**Invited speakers: **

*Ted Chinburg (U Penn): *Crypto Capacity Theory

Suppose f(x) is a monic integral polynomial of degree d and that N is a positive integer. With cryptographic applications in mind, Coppersmith showed in the 1990's that there is a polynomial time algorithm for finding all integers r such that f(r) is congruent to 0 mod N and |r| < N^{1/d}. In this talk I will discuss how Coppersmith's method is in fact a baby case of adelic capacity theory but with the additional input of the LLL theorem. This leads to a new approach to many results of this kind as well as to new variants of capacity theory. This is joint work with N. Heninger and Z. Scherr.

*Francois Escriva (VU Amsterdam): *Frobenius lifts, cup products, and point counting

Since Kedlaya published his algorithm for point counting on hyperelliptic curves in 2001, most research in point counting has focused on extending his algorithm to more general classes of curves. In this talk, we first present his algorithm, together with some necessary notions of p-adic analysis. Then we explain the new ideas and techniques developed in the preprint arxiv:1306.5102 (joint work with Amnon Besser and Rob de Jeu), which apply to any curve for which some auxiliary data are given.

*Antoine Joux (Paris):*

*David Lubicz (Rennes): *AGM, application to point counting and generalisations

An algorithm of Mestre uses a p-adic version of the AGM in order to count rational points on an elliptic over a finite field of characteristic 2. In this talk, we will give a brief survey of Mestre's algorithms and subsequent developments. Then we will present more recent results on generalisations of the AGM for abelian varieties of higher dimension.

*Jan Tuitman (Leuven): *Some recent developments in p-adic point counting.

First, I will speak about an extension of Kedlaya's algorithm to a very general class of curves (potentially any curve) introduced in my recent preprint. This will include a demonstration of my implementation of this algorithm (the pcc_p and pcc_q MAGMA packages that can be found on my webpage). Second, if time permits, I will talk about the work of Harvey and Harvey-Sutherland on p-adic point counting in average polynomial time on hyperelliptic curves.

*Laurence Wolsey (Louvain-la-Neuve):* Cutting Planes and Extended Formulations for Single and Multi-Vehicle Inventory Routing Problems

The Inventory Routing Problem (IRP) involves the distribution of one or more products from a supplier to a set of customers over a discrete planning horizon. Each customer has a known demand to be met in each period and can hold a limited amount of stock. The product is shipped through a distribution network by a fleet of vehicles of limited capacity.

The version treated here, the so-called Vendor Managed Inventory Routing Problem (VMIRP) is the Inventory Routing problem arising when replenishment policies are decided in advance. We consider two replenishment policies. The first is known as Order-up (OUP): if a customer is visited in a period, then the amount shipped to a client must bring the stock level up to the upper bound. The latter is called Stock Upper Bound (SUB): the stock level in each period cannot exceed the upper bound. The objective is to find replenishment decisions minimizing the sum of the storage and of the distribution costs.

VMIRP contains two important subproblems: a lot-sizing problem for each client and an (almost) classical routing problem. First we introduce some new inequalities for the single period routing problem, variants that take inventory into account and multiperiod extensions. Secondly we introduce reformulations of OUP and SUB derived from the single-item lot-sizing substructure under both constant and time-varying demands. Computational results on benchmark instances with a single product and single and multiple vehicles are presented.

This is joint work with Pasquale Avella and Maurizio Boccia.

**Contributed speakers:**

*Krzysztof Dorobisz*: Deformations of finite group representations

I will talk about the problem of lifting linear group representations over fields to representations over certain types of local rings. During last year's Symposium I discussed the question which rings can represent the associated deformation functors. As a follow-up, this time I want to address a similar problem, but restricting to representations of finite groups. Surprisingly, this slight modification requires developing completely new techniques and leads to different results.

*Sam van Gool:*

*Ariyan Javanpeykar:* Integral points on moduli stacks

Faltings proved the finiteness of the set of integral points on the moduli stack of abelian varieties. In a joint work with Daniel Loughran, we investigate the finiteness of the set of integral points on moduli stacks of complete intersections. It turns out that for complete intersections of small Hodge level, similar finiteness statements can be proven.