Past Junior Number Theory Seminar

31 January 2011
16:00
Frank Gounelas
Abstract
The main aim of this talk will be to present a proof of the Tsen-Lang theorem on the existence of points on complete intersections and sketch a proof of the Grabber-Harris-Starr theorem giving the existence of a section to a fibration of a rationally connected variety over a curve. Time permitting, recent work of de Jong and Starr on rationally simply connected varieties will be discussed with applications to the number theory of hypersurfaces.
  • Junior Number Theory Seminar
17 January 2011
16:00
Lillian Pierce
Abstract
<p>An old conjecture of Hardy and Littlewood posits that on average, the number of representations of a positive integer N as a sum of k, k-th powers is "very small." Recently, it has been observed that this conjecture is closely related to properties of a discrete fractional integral operator in harmonic analysis. This talk will give a basic introduction to the two key problems, describe the &nbsp;correspondence between them, and show how number theoretic methods, in particular the circle method and mean values of Weyl sums, can be used to say something new in abstract harmonic analysis.</p>
  • Junior Number Theory Seminar
8 November 2010
16:00
Frank Gounelas
Abstract
In this talk I will introduce some of the basic ideas linking the theory of complex multiplication for elliptic curves and class field theory. Time permitting, I'll mention Shimura and Taniyama's work on the case of abelian varieties.
  • Junior Number Theory Seminar
1 November 2010
16:00
James Maynard
Abstract
The Siegel-Walfisz theorem gives an asymptotic estimate for the number of primes in an arithmetic progression, provided the modulus of the progression is small in comparison with the length of the progression. Counting primes is harder when the modulus is not so small compared to the length, but estimates such as Linnik's constant and the Brun-Titchmarsh theorem give us some information. We aim to look in particular at upper bounds for the number of primes in such a progression, and improving the Brun-Titchmarsh bound.
  • Junior Number Theory Seminar

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