Solving systems of linear equations Ax = b is easy, but how can we solve such a system when given a "noisy" version of b? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties. The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.
Let C be a curve over the rationals of genus g at least 2. By Faltings' theorem, we know C has finitely many rational points. When the Mordell-Weil rank r of the Jacobian of C is less than g, the Chabauty-Coleman method can be used to find these rational points through the construction of certain p-adic integrals. I will describe a moderate extension of these results to the case when r = g. The main tool is the theory of p-adic height pairings, which give rise to p-adic double integrals that allow us to find integral points on curves. In particular, I will discuss how to carry out this ``quadratic Chabauty'' method on hyperelliptic curves over number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).
Chebychev was the first to point out that for ``most'' real numbers x the number of prime numbers up to x and congruent to 3 mod 4 exceeds that of primes up to x and congruent to 1 mod 4. This kind of phenomenon and its generalizations have in the recent years been called Chebychev's bias. In the 1990's Rubinstein and Sarnak gave a general framework to study that question. They notably emphasized the relevance of related Dirichlet L-functions and remarked that much can be deduced about Chebychev's bias if one is willing to to believe that both the Riemann Hypothesis and a strong form of simplicity of the zeros holds for these L-functions. The aim of the talk is to present an analogous study in the setting of elliptic curves over the rational function field Fq(t). Given such an elliptic curve E/Fq(t) we will see how to study sums of type ∑deg v<x cos(θv) where v runs over the places of good reduction of E/Fq(t) and ±θv are the arguments to the inverse roots of the numerator of the zeta function of the reduced elliptic curve Ev over the residual field at v. An important feature of this geometric setting is that the conjectures on zeros of L-functions in the classical case can often be replaced by theorems. (Joint work with B. Cha and D. Fiorilli.)
Let V be an an affine algebraic variety defined over ℚ alg ∩ ℝ, and suppose that V (ℝ) is non-empty. Then there exists a point in V (ℚ alg ∩ ℝ) whose height is bounded by a polynomial in terms of the heights of the coordinates of a set of equations defining V . In this talk, I will discuss the application of the above fact in the proof of the André-Oort conjecture for general Shimura varieties assuming the Generalised Riemann Hypothesis. I will also discuss the proof of the above fact, which is mostly elementary with one application of the theory of Chow forms.
A linear [n,k,d]-code C over the finite field fq, q=ph, is the set of vectors of a k-dimensional subspace of the n-dimensional vector
space V(n,q) over fq. The Hamming distance between two codewords is the number of positions in which they differ. The minimum
distance d is the minimum of the distances between all pairs of codewords of C. The Singleton bound for linear codes is the following
relation between the parameters k,n and d of a linear code:
This theorem was obtained by further elaborating on essential ideas found in Ball 2012, in which the MDS-conjecture was shown for
r < p+1. In this talk we report on recent joint work with Simeon Ball and Ameera Chowdury on the MDS conjecture. The talk will include the essential
parts of Ball 2012 and Ball and de Beule 2012, together with the ideas of Ameera Chowdury that lead to a short proof of the theorem so far
which is based on the use of inclusion matrices. We will particularly focus on the connection between the rank of the matrix, the weight of the vectors in its columnspace and
the extendability of an arc, and how evidence based on computational results indicates that this connection might lead to an improvement of
the theorem. Finally, if time permits, we will also discuss the connection of this problem with algebraic hypersurfaces.