Number Theory Seminar
The Number Theory Seminar is a 55-minute in-person research seminar, including questions. Talks are suitable for PhD students (including new PhD students) from all fields of number theory.
The seminar is at 15:00 on Mondays (except bank holidays). On seminar days we meet for lunch at 12:30 and coffee at 16:00 in the common room. Seminars are held in B3.02 of the Zeeman building. Sometimes a different room is used, see the entry below.
Colleagues and especially number theory group members are warmly encouraged to suggest speakers by emailing the organisers: Simon Rydin Myerson (term 1), Adam Harper (terms 2-3), Harry Schmidt (all terms).
We kindly remind members that it is polite to the speaker, to come to talks we might not personally expect to be interested in, and in compensation there will be a good audience for the speakers each of us is interested in!
A list of members of the group and research interests is available.
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06 January 2025 at 15:00 in B3.02
Speaker: Sean Eberhard (Warwick)
Title: TBA
Abstract: TBA
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13 January 2025 at 15:00 in B3.02
Speaker: Max Xu (Simons/Courant Institute)
Title: TBA
Abstract: TBA
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20 January 2025 at 15:00 in B3.02
Speaker: Robin Bartlett (Glasgow)
Title: TBA
Abstract: TBA
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27 January 2025 at 15:00 in B3.02
Speaker: Abhishek Saha (QMUL)
Title: TBA
Abstract: TBA
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03 February 2025 at 15:00 in B3.02
Speaker: Tim Dokchitser (Bristol)
Title: TBA
Abstract: TBA
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10 February 2025 at 15:00 in B3.02
Speaker: Jessica Alessandri (Bath)
Title: TBA
Abstract: TBA
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17 February 2025 at 15:00 in B3.02
Speaker: Robert Wilms (Caen)
Title: TBA
Abstract: TBA
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24 February 2025 at 15:00 in B3.02
Speaker: Shin-ya Koyama (Toyo University)
Title: TBA
Abstract: TBA
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07 March 2025 at 13:00 in B3.02
Speaker: David Masser (Basel)
Title: TBA
Abstract: TBA
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10 March 2025 at 15:00 in B3.02
Speaker: Oleksiy Klurman (Bristol)
Title: TBA
Abstract: TBA
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02 December 2024 at 15:00 in B3.02
Speaker: Nikolaos Diamantis (Nottingham)
Title: L-series of half-integral weight cusp forms and an analogue of the period polynomial
Abstract: We construct a polynomial expressed in terms of values of the L-series attached to a half-integral weight cusp form. This polynomial can be thought of as an analogue of the classical period polynomial since it also satisfies certain "period relations". We show how it induces a lift of half-integral weight cusp forms to integral weight forms which is compatible with the L-series of the respective forms. This lift is explicit thanks to a result by Pasol and Popa. (Joint work with Branch, Raji and Rolen).
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25 November 2024 at 15:00 in B302
Speaker: Elvira Lupoian (Imperial)
Title: Runge’s Method and Integral Points on Modular Curves
Abstract: The study of integral points on curves dates to Siegel’s theorem in 1929, which has been historically studied by many due to its connections to Mordell’s conjecture (Faltings’ theorem) . More recently, Bilu and Parent were able to prove Serre’s uniformity conjecture in the split Cartan case by studying integral points on the corresponding modular curve. Their proof relies on efficiently determining the integral points using the so -called Runge’s method. In this talk, we review this method and discuss how the work of Bilu and Parent can be adapted to efficiently bounds heights of integral points on certain covers of the classical modular curve $X_{0} \left( p \right)$.
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18 November 2024 at 15:00 in B3.02
Speaker: Emma Bailey (Bristol)
Title: Large deviations of Selberg’s CLT: upper and lower bounds
Abstract: Selberg’s CLT informs us that the logarithm of the Riemann zeta function evaluated on the critical line behaves as a complex Gaussian. It is natural, therefore, to study how far this Gaussianity persists. This talk will present conditional and unconditional results on atypically large values, and is joint with Louis-Pierre Arguin and Asher Roberts.
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11 November 2024 at 15:00 in B.302, Zeeman building
Speaker: Chris Daw (University of Reading)
Title: Large Galois orbits under multiplicative degeneration
Abstract: The Pila-Zannier strategy is a powerful technique for proving results in unlikely intersections. In this talk, I will recall the Zilber-Pink conjecture for Shimura varieties and describe how Pila-Zannier works in this setting. I will highlight the most difficult outstanding obstacle to implementing the strategy — the so-called Large Galois Orbits conjecture — and I will explain recent progress towards this conjecture, building on the works of André and Bombieri. This is joint with Martin Orr (Manchester).
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04 November 2024 at 15:00 in B.302, Zeeman building
Speaker: Andrei Yafaev (University College London)
Title: TBA
Abstract: TBA
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28 October 2024 at 15:00 in B3.02
Speaker: Akshat Mudgal (Warwick)
Title: Recent progress towards the sum–product conjecture and related problems
Abstract: An important open problem in combinatorial number theory is the Erdős–Szemerédi sum–product conjecture, which suggests that for any positive integers s, N, and for any set A of N integers, either there are many s-fold sums of the form $a_1 + … + a_s$ or there are many s-fold products of the form $a_1\dots a_s$. While this remains wide open, various generalisations of this problem have been considered more recently, including the question of finding optimal variations of the so-called low energy decompositions.<br><br>In this talk, I will outline some recent progress towards the above questions, as well as highlight how these connect very naturally to other key conjectures in additive combinatorics.
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21 October 2024 at 15:00 in B302
Speaker: Tobias Berger (University of Sheffield)
Title: Pseudomodularity of residually reducible Galois representations
Abstract: After a survey of previous work I will present new results on pseudomodularity of residually reducible Galois representations with 3 residual pieces. I will discuss applications to proving modularity of Galois representations arising from abelian surfaces and Picard curves. This is joint work with Krzysztof Klosin (CUNY).
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14 October 2024 at 15:00 in B3.02
Speaker: Maleeha Khawaja (Sheffield)
Title: Galois groups of low degree points on curves
Abstract: Whilst the study of low degree algebraic points on curves is an active area of research, there has been little emphasis on the Galois-theoretic description of these points. In this talk, we focus on the behaviour of low degree points whose Galois group is primitive. Furthermore, we shall see that the behaviour changes if the degree is large (with respect to the genus of the curve). This talk is based on joint work with Frazer Jarvis (Sheffield) and Samir Siksek (Warwick).
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07 October 2024 at 15:00 in B.302, Zeeman building
Speaker: Han Yu (University of Warwick)
Title: TBA
Abstract: TBA
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30 September 2024 at 15:00 in B3.02
Speaker: Tim Browning (IST Austria)
Title: Pairs of commuting matrices
Abstract: I'll discuss commuting varieties and a new upper bound for the number of pairs of commuting $n \times n$ matrices with integer entries and height at most $T$, as $T \to \infty$. Our approach uses Fourier analysis and mod $p$ information, together with a result about the flatness of the commutator Lie bracket, which we also solve. This is joint work with Will Sawin and Victor Wang.
Click on a title to view the abstract!