DIAMANT Symposium Spring 2021

Wild Galois Representations

In this talk we will investigate the Galois action on a certain family of hyperelliptic curves defined over local fields. In particular we will look at curves with potentially good reduction, which acquire good reduction over a wildly ramified “large” extension. We will first clarify what large means and then show how to determine the Galois representation in the case considered.

Elliptic curves and L-functions

The Birch-Swinnerton-Dyer conjecture provides a way of studying rational points on elliptic curves. Curiously, some basic properties of L-functions translate into out-of-reach statements about rational points, and vice versa. I will discuss several unexplained consequences that rational points and L-functions imply about each other.

Multivariate cryptography and the complexity of solving multivariate polynomial systems

The security of multivariate cryptographic primitives relies on the hardness of computing the solutions of multivariate polynomial systems. Since we can compute the solutions of a polynomial system from its Groebner basis, bounds on the complexity of Groebner bases computations provide bounds on the security of the corresponding multivariate cryptographic primitives. After recalling the basics on multivariate cryptography and on linear algebra methods to compute Groebner bases, I will introduce and compare some measures of complexity currently used in cryptography.

The adjoint of a polytope

This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, physics, and geometric modeling. First, we recall the definitions of the adjoint of a polytope given Wachspress in 1975 and by Warren in 1996 in the context of geometric modeling. They defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics (as the numerator of a generating function over all moments of the uniform probability distribution on a polytope), in intersection theory (as the central piece in Segre classes of monomial schemes), and in physics (when studying scattering amplitudes). Thirdly, we show that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3- or elliptic surfaces.

Optimizing a graph-based polynomial from queuing theory

For given integers n and d, both at least 2, we consider a homogeneous multivariate polynomial of degree d in variables indexed by the edges of the complete graph on n vertices and coefficients depending on cardinalities of certain unions of edges.
Cardinaels, Borst, van Leeuwaarden [arXiv:2005.14566, 2020] asked whether this polynomial (which arises in a model of job-occupancy in redundancy scheduling) attains its minimum over the standard simplex at the uniform probability vector. Brosch, Laurent and Steenkamp [arXiv 2009.04510, 2020] showed that the polynomial is convex if d=2 and d=3, implying the desired result for these d.
We use elementary representation theory to show that for fixed d, the polynomial is convex (for all n at least 2) if and only if a constant number of constant matrices (with size and coefficients independent of n) are positive semidefinite. This result is then used in combination with a computer-assisted verification to show that the polynomial is convex for d at most 8.

Generalised Fermat equations with signature (2,5,7)

In this talk we will discuss a method for attacking generalised Fermat equations with signature (2,5,7). We first give an overview of the method which involves Belyi maps, Hunter searches and the computation of p-adic étale algebras. We will then go a bit more in depth on how we compute the p-adic étale algebras.

Galois groups of differential equations

The differential Galois group of a linear differential equation is a linear algebraic group that measures the algebraic relations among the solutions. Similarly to the Galois group of a polynomial, it can be defined as the automorphism group of an appropriate splitting field. The absolute Galois group of a field can be described as the projective limit of the Galois groups of all polynomials with coefficients in that field. Likewise, the absolute differential Galois group of a differential field (i.e., a field equipped with a derivation) can be described as the projective limit of all differential Galois groups of all linear differential equations with coefficients in that differential field. A classical result in the Galois theory of polynomials states that the absolute Galois group of the field of rational functions over an algebraically closed field is a free profinite group. In this talk we will explore a differential analog of this result.

This is joint work with David Harbater, Julia Hartmann and Annette Bachmayr.

Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

The k-th power of a graph G=(V,E), denoted G^k, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This talk will present eigenvalue bounds for the independence number and chromatic number of G^k which purely depend on the spectrum of G, together with a method to optimize them. Some of the new bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general. Infinite families of graphs where the bounds are sharp are presented as well.

This is joint work with A. Abiad, G. Coutinho, M.A. Fiol and B.D. Nogueira.