Past Number Theory Seminar

13 November 2014
16:00
Abstract

We discuss a new method to bound the number of primes in certain very thin sets. The sets $S$ under consideration have the property that if $p\in S$ and $q$ is prime with $q|(p-1)$, then $q\in S$. For each prime $p$, only 1 or 2 residue classes modulo $p$ are omitted, and thus the traditional small sieve furnishes only the bound $O(x/\log^2 x)$ (at best) for the counting function of $S$. Using a different strategy, one related to the theory of prime chains and Pratt trees, we prove that either $S$
contains all primes or $\# \{p\in S : p\le x \} = O(x^{1-c})$ for some positive $c$. Such sets arise, for example, in work on Carmichael's conjecture for Euler's function.

  • Number Theory Seminar
6 November 2014
16:00
Jack Thorne
Abstract

Let f be an elliptic modular newform of weight at least 2. The 
problem of the automorphy of the symmetric power L-functions of f is a 
key example of Langlands' functoriality conjectures. Recently, the 
potential automorphy of these L-functions has been established, using 
automorphy lifting techniques, and leading to a proof of the Sato-Tate 
conjecture. I will discuss a new approach to the automorphy of these 
L-functions that shows the existence of Sym^m f for m = 1,...,8.

  • Number Theory Seminar
12 June 2014
16:00
Christopher Lazda
Abstract
If X/F is a smooth and proper variety over a global function field of characteristic p, then for all l different from p the co-ordinate ring of the l-adic unipotent fundamental group is a Galois representation, which is unramified at all places of good reduction. In this talk, I will ask the question of what the correct p-adic analogue of this is, by spreading out over a smooth model for C and proving a version of the homotopy exact sequence associated to a fibration. There is also a version for path torsors, which enables me to define an function field analogue of the global period map used by Minhyong Kim to study rational points.
  • Number Theory Seminar
5 June 2014
16:00
Andrew Granville
Abstract
For the last few years Soundararajan and I have been developing an alternative "pretentious" approach to analytic number theory. Recently Harper established a more intuitive proof of Halasz's Theorem, the key result in the area, which has allowed the three of us to provide new (and somewhat simpler) proofs to several difficult theorems (like Linnik's Theorem), as well as to suggest some new directions. We shall review these developments in this talk.
  • Number Theory Seminar
29 May 2014
16:00
Miguel Walsh
Abstract
We will discuss some connections between the polynomial method, sieve theory, inverse problems in arithmetic combinatorics and the estimation of rational points on curves. Our motivating questions will be to classify those sets that are irregularly distributed in residue classes and to understand how many rational points of bounded height can a curve of fixed degree have.
  • Number Theory Seminar

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