Past Junior Number Theory Seminar

16 February 2015
16:00
Haden Spence
Abstract

In this talk I will discuss the notion of o-minimality, which can be approached from either a model-theoretic standpoint, or an algebraic one.  I will exhibit some o-minimal structures, focussing on those most relevant to number theorists, and attempt to explain how o-minimality can be used to attain an assortment of results.

  • Junior Number Theory Seminar
17 November 2014
16:00
Alex Betts
Abstract
We give an overview of Amnon Neeman's proof of Grothendieck's duality, working in the unbounded derived category and constructing the exceptional inverse image functor by appealing to an abstract adjoint functor theorem. The focus will be on developing the theory of the unbounded derived category and Spaltenstein's techniques for applying this theory in the algebro-geometric framework.
  • Junior Number Theory Seminar
10 November 2014
16:00
Jan Vonk
Abstract

We will discuss Raynaud's classical theory on Néron models of Jacobians of curves, and mention some tropical aspects of the theory that help us understand modular curves from a modern non-Archimedean viewpoint. There will be an annoyingly large number of examples illustrating the key principles throughout. 

  • Junior Number Theory Seminar
3 November 2014
16:00
James Maynard
Abstract

Cramer conjectured a random model for the distribution of the primes, which would suggest that, on the scale of the average prime gap, the primes can be modelled by a Poisson process. In particular, the set of limit points of normalized prime gaps would be the whole interval $[0,\infty)$. I will describe joint work with Banks and Freiberg which shows that at least 1/8 of the positive reals are in the set of limit points. 

  • Junior Number Theory Seminar
27 October 2014
16:00
Simon Rydin Myerson
Abstract

Consider a nonsingular projective variety $X$ defined by a system of $R$ forms of the same degree $d$. The circle method proves the Hasse principle and Manin's conjecture for $X$ when $\text{dim}X > C(d,R)$. I will describe how to improve the value of $C$ when $R$ is large. I use a technique for estimating mean values of exponential sums which I call a ``moat lemma". This leads to a novel and intriguing system of auxiliary inequalities.

 

  • Junior Number Theory Seminar

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