Discrete, Interactive and Algorithmic Mathematics, Algebra and Number Theory
Organizer & Chair: Robin de Jong (Universiteit Leiden)
Waaier 3, 14:15 - 16:15
The generalized Fermat equation
Sander Dahmen (Universiteit Utrecht)
In this talk we consider the equation xp + yq = zr 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.
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.
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 > d2 variables has a non-trivial zero in Kn. 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).
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)