A generalisation of the classical Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, very little is known about the corresponding formula for complete or singular Riemannian manifolds. In this talk, we explain a new Chern-Gauss-Bonnet theorem for a class of manifolds with finitely many conformally flat ends and singular points. More precisely, under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a Chern-Gauss-Bonnet type formula with error terms that can be expressed as isoperimetric deficits. This is joint work with Huy Nguyen. 

Gauss was the first to give a formula for the number of monic irreducible polynomials of degree n over a finite field. A natural problem is to determine the number of such polynomials for which certain coefficients are prescribed. While some asymptotic and existence results have been obtained, very few exact results are known. In this talk I shall present an algorithm which for any finite field GF(q) of characteristic p expresses the number of monic irreducibles of degree n for which the first l < p coefficients are prescribed, for n >= l and coprime to p, in terms of the number of GF(q^n)-rational points of certain affine varieties defined over GF(q). 
The GF(2) base field case is related to the distribution of binary Kloosterman sums, which have numerous applications in coding theory and cryptography, for example via the construction of bent functions. Using a variant of the algorithm, we present varieties (which are all curves) for l <= 7 and compute explicit formulae for l <= 5; before this work such formulae were only known for l <= 3. While this connection motivates the problem, the talk shall focus mainly on computational algebraic geometry, with the algorithm, theoretical questions and computational challenges taking centre stage.

The challenge is to develop an automated process that transforms an initial desired design of turbine rotor and blades in to a close approximation having eigenfrequencies that avoid the operating frequency (and its first harmonic) of the turbine.

Isogenies are algebraic group morphisms of elliptic curves. Let E, E' be two (ordinary) elliptic curves defined over a finite field of characteristic p, and suppose that there exists an isogeny ψ between E and E'. The explicit isogeny problem asks to compute a rational function expression for ψ. Various specializations of this problem appear naturally in point counting and elliptic curve cryptography. There exist essentially two families of algorithms to compute isogenies. Algorithms based on Weierstraß' differential equation are very fast and well suited in the point count setting, but are clumsier in general. Algorithms based on interpolation work more generally, but have exponential complexity in log(p) (the characteristic of the finite field). We propose a new interpolation-based algorithm that solves the explicit isogeny problem in polynomial time in all the involved parameters. Our approach is inspired by a previous algorithm of Couveignes', that performs interpolation on the p-torsion on the curves. We replace the p-torsion in Couveignes' algorithm with the ℓ-torsion for some small prime ℓ; however this adaptation requires some non-trivial work on isogeny graphs in order to yield a satisfying complexity. Joint work with Cyril Hugounenq, Jérôme Plût and Éric Schost.

I will report on the results of my recent work with Dmitri Yafaev (Univeristy of Rennes-1). We consider functions $\omega$ on the unit circle with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions in the BMO norm. We find explicitly the leading term of the asymptotics of the distance in the BMO norm between $\omega$ and the set of rational functions of degree $n$ as $n$ goes to infinity. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.
 

We prove higher differentiability of bounded local minimizers to some degenerate functionals satisfying anisotropic growth conditions. In the two-dimensional case we also study the Lipschitz regularity of such minimizers without any limitation on the exponents of anisotropy.

Tbd