**Talk abstracts, DIAMANT/EIDMA Symposium, May 26/27, Hotel Kapellerput, Heeze**

__Invited speakers:__

*Wouter Castryck (KU Leuven) - Newton polygons and curve gonalities*

It is well-known that an irreducible bivariate polynomial

*f*in

**C**

*[x,y]*generically defines a curve whose genus equals the number of

**Z**

^{2}-valued points that are contained in the interior of the Newton polygon NP(

*f*). The central question of this talk is: does there exist a similar combinatorial interpretation for the gonality of the curve defined by

*f*, i.e. the minimal degree of a rational map to the projective line? This is joint work with Filip Cools.

*Douglas Hensley (Texas A&M University) - Computing key parameters of continued fraction dynamical systems*

Many questions about continued fractions boil down to questions about the spectrum of the transfer operator

*L*acting on the space of 'reasonable' functions on [0,1] for the dynamical system given by the state space E=[0,1) -

**Q**and the step function

*T : E -> E*with

*Tx=1/x - [1/x]*. What happens if we think of linear operators as given by matrices, use the polynomials

*(t-1)*as a basis for

^{n}*V*, and try to get the coefficients of the matrix for

*L*and extract eigenvalues and eigenvectors?

One can get away with this, and justify it, and compute several key paramters to high precision in this fashion. The idea works nicely on variants of the classical continued fraction algorithm, including variants for which no clowed form answers to these questions are known. Some of this is the fruit of joint work with Dajani, Kraaikamp, and Masarotto. A particularly difficult case, on the edge of what is possible, is the Hurwitz complex continued fraction.

*Michel Lavrauw - Semifields from skew polynomial rings*

(joint work with John Sheekey) Skew polynomial rings are used to construct finite semifields, and it is shown that these semifields are isotopic to the so-called {\it cyclic} semifields obtained from semilinear transformations. We determine the nuclei of these semifields and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in [1] and implicitly by Dempwolff in [2].

[1] Kantor, William M.; Liebler, Robert A.: Semifields arising from irreducible semilinear transformations.

*J. Aust. Math. Soc.*85 (2008), no. 3, 333--339.

[2] Dempwolff, Ulrich: On Irreducible Semilinear Transformations. Preprint.

*Marco Streng (University of Warwick) - Algebraic values of complex functions*

At first sight, transcendental complex analytic functions have no reason to take algebraic values on algebraic inputs. But there exist surprising examples where this does happen.

The easiest (and perhaps least surprising) example is the function that maps

*z*to exp(

*2πiz*), which at rational numbers

*z = a/b*takes values that are roots of the polynomials

*X*. In fact, the Kronecker-Weber theorem states that these special values exp(

^{b}- 1*2πia/b*) generate the union of all so-called "abelian extensions" of the field

**Q**of rational numbers.

Problem twelve on Hilbert's famous list of 23 problems asks for complex functions that generalize the Kronecker-Weber theorem to abelian extensions of arbitrary number fields. Despite much work on this problem over the 20th century, it is mostly still wide open.

I will give an introduction to this problem, and near the end of the talk, will link it to my own current research.

__Contributed talks:__

*Let*

Samuele Anni (UL) - Images of residual modular Galois representations

Samuele Anni (UL) - Images of residual modular Galois representations

*l*be a prime number, to any mod-

*l*modular form, which is an eigenform for all Hecke operators, is associated a 2-dimensional mod-

*l*representation of the absolute Galois group of the rationals. Our aim is to make an algorithm to compute the image of such a representation, using modular curves and twists between modular forms.

Murat Firat (TU/e) - Vehicle refueling problem with limited resources

Murat Firat (TU/e) - Vehicle refueling problem with limited resources

In our vehicle refueling problem, there is a fixed route on which a series of fuel stations are located. Various amounts of fuel are needed between successive stations. At the stations, limited amounts of fuel can be bought with varying prices. The capacity of the fuel tank may also vary between adjacent fuel stations. The main goal is to reach the final destination with cheapest cost of fuel. The vehicle refueling problem corresponds to an inventory-capacitated lot-sizing problem with zero setup costs, zero inventory holding costs, linear cost functions for production, and non-stationery inventory capacity. The movements of the vehicle between adjacent stations correspond to stages in time and the fuel is the commodity in demand for each stage. The fuel available at stations corresponds to production capacities and the fuel carried in the tank corresponds to keeping some commodity stock for later demands. In the talk an O(n log n) time algorithm for this vehicle refueling problem will be presented.

*Bart Frenk (TU/e) - TBA*

*In a not yet published article Bhargava and Satriano define a generalization of the Galois closure for field extensions to the case of a commutative algebra over a commutative ring. We will present this definition and some of its basic properties and discuss a more general construction.*

Alberto Gioia (UL) - Bhargava's Galois closure

Alberto Gioia (UL) - Bhargava's Galois closure

*Frederik von Heymann (TU Delft) - Ellipsoids to the rescue: finding structure in non-fulldimensional lattices*

Integer linear problems are most commonly solved by first solving the linear relaxation and then applying certain strategies to obtain integer solutions from this. Among these strategies, cutting planes are very popular and successful for a large class of ILPs.

We want to investigate under which circumstances these methods are slower than the size of the problem seems to indicate. One observation is that the presence of equalities can have unwanted effects, as here the feasible region of the problem (and its relaxation) lies in an affine subspace, while the cutting planes also take integer points outside this subspace into account. It therefore seems advisable to explore reformulations of the problem in the subspace. As the lattice restricted to this subspace is not the standard integer lattice, it becomes important to find a good basis for it, which ideally captures the shape of the feasible region, relative to the lattice. It appears that this structure can be described very well by ellipsoids, which I will try to explain in my talk.

Wouter Lueks (RUG) - TBA

Wouter Lueks (RUG) - TBA

*Rene Pannekoek (UL) - Torsion points on elliptic curves over*

**Q**_{p}with additive reductionLet

*p*be a prime and

*E*an elliptic curve over

*with additive reduction. For*

**Q**_{p}*p*at least 11,

*E(*has no

**Q**_{p})*p*-torsion. For

*p*less than 11, we will give examples of

*E*where

*E(*has

**Q**_{p})*p*-torsion. We will give an easy criterion for deciding whether

*E(*has

**Q**_{p})*p*-torsion, given a Weierstrass equation for

*E*with all coefficients in

*p*

**Z**_{p.}

*Guus Regts (CWI) - The rank of edge connection matrices and the dimension of algebras of invariant tensors*

In this talk I will gie a characterization of the reank of edge connection matrices of partition functions of real vertex models. The rank is equal to the dimension of the homogeneous components of a tensor subalgebra, which is the invariant algebra of a subgroup of the orthogonal group. The subgroup is completely determined by the vertex model. The result is analogous to a result of L. Lovasz characterizing the ranks of vertex connection matrices of partition functions of real weighted spin models. This result answers a question of B. Szegedy.

*Dan Roozemond (University of Sydney) - On Lie algebras generated by few extremal elements*

An element

*x*of a Lie algebra

*L*is called extremal if

*[x,[x,y]]*is a scalar multiple of

*x*for all

*y*in

*L*. If

*L*is a Lie algebra of Chevalley type, then the extremal elements are precisely the long root elements, so they are a quite natural object to study. We consider the setup where a Lie algebra

*L*is generated by a small number of extremal elements whose commutation relations are prescribed by a given graph. In particula, for any finite graph

*G*and any field

*K*of characteristic not 2, we consider an algebraic variety

*X*over

*K*whose

*K*-points parametrize Lie algebras generated by extremal elements. Here the generators correspond to the vertices of the graph, and we prescribe commutation relations corresponding to the non-edges of

*G*.

Using extensive computer computations (the algorithms developed may be of independent interest) we found that for all connected undirected finite graphs on at most 5 vertices

*X*is a finite-dimensional affine space. Furthermore, we found that for maximal-dimensional Lie algebras generated by 5 extremal elements

*X*is a point, so that the only Lie algebra that occurs is nilpotent. The latter is the first known case for which

*X*is trivial.

*Stefan van Zwam (CWI) - Fragility in matroid theory*

Matroids are combinatorial structures abstracting the notion of linear dependence. Matroids consist of a finite set called the ground set (think of a set of vectors), and a partition of its subsets into dependent and independent ones. There are two operations to remove an element, called deletion and contraction. These operations generate a partial order, and a matroid

*N*is said to be a minor of

*M*if

*N*is below

*M*in this partial order.

A matroid

*M*is

*N*-fragile if, for every element

*e*of

*M*, either the deletion or contraction of

*e*results in a matroid without minor isomorphic to

*N*. In this talk I will discuss some aspects of fragility, why we are interested, what the big open problems are, and what we know. I will assume that my audience has had no prior exposure to matroid theory.