# Minisymposium Mathematics Cluster Diamant

## Discrete, Interactive and Algorithmic Mathematics, Algebra and Number Theory

**Organizer & Chair: Robin de Jong (Universiteit Leiden)**

Waaier 3, 14:15 - 16:15

**14:15-14:45The generalized Fermat
equation**

*Sander Dahmen (Universiteit Utrecht)*

In this talk we consider the equation x^{p} + y^{q} = z^{r} in nonzero
coprime integers x,y,z and integers p,q,r > 1. We will give an overview of
current techniques to attack this equation and describe their successes and
limitations.

**14:45-15:15 Random geometric graphs**

*Tobias Mueller (CWI Amsterdam)*

If we pick n points at random from d-dimensional space (i.i.d. according to some probability measure) and fix an r > 0, then we obtain a random geometric graph by joining two points by an edge whenever their distance is at most r. I will give a brief overview of some of the main results on random geometric graphs and then describe my own work on Hamilton cycles and the chromatic number of random geometric graphs.

**15:15-15:45 Rational points and zero-cycles on
p-adic varieties **

*Arne Smeets (Université Paris-Sud/Orsay, France)*

Let p be a prime number and let K be a p-adic field. Artin’s conjecture
predicts that any homogeneous form of degree d over K in n > d^{2} variables has a
non-trivial zero in K^{n}. Special cases of this conjecture are known, and Ax and
Kochen proved an “asymptotic version”, but a counterexample was constructed by
Terjanian. However Kato and Kuzumaki formulated a modified version of
Artin’s conjecture which, when translated into the language of algebraic
geometry, predicts the existence of zero-cycles of degree 1 on certain p-adic
varieties in projective space - rather than the existence of rational points. They
have proved their conjecture for hypersurfaces of prime degree. Recently,
Heath-Brown proved the Kato-Kuzumaki conjecture for intersections of quadric
hypersurfaces. His result has led to interesting progress on the u-invariant
problem for p-adic function fields. In this talk I will present an overview of the
recent results and their consequences. If time permits, I will also discuss some
interesting open cases of the conjecture (and possible strategies to tackle these
cases).

**15:45-16:15 Grothendieck inequalities for rank
constraint semidefinite programs**

*Frank Vallentin (Technische Universiteit Delft)*

In 1953 Grothendieck worked on the theory of Banach spaces where he proved the ‘fundamental theorem in the metric theory of tensor product’, nowadays called Grothendieck inequality. This inequality is a fundamental and unifying tool in many areas of mathematics and computer science (functional analysis, combinatorics, machine learning, system theory, quantum information theory, numerical linear algebra). With hindsight one can view Grothendieck’s inequality and its proof (which is algorithmic) as the first randomized approximation algorithm based on semidefinite programming. In the talk I want to extend Grothendieck’s inequality so that it can be used to give approximation algorithms for finding ground states of the n-vector model in statistical mechanics. Grothendieck’s inequality itself together with the best known constant (due to Krivine) gives a 0.56 approximation algorithm for the Ising model on the integer lattice. For the three-dimensional Heisenberg model the algorithm achieves a ratio of 0.78.

(based on joint work with Jop Briet, Fernando de Oliveira Filho http://arxiv.org/abs/1011.1754)