Forthcoming Seminars › Seminar Series

Junior Number Theory Seminar

Mon, 18/01/2010
16:00 		Timothy Trudgian (Oxford) Junior Number Theory Seminar Add to calendar SR1
An Round-Up of the Circle Problem
How many integer-points lie in a circle of radius $ \sqrt{x} $? A poor man's approximation might be $ \pi x $, and indeed, the aim-of-the-game is to estimate
$$P(x) = \sharp\{(m, n) \in\mathbb{Z}: \;\; m^{2} + n^{2} \leq x\} -\pi x,$$
Once one gets the eye in to show that $ P(x) = O(x^{1/2}) $, the task is to graft an innings to reduce this bound as much as one can. Since the cricket-loving G. H. Hardy proved that $ P(x) = O(x^{\alpha}) $ can only possible hold when $ \alpha \geq 1/4 $ there is some room for improvement in the middle-order. In this first match of the Junior Number Theory Seminar Series, I will present a summary of results on $ P(x) $.
Mon, 25/01/2010
16:00 		James Maynard (Mathematical Institute Oxford) Junior Number Theory Seminar Add to calendar SR1
The Hardy–Littlewood Conjectures
Mon, 01/02/2010
16:00 		Damiano Testa (Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
Intersections of two cubics and Artin's conjecture
Suppose that $ C $ and $ C' $ are cubic forms in at least 19 variables over a $ p $-adic field $ k $. A special case of a conjecture of Artin is that the forms $ C $ and $ C' $ have a common zero over $ k $. While the conjecture of Artin is false in general, we try to argue that, in this case, it is (almost) correct! This is still work in progress (joint with Heath-Brown), so do not expect a full answer. As a historical note, some cases of Artin's conjecture for certain hypersurfaces are known. Moreover, Jahan analyzed the case of the simultaneous vanishing of a cubic and a quadratic form. The approach we follow is closely based on Jahan's approach, thus there might be some overlap between his talk and this one. My talk will anyway be self-contained, so I will repeat everything that I need that might have already been said in Jahan's talk.
Mon, 08/02/2010
16:00 		Sebastian Pancratz (Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
Fast reduction in the de Rham cohomology groups of projective hypersurfaces
Let $ X $ be a smooth hypersurface in projective space over a field $ K $ of characteristic zero and let $ U $ denote the open complement. Then the elements of the algebraic de Rham cohomology group $ H_{dR}^n(U/K) $ can be represented by $ n $-forms of the form $ Q \Omega / P^k $ for homogeneous polynomials $ Q $ and integer pole orders $ k $, where $ \Omega $ is some fixed $ n $-form. The problem of finding a unique representative is computationally intensive and typically based on the pre-computation of a Groebner basis. I will present a more direct approach based on elementary linear algebra. As presented, the method will apply to diagonal hypersurfaces, but it will clear that it also applies to families of projective hypersurfaces containing a diagonal fibre. Moreover, with minor modifications the method is applicable to larger classes of smooth projective hypersurfaces.
Mon, 15/02/2010
16:00 		TBA (Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
To Be Announced
Mon, 22/02/2010
16:00 		TBA (Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
To Be Announced
Mon, 22/02/2010
16:00 		James Maynard (Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
Prime gaps
Mon, 01/03/2010
16:00 		(Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
No Seminar

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