London Number Theory Seminar 



The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL.
KINGS This term (Summer 2016), the seminar will be hosted by Kings College, and will be held on Wednesdays starting at 4pm. The talks will be in room S4.23 of the Department of Mathematics, starting on Wed 27th April and finished on 29th June. The talks will be preceded by tea at 3:30 in S5.21.A preliminary list of speakers is below; for full details including up to date titles and abstracts please go to Mahesh's web page here.
27/4/16 Fernando Shao (Oxford)
Title: Vinogradov's three primes theorem with almost twin primes
Abstract: The general theme of this talk is about solving linear equations in sets of number theoretic interest.
Specifically I will discuss the problem with the linear equations being N = x+y+z (for a fixed large N) and the set being "almost twin primes".
The focus will be on the underlying ideas coming from both additive combinatorics and sieve theory. This is joint work with Kaisa Matomaki.
4/5/16 No seminar because of conference RandomWavesInLondon.
11/5/16 Oleksiy Klurman (Universite de Montreal/UCL)
Title: Correlations of multiplicative functions and applications.
Abstract: A deep problem in analytic number theory is to understand correlations of general multiplicative functions. In this talk, we derive correlations formulas for socalled bounded "pretentious" multiplicative functions. This has a number or desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\to \{1,1\}$ with bounded partial sums. This answers a question of Erd\H{o}s from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $f(n)$ is small on average. This settles an old conjecture of K\'atai. If time permits, we discuss some further applications to the related problems.
18/5/16 Giovanni Rosso (Cambridge)
Title: Trivial zero for a padic Lfunction associated with Siegel forms
Abstract: We shall begin with an introduction to Lfunctions in arithmetic, their padic interpolation and trivial zero. We shall state a conjecture of Greenberg and Benois which predicts the order and the leading coefficient of padic Lfunctions when trivial zeros appear. We shall then explain how one calculates the first derivative of the standard padic Lfunction of an ordinary Siegel form with level at p.
25/5/16 Gergely Zábrádi (Eötvös Loránd)
Title: Smooth mod p^n representations and direct powers of Galois groups.
Abstract: Let G be a Qpsplit reductive group with connected centre and Borel subgroup B=TN.
We construct a right exact functor D from the category of smooth modulo p^n representations of B to the category of projective limits of continuous mod p^n representations of a direct power of the absolute Galois group Gal(Qpbar/Qp) of Qp indexed by the set of simple roots. The objects connecting the two sides are (phi,Gamma)modules over a multivariable (commutative) Laurent series ring which correspond to the Galois side via an equivalence
of categories. Parabolic induction from a subgroup P = L_P N_P amounts to the extension of the representation on the Galois side to the copies of Gal(Qpbar/Qp) indexed by the simple roots alpha not contained in the Levi component L_P using the action of the image of the cocharacter dual to alpha and local class field theory. D is exact and yields finite dimensional representations on the category SP of finite length representations with subquotients of principal series as JordanHölder factors. Using the Gequivariant sheaf of Schneider, Vigneras, and the author on the flag variety G/B corresponding to the Galois representation we show that D is fully faithful on the full subcategory of SP with JordanHölder factors isomorphic to irreducible principal series. Breuil has (preliminary) conjectures for
the values of D at certain representations of GL_n(Qp) built out from some mod p Hecke isotypic subspaces of global automorphic representations.
1/6/16 Trevor Wooley (Bristol)
Title: TBA
6th and 7th June  LondonParis Number Theory Seminar.
8/6/16 Adam Harper (Cambridge)
Title: TBA
15/6/16 Two seminars
2:303:30 S4.23 (note unusual time) Ben Green (Oxfrod)
Title: A note on multiplicative functions on progressions to large moduli.
45 Jens Marklof (Bristol)
Title: TBA
22/6/16 Maria Valentino (King’s)
Title: TBA
29/6/16 Nina Snaith (Bristol)
Title: TBA
The seminar will be preceded by the Study Groups, and this term there seem to be three: Classification of mod p representations of padic groups (12451415), an informal study group on Scholze's LubinTate tower paper (14301530) and an analytic study group (14301530).
A list of previous seminar talks is here.
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