Tentative programme of the DIAMANT/EIDMA symposium, 26-27 November 2009

Thursday 26

Arrival and coffee 
10:50-11:00  Opening
11:00-11:55  Karen Aardal
Mixed-integer optimization, lattices, and lattice free sets
In mixed integer optimization we maximize a linear function over a polyhedron, with the additional restriction that a subset of the variables are allowed integer values only. We describe how the structure of the underlying lattice can be used to enhance optimization algorithms. We also point to some recent developments in cutting plane theory that naturally lead to the study of lattice free sets.
Arjen Stolk
Discrete tomography for periodic base sets
Consider an (unknown) finite subset of the integer lattice Z2. Such a shape can be `measured' by taking a straight line through the lattice and counting the number of points of the shape that lie on the line. Suppose these numbers are given for all lines in a finite set of directions, what can one say about the unknown shape? This is the problem of discrete parallel X-rays. Interesting variations of this problem arise by restricting the base set from which the shape is chosen to a subset of Z2.
12:30-13:55  Lunch
Fernando Máriode Oliveira Filho Exponential density decay in compact rank one symmetric spaces
  (CWI) Falconer (1986) has shown that if d1, d2, ... is a sequence of distances converging to 0, then the maximum density m(d1, ..., dN) of a subset of Rn that does not contain pairs of points at distance d1, ..., dN from each other goes to 0 as N goes to infinity.
We show a stronger version of this result for compact rank one symmetric spaces, among which the sphere. More specifically, we show that in these spaces the density decays exponentially with N.
(Joint work with Frank Vallentin.)
Gabriele Dalla Torre
The Tate pairing and number fields
The Tate pairing of curves over finite fields is a non-degenerate paring from a subgroup and a quotient group of the divisor class group onto a quotient group of the multiplicative group of the finite field. The goal of this talk is to study the unit-residue group, which is an obstruction to a similar pairing on number fields.
15:00-15:25 Cecilia Salgado
Comparing the rank of the generic and the special fibres on rational elliptic surfaces
We prove, for a large class of rational elliptic surfaces, that there are infinitely many fibres with rank at least equal to the generic rank plus two.
Tea break
Vincent Moulton
A Brief Introduction to Phylogenetic Combinatorics
  (University of East Anglia) It has now been 150 years since Charles Darwin presented his theory on the origin of species asserting that all organisms are related to each other by common descent via a “tree of life”. Since then, biologists have been able to piece together a great deal of information concerning this tree – relying in particular in more recent times on the advent of ever cheaper and faster DNA sequencing technologies. Even so, there remain many fascinating open problems concerning the tree of life and the evolutionary processes underlying it, problems whose solution often require sophisticated techniques from areas such as mathematics, computer-science and statistics.
Phylogenetic combinatorics can be regarded as a branch of discrete applied mathematics concerned with the combinatorial description and analysis of phylogenetic or evolutionary trees and related mathematical structures such as phylogenetic networks and tight spans. In this talk, we present a brief overview of phylogenetic combinatorics, mentioning some classical as well as some new results and directions in this area.

Friday 27

9:00-9:55 Marta Casanellas  
Applying algebraic geometry in phylogenetics
  (Universitat Politècnica de Catalunya) Many statistical models of evolution can be viewed as algebraic varieties. As a consequence, algebraic geometry has a lot to do in phylogenetic reconstruction. Indeed, the generators of the ideal of a statistical model of evolution should allow to determine the tree formed by a set of living species.
We will present methods of phylogenetic inference based on algebraic geometry and we will discuss why algebraic geometry should be considered as a powerful tool for phylogenetic reconstruction. We will also see how an in-depth study of the geometry of the model leads to major improvements on the methods based on algebraic geometry.
Marco Streng Igusa class polynomials
Igusa class polynomials are the genus 2 analogue of the classical Hilbert class polynomial. We explain both notions and discuss the differences between the classical (elliptic) case and the genus 2 case, mostly from a computational perspective.
10:30-11:00 Coffee break
Rene Pannekoek
Parametrizing cubic surfaces and constructing Severi-Brauer surfaces over number fields
Let S/K be a smooth projective cubic surface in P3 over a perfect field. First I review a known necessary and sufficient criterion for the existence of a K-birational map from P2 to S. Also, assuming that no birational map exists, I show that a K-rational map with degree at most 6 still exists under the condition that S(K) is non-empty and K has cardinality at least 37. This has all been known since the 70s and follows from work by Manin and Swinnerton-Dyer. Let now S/K be an explicitly given cubic curface as above, and K a number field. I will then show how to verify in practice the existence of a birational map from P2 to S. Lastly, I will construct two cubic surfaces over Q which satisfy the said criterion but for the existence of a Q-rational point. This comes down to constructing an explicit Severi-Brauer surface over Q, for which I will also outline a method.
11:30-12:25 Ronald Cramer
Towers of Algebraic Functions Fields in Secure Computation
Since the early 1980s towers of algebraic functions fields have played a major role in the theory of error correcting codes. Recently, it has been discovered that towers also have an important bearing on secure multi-party computation. In this talk I will elaborate on this connection.
12:30-13:55  Lunch
14:00-14:25 Filip Cools
A Brill-Noether general graph in every genus
Linear systems on graphs and metric graphs have been introduced by M. Baker and S. Norine in their paper "Riemann-Roch and Abel-Jacobi theory on a finite graph", where they show that these systems obey the counterparts of some well-known properties for linear systems on algebraic curves.  In  "Specialization of linear systems from curves to graphs", M. Baker conjectures that also the counterpart of the Brill-Noether Theory should hold for linear systems on graphs and metric graphs.
In my talk, I will give an explicit sequence of Brill-Noether general graphs in every genus. This is joint work with J. Draisma, E. Mihaylova Robeva and S. Payne.
Matthias Mnich
Chromatic Coding and Subexponential Parameterized Complexity
We study the probabilistic method of Chromatic Coding for the design of subexponential fixed-parameter (FPT) algorithms. This method yields the first subexponential FPT-algorithms for dense parameterized problems such as Feedback Arc Set in Tournaments or Dense Betweenness. Here we consider the Dense Rooted Triplet Inconsistency (Dense RTI) problem which for a dense set R of rooted triplets seeks a distinctly leaf-labeled tree that contains subdivisions of all but k triplets in R as subgraph. We derive a kernel for Dense RTI with a quadratic number of labels and a subexponential FPT algorithm with running time O(n4 + 2O(k^{1/3} log k), where n denotes the number of labels appearing in R.
Gerhard Woeginger
Algorithmic problems under polyhedral norms
The talk discusses some algorithmic problems in Rn that are hard/intractable if distances are measured according to the Euclidean norm, but become easy/easier if distances are measured according to norms with polyhedral unit balls.
16:00 Closing

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