Past Junior Number Theory Seminar

19 October 2015
16:00
Benjamin Green
Abstract

In this talk I will define algebraic automorphic forms, first defined by Gross, which are objects that are conjectured to have Galois representations attached to them. I will explain how this fits into the general picture of the Langlands program and, giving some examples, briefly describe one method of proving certain cases of the conjecture. 

  • Junior Number Theory Seminar
1 June 2015
16:00
Alex Betts
Abstract

We will give a sketch overview of Scholze's theory of perfectoid spaces and the tilting equivalence, starting from Huber's geometric approach to valuation theory. Applications to weight-monodromy and p-adic Hodge theory we will only hint at, preferring instead to focus on examples which illustrate the philosophy of tilting equivalence.
 

  • Junior Number Theory Seminar
18 May 2015
16:00
Michael Tyler
Abstract

After some generalities on étale fundamental groups and anabelian geometry, I will explore some of the current results on the section conjecture, including those of Koenigsmann and Pop on the birational section conjecture, and a recent unpublished result of Mohamed Saidi which reduces the section conjecture for finitely generated fields over the rationals to the case of number fields.

  • Junior Number Theory Seminar
27 April 2015
16:00
Benjamin Green
Abstract

In this talk I will describe two instances of Langlands functoriality concerning the group $\mathrm{Sp}_{2n}$. I will then very briefly explain how this enables one to attach Galois representations to automorphic representations of (inner forms of) $\mathrm{Sp}_{2n}$. 

  • Junior Number Theory Seminar
23 February 2015
16:00
Aled Walker
Abstract

In 1937 Vinogradov showed that every sufficiently large odd number is the sum of three primes, using bounds on the sums of additive characters taken over the primes. He was improving, rather dramatically, on an earlier result of Schnirelmann, which showed that every sufficiently large integer is the sum of at most 37 000 primes. We discuss a natural analogue of this question in the multiplicative group (Z/pZ)* and find that, although the current unconditional character sum technology is too weak to use Vinogradov's approach, an idea from Schnirelmann's work still proves fruitful. We will use a result of Selberg-Delange, an application of a small sieve, and a few easy ideas from additive combinatorics. 

  • Junior Number Theory Seminar

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