University of Bordeaux

Can we get rigorous answers when computing with real and complex numbers? There are now many applications where this is possible thanks to a combination of tools from computer algebra and traditional numerical computing. I will give an overview of such methods in the context of two projects I'm developing. The first project, Arb, is a library for arbitrary-precision ball arithmetic, a form of interval arithmetic enabling numerical computations with rigorous error bounds. The second project, Fungrim, is a database of knowledge about mathematical functions represented in symbolic form. It is intended to function both as a traditional reference work and as a software library to support symbolic-numeric methods for problems involving transcendental functions. I will explain a few central algorithmic ideas and explain the research goals of these projects.

This talk is concerned with quantitative periodic homogenization in domains with boundaries. The quantitative analysis near boundaries leads to the study of boundary layers correctors, which have in general a nonperiodic structure. The interaction between the boundary and the microstructure creates geometric resonances, making the study of the asymptotics or continuity properties particularly challenging. The talk is based on work with S. Armstrong, T. Kuusi and J.-C. Mourrat, as well as work by Z. Shen and J. Zhuge

Classical modular functions and forms may be evaluated numerically using truncations of the q-series of the Dedekind eta-function or of Jacobi theta-constants. We show that the special structure of the exponents occurring in these series makes it possible to evaluate their truncations to N terms with N+o(N) multiplications; the proofs use elementary number theory and sometimes rely on a Bateman-Horn type conjecture. We furthermore obtain a baby-step giant-step algorithm needing only a sublinear number of multiplications, more precisely O (N/log^r N) for any r>0. Both approaches lead to a measurable speed-up in practical precision ranges, and push the cross-over point for the asymptotically faster arithmetic- geometric mean algorithm even further.

(joint work with William Hart and Fredrik Johansson) ​

Additive combinatorics enable one to characterise subsets S of elements in a group such that S+S has

small cardinality. In particular a theorem of Vosper says that subsets of integers modulo a prime p

with minimal sumsets can only be arithmetic progressions, apart from some degenerate cases. We are

interested in q-analogues of these results, namely characterising subspaces S in some algebras such

that the linear span of its square S^2 has small dimension.

Analogues of Vosper's theorem will imply that such spaces will have bases consisting of elements in

geometric progression.

We derive such analogues in extensions of finite fields, where bounds on codes in the space of

quadratic forms play a crucial role. We also obtain that under appropriately formulated conditions,

linear codes with small squares for the component-wise product can only be generalized Reed-Solomon

codes.



Based on joint works with Christine Bachoc and Oriol Serra, and with Diego Mirandola.