**Invited talks:**

*Jop Briët (CWI Amsterdam) - Hardness of Grothendieck Problems*

Grothendieck’s inequality has its origin in an obscure 1953 paper on tensor products of Banach spaces. The inequality not only proved to be fundamental for that field, it also had striking applications in computer science. It implies that natural hard optimization problems admit efficient approximation algorithms whose quality is given by the Grothendieck constant, a still mysterious number associated with the inequality. Recently, Naor, Regev, and Vidick showed that the same is true for the non-commutative extension of the inequality, whose associated constant is known to equal 2. The new results in this talk are about the other side of the coin: hardness of approximation. The Grothendieck constants also turn out to precisely mark the threshold for efficient approximation algorithms, showing that the algorithms implied by the inequalities are also likely to be optimal. The core of our proof is a geometric lemma asserting the existence of an embedding of a finite-dimensional Hilbert space into a normed space of matrices such that that the image of standard basis vectors is longer than that of unit vectors with no large coordinates.

*William Cook (U Waterloo) - The traveling salesman problem *

The traveling salesman problem is easy to state: given a number of cities, along with the cost to travel between each pair of them, find the cheapest way to visit them all and return to your starting point. Easy to state, but difficult to solve. I will use the TSP to discuss mathematics applied to solving specific instances of seemingly intractable computational models.

*Iwan Duursma (U Illinois) - Coding theory for distributed storage and networks*

A main problem affecting data storage in large clusters of disks is the frequent failure of one or more disks. Classic error-correcting codes are inefficient in this new setting. New code constructions allow faulty disks to be repaired using small amounts of data from healthy disks. Except for special cases the optimal allocation of disk space to primary storage and to repair data is unsolved. We discuss the trade-off between maximizing storage space and minimizing repair bandwith, presenting effective algebraic and combinatorial solutions as well as theoretical upper bounds.

*Viresh Patel (UvA) - Connectivity and Hamilton cycles in graphs and directed graphs*

A Hamilton cycle is one of the most basic global structures one might wish to find in a graph. It is a cycle that visits every vertex of the graph exactly once making it the structure of interest, for example, in the famous Travelling Salesman Problem. Much research has been carried out in finding useful graph conditions that guarantee the presence of a Hamilton cycle. In this talk, I consider different measures of how well connected a graph is (such as its expansion) and how this relates to the presence of one or more Hamilton cycles. Along the way, I give a flavour of some of the important notions in extremal combinatorics such as randomness, quasi-randomness, and the celebrated Szemerédi Regularity Lemma.

**Special afternoon "Lattices and automorphic forms":**

*Neil Dummigan (U Sheffield) - L-values and Hecke eigenvalue congruences*

I will look at examples of experimental congruences involving Hecke eigenvalues of classical modular forms, Siegel modular forms of genus 2, and automorphic forms for SO(4,3), the latter calculated by Megarbane using algebraic modular forms on the compact form SO(7). The moduli appear to be factors in critical L-values (suggested by the Bloch-Kato conjecture). I will consider the problems that arise in computing or approximating these L-values to support this, and the relevance of work

of Farmer and Ryan.

*Sebastian Schoennenbeck (U Aachen) - Algorithmic Treatment of Algebraic Modular Forms*

Algebraic modular forms are certain objects in the theory of automorphic forms that are particularly well suited for computations. The space of modular forms is equipped with a natural set of linear operators, the

so-called Hecke algebra, and one of the main problems is to explicitly compute the action of these operators in given examples. After introducing the basic concepts of the theory I will talk about how one can tackle the various tasks arising in this situation in an algorithmic way. In the end I want to explain an alternative way of computing certain Hecke operators by adapting an idea originally due to Venkov.

*Rainer Schulze-Pillot (U Saarbrucken) - Theta correspondence and connections between algebraic automorphic forms and classical modular forms*

I will show how the concept of algebraic automorphic forms arose from the study of linear combinations of theta series of quadratic forms and of the action of Hecke operators on them and put this into the framework of Howe’s dual pair correspondences of automorphic representations. I will then study various approaches to a particular such correspondence, the Yoshida lifting, and some of its applications.

**Contributed talks:**

*Guus Bollen (TUe) - Algebraic equivalence of matroid representations*

I will present an equivalence relation on algebraic representations of matroids, which generalises the notion of linear equivalence of linear representations. Using a mixture of algebraic and geometric techniques, I will discuss properties of this relation. The focus of this talk will be on algebraic equivalence of linear matroid representations.

*Hao Chen (TUe) - Chromatic number of ball packings and the Borsuk conjecture*

A ball packing is a set of balls with disjoint interiors. I will talk about a surprisingly recent problem that deserves more attention: What is the maximum chromatic number for the tangency graph of a ball packing in dimension d? The current upper bound is exponential (w.r.t. the dimension), while the lower bound is linear. When the balls are of the same radius, the problem is the "opposite" of the Borsuk conjecture. Recent progress on the Borsuk conjecture lead to a slight improvement on the lower bound, and the approach makes use of strongly regular graphs.

*Johan Commelin (RUN) - On the Mumford-Tate conjecture for the product of an abelian surface and a K3 surface*

The Mumford-Tate conjecture is a precise way to say that two invariants of an algebraic variety X (over a number field) convey the same information. The two invariants in question are (1) the Hodge structure on the singular cohomology of the complex analytic variety associated with X; and (2) the Galois representation on the l-adic etale cohomology of X. The conjecture fits in a bigger framework of conjectures, like the Hodge conjecture and the Tate conjecture; but the factual evidence is very small. In this talk I will discuss the Mumford-Tate conjecture for the product of an abelian surface and a K3 surface.

*Dino Festi (UL) - Picard lattice of certain double covers of P^2*

A K3 surface is a smooth projective surface with trivial canonical divisor and trivial first cohomology group. A double cover of P^2 ramified along a smooth sextic curve is an example of K3 surface. To every K3 surface it is possible to associate a lattice, called Picard (or Néron-Severi) lattice. The Picard lattice encodes important information about the arithmetic and the geometry of the surface, and it is often not easy to compute. In this talk I am going to show how we computed the Picard lattice of the surfaces in a 1-dimensional family of double covers of P^2 and its Galois module structure. This is joint work with Florian Bouyer, Edgar Costa, Eric Larson, Chris Nicholls, and Mckenzie West.

*Chloe Martindale (UL) - Isogeny graphs*

Isogeny graphs for elliptic curves are graphs whose vertices are given by isomorphism classes of elliptic curves defined over a finite field, and whose edges are given by p-isogenies, where p is a prime. I will explain their structure and give some applications to cryptography, and then give some different approaches for drawing isogeny graphs for genus 2 curves over finite fields. Finally, I will present some results on the structure of isogeny graphs for genus 2 curves over finite fields.

*Mima Stanojkovski (UL) - Intense triples*

Let G be a finite group. A good strategy for understanding the structure of G is that of studying its group of symmetries, Aut(G). Let Int(G) be the subgroup of Aut(G) consisting of those automorphisms sending each subgroup of G to a conjugate. When G is a p-group, Int(G) can be written as the semidirect product of P with C, where P is a (normal) p-subgroup and C is cyclic of order coprime to p. If α is a non-trivial element of C, the triple (p,G,α) is called intense and the structure of G is (surprisingly!) almost completely determined by its nilpotency class.

*Erik Visse (UL) - Bounds on Brauer groups of Kummer surfaces*

Ever since Manin formulated the Brauer-Manin obstruction to the existence of rational points, Brauer groups have become interesing objects in the study of rational points on varieties. In the case of K3 surfaces over number fields, Skorobogatov and Zarhin have proven that these groups are always finite. During the most recent Arizona Winter School we have started working on effectively bounding the size of the Brauer group when the K3 surface under consideration is a Kummer surface. In particular after an effectively computable field extension, we can (so far) give a formulaic bound for the transcendental part of the Brauer group only in terms of the field extension degree and the Faltings height of the associated abelian surface. In this talk I will recall these notions, explain the main idea of Skorobogatov and Zarhin that launched our work and treat our methods. This is joint work with Victoria Cantral Farfan, Yunqing Tang and Sho Tanimoto.