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 MordellWeil rank r of the Jacobian of C is less than g, the ChabautyColeman method can be used to find these rational points through the construction of certain padic integrals. I will describe a moderate extension of these results to the case when r = g. The main tool is the theory of padic height pairings, which give rise to padic 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 Lfunctions 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 Lfunctions. 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(θ_{v}) where v runs over the places of good reduction of E/Fq(t) and ±θ_{v} are the arguments to the inverse roots of the numerator of the zeta function of the reduced elliptic curve E_{v} over the residual field at v. An important feature of this geometric setting is that the conjectures on zeros of Lfunctions 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 nonempty. 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 AndréOort 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 f_{q}, q=p^{h}, is the set of vectors of a kdimensional subspace of the ndimensional vector
space V(n,q) over f_{q}. 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:
If C reaches the Singleton bound, then nk=d1. By the fundamental theorem of linear codes, every d1 columns of the parity check matrix of C are linearly independent. Hence, its columns represent a set of vectors in V(nk,q) with the property that every subset of size nk is a basis, and vice versa. Such a set of columns is called an arc. The following statement is known as the MDSconjecture.
