16:00
16:00
C3
Monochromatic non-commuting products
Matt Bowen
(University of Oxford)
Abstract
We show that any finite coloring of an amenable group contains 'many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables. This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem. Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.
Thu, 13 Mar 2025
16:00
16:00
Lecture Room 4
Fourier Asymptotics and Effective Equidistribution
Subhajit Jana
(Queen Mary University of London)
Abstract
We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. This is a joint work with Shreyasi Datta.
Mon, 14 Oct 2024
16:00
16:00
C3
Self-Similar Sets and Self-Similar Measures
Constantin Kogler
(University of Oxford)
Abstract
We give a gentle introduction to the theory of self-similar sets and self-similar measures. Connections of this topic to Diophantine approximation on Lie groups as well as to additive combinatorics will be exposed. In particular, we will discuss recent progress on Bernoulli convolutions. If time permits, we mention recent joint work with Samuel Kittle on absolutely continuous self-similar measures.
Thu, 13 Feb 2025
16:00
16:00
Lecture Room 4
On the exceptional set in the abc conjecture
Joni Teräväinen
(University of Cambridge)
Abstract
The well known abc conjecture asserts that for any coprime triple of positive integers satisfying $a+b=c$, we have $c<K_{\varepsilon} \mathrm{rad}(abc)^{1+\varepsilon}$, where $\mathrm{rad}$ is the squarefree radical function.
In this talk, I will discuss a proof giving the first power-saving improvement over the trivial bound for the number of exceptions to this conjecture. The proof is based on a combination of various methods for counting rational points on curves, and a combinatorial analysis to patch these cases together.
This is joint work with Tim Browning and Jared Lichtman.
Mon, 28 Oct 2024
16:00
16:00
C3
An introduction to modularity lifting
Dmitri Whitmore
(University of Cambridge)
Abstract
The (global) Langlands programme is a vast generalization of classical reciprocity laws. Roughly, it predicts a correspondence between:
1) modular forms (and their generalizations, automorphic forms)
2) representations of the Galois group of a number field.
While many constructions of Galois representations from automorphic forms exist, the converse direction is often harder to establish. The main tools to do so are modularity lifting theorems and are proved via the Taylor-Wiles method, originating from Wiles' proof of Fermat's Last Theorem.
I will introduce these ideas and their applications, focusing particularly on the problem of modularity of elliptic curves. I will then briefly discuss a generalization of the Taylor-Wiles method developed in my thesis which led to new modularity theorems in the setting of quadratic extensions of totally real fields by building of work of Boxer-Calegari-Gee-Pilloni.