| Tuesday 30th September |
Jan-Christoph Schlage-Puchta Freiburg
Modular subgroup arithmetic for surface groups | |
| Monday 6th October |
Christian Elsholtz Sums of unit fractions ABSTRACT This is a survey on the solutions of the diophantine equation \frac{m}{n}= \frac{1}{x_1} + \cdots +\frac{1}{x_k}. We use methods from elementary, combinatorial and analytic number theory and some basic group theory. We develop a method that could be useful for other diophantine equations. | |
| Tuesday 21st October |
Christian Elsholtz Additive decompositions of the set of primes ABSTRACT: This is a survey on Ostmann's problem. Ostmann asked whether there exist two sets A and B (with at least two elements each) so that their sumset A+B equals the set of primes, for sufficiently large primes. Using a new version of the large sieve method I can show, that such sets A and B would need to have counting functions of size N^(1/2 +o(1)), whereas previously only a lower bound of N^(o(1)) and an upper bound of N^(1+o(1)) was known. This implies, for example, that the set of primes cannot be decomposed into three such sets. |
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| Monday 27th October |
James McKee A killer exponent for maximal torsion cosets ABSTRACT: Results of Laurent, Bombieri and Zannier, and Schmidt state that for any variety V defined over a number field, the union of all torsion cosets contained in V is in fact contained in a union of finitely many maximal torsion cosets. The first part of this talk will attempt to explain what all these words mean. The finiteness of the number of maximal torsion cosets in V immediately implies the existence of a single "killer" exponent for all these cosets. The second part of the talk will describe an effective version of this result, giving an explicit bound for the size of the killer exponent. This is joint work with Chris Smyth (Edinburgh), motivated by the problem of showing that there exist Salem numbers of every trace. | |
| Monday November 10th |
Igor Shparlinski Quadratic residues and non-residues in the sequence n! | |
| Tuesday 18th November |
James McKee
Salem numbers via interlacing | |
| Monday 24th November |
Igor Shparlinski
Pseudorandom Points on Elliptic Curves Abstract: Period, Distribution and other properties of sequences of the form P_n = P_{n-1} + G (= n G) and P_n = e P_{n-1} = (e^n G) where P_0 = O on elliptic curves. Survey with some sketches of proofs. | |
| Saturday 29th November |
SECANTS at Royal Holloway. | |
| Monday 8th December |
Igor Shparlinski
Arithmetic Functions on Sparse Integers Abstract: We evaluate the average value \sum_{s \in S, s \le x} \phi(s)/s taken over various sets S of integers with restricted g-ary expansions. In particular we settle an open question of Erdos, Mauduit and Sarkozy, and improve another result of Mauduit and Sarkozy. |