Past Junior Number Theory Seminar

20 October 2014
16:00
Netan Dogra
Abstract
For a finite set of primes S, the S-unit equation asks for solutions to a+b=1, with
a and b rational numbers which are units at all primes not in S. By a theorem of Siegel,
for any given S this equation will only have finitely many solutions. This talk will review
the relation between this equation and other Diophantine problems, and will explain a
Galois-theoretic approach to proving Siegel's theorem.
  • Junior Number Theory Seminar
16 June 2014
16:00
Tom Lovering
Abstract
Since their introduction, Shimura varieties have proven to be important landmarks sitting right at the crossroads between algebraic geometry, number theory and representation theory. In this talk, starting from the yoga of motives and Hodge theory, we will try to motivate Deligne's construction of Shimura varieties, and briefly survey some of their zoology and basic properties. I may also say something about the links to automorphic forms, or their integral canonical models.
  • Junior Number Theory Seminar
26 May 2014
16:00
Przemysław Mazur
Abstract
The Balog-Szemeredi-Gowers theorem states that, given any finite subset of an abelian group with large additive energy, we can find its large subset with small doubling constant. We can ask how this constant depends on the initial additive energy. In the talk, I will give an upper bound, mention the best existing lower bound and, if time permits, present an approach that gives a hope to improve the lower bound and make it asymptotically equal to the upper bound from the beginning of the talk.
  • Junior Number Theory Seminar
19 May 2014
16:00
Javier Fresán
Abstract
At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are always products of values of the gamma function at rational numbers, with exponents determined by the Hodge decomposition. I will explain a proof of an alternating variant of this conjecture for the cohomology groups of smooth, projective varieties over the algebraic numbers acted upon by a finite order automorphism.
  • Junior Number Theory Seminar
5 May 2014
16:00
Simon Myerson
Abstract
Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong. I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.
  • Junior Number Theory Seminar

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