DIAMANT Symposium Spring 2016

Systems of quadratic forms

In this talk we discuss some aspects concerning the arithmetic of systems of quadratic forms. This includes a result on the frequency of counterexamples to the Hasse principle for del Pezzo surfaces of degree four (joint work with J. Jahnel), and a result on the representability of integers by systems of three quadratic forms (joint work with L. B. Pierce and M. M. Wood).

Approximation Algorithms for Traveling Salesmen

For the famous traveling salesman problem (TSP), Christofides’ 1976 algorithm with approximation ratio 3/2 is still the best we know. But recently there has been progress on interesting variants. We will review the state of the art. In particular, we focus on the s-t-path TSP, in which start and end of the tour are given and not identical.

Squarefree discriminants

The question as to whether a positive proportion of monic irreducible integer polynomials of degree n have squarefree discriminant is an old one. (The interest in such polynomials f having squarefree discriminant comes from the fact that in such cases it is immediate to construct the ring of integers in the number field Q[x]/(f(x)), namely, the ring of integers is Z[x]/(f(x)).) An exact formula for the density was conjectured by Jos Brakenhoff and Hendrik Lenstra.

In this pair of talks, we describe recent joint work with Arul Shankar that allows us to determine the exact probability that a random monic integer polynomial has squarefree discriminant, thus establishing the conjectured density of Brakenhoff and Lenstra.

Algebraic matroids and Frobenius flocks

In characteristic zero, any algebraic matroid has a linear representation. In positive characteristic, this no longer holds. As an alternative, we introduce the notion of a Frobenius flock. Each algebraic matroid can be represented by a Frobenius flock. Eventually, we want to use Frobenius flocks to help determine whether or not certain matroids are algebraic over certain fields.

Statistics for the number of points on biquadratic curves over finite fields

A basic question about a curve over a finite field is how many points it has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and compute the distribution law for the concrete family of biquadratic curves.

On the 16-rank of class groups of quadratic number fields, via the negative Pell equation

Let CL(D) denote the class group of the quadratic number field of discriminant D. We will briefly explain the link between the negative Pell equation x^2-2py^2 = -1 and the 16-ranks of the class groups CL(8p), CL(-4p), and CL(-8p). We will then give a density result for the 16-rank of CL(-8p) as p ranges over primes congruent to -1 modulo 4. Finally, we discuss progress on a density result for the 16-rank of CL(-4p).

Jump sets in local fields

We will see how jump sets, easy combinatorial objects, turn out to be very useful when dealing with local fields. We will discuss the following main subjects: (1) The principal units as a filtered module. (2) Jumps of characters. Moreover I will provide hints about how jump sets allow to (a) Predict exact relations between (1) and classical invariants coming from ramification theory. (b) Deduce finer invariants of Eisenstein polynomials, with respect to the ones coming from the Newton polygon of ramification, and have a fast procedure to compute them. (c) Predict a mass formula for (1).

Completely positive semidefinite cone

We investigate structural properties of the completely positive semidefinite cone, consisting of all the n × n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. We will give an overview of the structure of the completely positive semidefinite cone and show how this cone can be used to study problems arising in quantum information theory. Joint work with Sabine Burgdorf and Monique Laurent.

Minimal Models

I will review the minimal model construction, a way to construct from representations of the Virasoro algebra a certain class of conformal field theories related to the coset construction on semisimple Lie algebras and the WZW model, as well as the critical Ising model.

Serre’s problem on the density of isotropic fibres in conic bundles

Let \pi:X\ –> P^1_Q be a non-singular conic bundle over Q having n non-split fibres and denote by N(\pi,B) the cardinality of the fibres of Weil height at most B that possess a rational point. Serre showed in 1990 that a direct application of the large sieve yields N(\pi,B) << B^2(\log B)^{-n/2} and raised the problem of proving that this is the true order of magnitude of N(\pi,B) under the necessary assumption that there exists at least one smooth fibre with a rational point. We solve this problem for all non-singular conic bundles of rank at most 3. Our method comprises the use of Hooley neutralisers, estimating divisor sums over values of binary forms, and an application of the Rosser-Iwaniec sieve.[/et_pb_accordion_item][et_pb_accordion_item title="Wouter Zomervrucht (U Leiden) " _builder_version="4.11.4" global_colors_info="{}" open="off"]

Bhargava’s cube law and cohomology

In his Disquisitiones Arithmeticae, Gauss describes a composition law on (equivalence classes of) integral binary quadratic forms of fixed discriminant. The resulting group is isomorphic to a quadratic class group. More recently, Bhargava explained Gauss composition as a consequence of a composition law on (equivalence classes of) 2x2x2-cubes of integers. The group structure is again related to class groups. Bhargava’s proof is arithmetic. We propose a new proof, obtaining Bhargava’s cube law instead from geometry, with class groups arising as flat cohomology.