Number Theory Seminar
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Thu, 20/01/2011 16:00 |
Jonathan Pila (Oxford) |
Number Theory Seminar  |
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Thu, 27/01/2011 16:00 |
Vladimir Dokchitser (Cambridge) |
Number Theory Seminar |
L3 |
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Thu, 03/02/2011 16:00 |
Jacob Tsimerman (Princeton University) |
Number Theory Seminar |
L3 |
We discuss the following question of Nick Katz and Frans Oort: Given an
Algebraically closed field K , is there an Abelian variety over K of
dimension g which is not isogenous to a Jacobian? For K the complex
numbers
its easy to see that the answer is yes for g>3 using measure theory, but
over a countable field like  new methods are required. Building on
work
of Chai-Oort, we show that, as expected, such Abelian varieties exist for
 and g>3 . We will explain the proof as well as its connection to
the
Andre Oort conjecture. |
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Thu, 10/02/2011 16:00 |
Ben Green (Cambridge) |
Number Theory Seminar |
L3 |
| I will introduce the notion of a nilsequence, which is a kind of "higher" analogue of the exponentials used in classical Fourier analysis. I will summarise the current state of our understanding of these objects. Then I will discuss a variety of applications: to solving linear equations in primes (joint with T. Tao), to a version of Waring's problem for so-called generalised polynomials (joint with V. Neale and Trevor Wooley) and to solving certain pairs of diagonal quadratic equations in eight variables (joint work with L. Matthiesen). Some of the work to be described is a little preliminary! | |||
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Thu, 17/02/2011 16:00 |
Jan Denef (Leuven) |
Logic Seminar Number Theory Seminar |
L3 |
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Thu, 17/02/2011 16:00 |
Jan Denef (Leuven) |
Logic Seminar Number Theory Seminar |
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We will sketch a new proof of the Theorem of Ax and Kochen that any projective hypersurface over the p-adic numbers has a p-adic rational point, if it is given by a homogeneous polynomial with more variables than the square of its degree d, assuming that p is large enough with respect to the degree d. Our proof is purely algebraic geometric and (unlike all previous ones) does not use methods from mathematical logic. It is based on a (small upgrade of a) theorem of Abramovich and Karu about weak toroidalization of morphisms. Our method also yields a new alternative approach to the model theory of henselian valued fields (including the Ax-Kochen-Ersov transfer principle and quantifier elimination). |
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Thu, 03/03/2011 16:00 |
Fred Diamond (King's College London) |
Number Theory Seminar |
L3 |
