**Invited speakers:**

*Britta Peis (U Aachen) - Sensitivity analysis for convex optimisation over polymatroids with applications to game theory*

Based on joint work with Tobias Harks and Max Klimm. We consider situations in which resources of limited capacity are commonly used by several participants. Such situations occur, for example, in scheduling, where jobs need to be assigned to machines, in routing, where traffic needs to be sent between certain origin-destination pairs through a network, or in broadcasting, where messages are sent along spanning trees in some undirected graph. We distinguish between the associated optimisation problem, in which a central authority decides on a feasible assignment with the goal

to minimise the resulting total cost of all participants, and its game-theoretic variant, in which each participant selfishly decides on an assignment with the goal to minimise his/her individual cost.

Throughout the talk, we will investigate the impact of changes in the input parameters, like resource capacity or demand, towards optimal solutions and equilibria. As it turns out, as long as the underlying feasibility regions form polymatroids, i.e., can be described by submodular functions (like in scheduling and broadcasting), almost everything behaves nicely: (i) small changes of the input parameter ensure fast re-optimization and new optimal solutions close to the old ones, (ii) the associated games admit equilibrium states, and (iii) anomalies like the Braess paradox cannot occur.

*Alexander May (U Bochum) - Recent advances in decoding random binary linear codes with implications to cryptography*

We will survey recent advances for decoding random linear codes, especially the so-called Representation technique and the use of Hamming Nearest Neighbor search. We will also discuss potential implications for the secure parameter choice of LPN and McEliece cryptosystems, both classically and quantumly.

*Wadim Zudilin (RU Nijmegen) - Hypergeometric motives for rigid hypergeometric Calabi--Yau threefolds*

In 2003, Fernando Rodriguez-Villegas conjectured fourteen congruences modulo p^3 that relate hypergeometric sums truncated at p-1 to the Fourier coefficients a(p) of weight 4 modular forms. Such "supercongruences" are now understood as particular instances of hypergeometric motives (HGMs). In my talk I will review some ingredients of the theory of HGMs and illustrate its features on the fourteen examples of the underlying rigid Calabi--Yau threefolds. I will further outline some ideas in the proofs of Villegas's conjectures given recently in my joint work with Ling Long, Fang-Ting Tu and Noriko Yui.

**Special afternoon "Diophantine problems and the Hardy-Littlewood method" on December 1:**

*Timothy Browning (U Bristol) - Rational curves on hypersurfaces and a geometric circle method*

Thanks to work by Ellenberg, Venkatesh and their collaborators, it has been known for several years that information about the cohomology groups of appropriate varieties over finite fields can be used to answer long-standing questions in analytic number theory over function fields. In an attempt to reverse the flow I will discuss the geometry of the space of rational curves on hypersurfaces using methods that are inspired by the Hardy-Littlewood circle method.

*Roger Heath-Brown (Oxford U) - Asymptotics for rational points on quadric hypersurfaces*

If Q(x_1,...,x_n) is an integral quadratic form, one can give an asymptotic formula for the number of integral zeros Q=0 in a box max|x_i|\le B, valid when B goes to infinity. In this talk we will be interested in the question - How large must B be compared to the height ||Q|| of the form, before this asymptotic formula takes effect? The case of diagonal forms in 4 variables is important for an application in joint work with Browning, and we will describe how a particularly good range for B can be obtained in this special situation.

*Pankaj Vishe (U Durham) - Quartic forms in 30 variables*

Given an integral homogeneous polynomial F, when it contains a rational point is a key problem in Diophantine Geometry. A variety is supposed to satisfy Hasse Principle if it contains a rational zero in the absence of any local obstructions. We will prove that smooth quartic (deg F=4) hypersurfaces satisfy the Hasse Principle as long as they are defined over at least 30 variables. The key tool here is employing a revolutionary idea of Kloosterman in our setting. The talk will start with an overview of Hasse principle and the circle method and Kloosterman's related work. This is a joint work with Oscar Marmon (Lund).

**Contributed speakers:**

*Garnet Akeyr (UL) - Higher-dimensional analogues of graphs for studying semistable varieties*

A natural object in the study of semistable curves is a dual graph, in which vertices correspond to irreducible components and edges to points of intersection. In this talk I will discuss the notion of a generalized Delta-complex, as defined in work by Chan, Galatius, and Payne. These are natural generalizations of objects in combinatorial topology, and in the best circumstances admit geometric realizations that are CW-complexes. I will discuss how these provide a good generalization of the dual graph of semistable curves, as well as difficulties that arise when one has non-trivial monodromy action on a related space. I will conclude by mentioning how maps between these dual complexes encode how the fibres of a semistable morphism change under specialisation/generisation.

*Arthur Bik (U Bern) - Euclidean distance degrees of orthogonally invariant varieties*

The Euclidean distance degree of an affine variety X in a complex vector space V counts the number of critical points on X of the distance to a sufficiently general point of V. Drusvyatskiy, Lee, Ottaviani and Thomas proved that the ED degree of an orthogonally invariant matrix variety X equals the ED degree of the set of diagonal matrices in X. In this talk, we generalize this result to stable varieties in a certain class of spaces with an orthogonal group action, called polar representations.

*Jan-Willem van Ittersum (UU) - Quantitative results on Diophantine equations in many variables*

A classical article by Birch (1962) implies that a system of polynomials of the same degree and with integer coefficients satisfies the smooth Hasse principle if the number of variables is large compared to the Birch’ singular locus. We discuss a quantitative version of this work, yielding asymptotics (in terms of the coefficient of maximal modulus of these polynomials) for the number of integer zeros of this system within a growing box. An application is quantitative strong approximation: assuming the existence of local zeros, we give an upper bound on the smallest non-trivial integer zero provided the variety corresponding to the system of polynomials is non-singular.

*Peter Koymans (UL) - Ternary Goldbach for Artin primes*

Fix an integer g which is neither a square nor -1. Artin's conjecture states that g is a primitive root modulo infinitely many primes. Currently this is known only under the assumption of GRH. In this talk I will discuss recent work with C. Frei and E. Sofos, where we prove, conditional on GRH, an asymptotic estimate for the number of representations of an odd integer n as the sum of three primes all with g as primitive root.

*Julian Lyczak (UL) - Explicit Brauer-Manin obstructions of order 5*

The simplest case in which a set of equations does not have an integral solution is when no solutions exist modulo a given integer or over the rationals. If this however is not the case then there are other possibilities to consider. For example, one can look at the Brauer group which is a group associated to this specific set of equations. Each element of this group can give a reason why there are no solutions. All known examples of such a Brauer-Manin obstruction are given by elements of order 2 or 3 in the respective Brauer groups. We will exhibit the first known example of such obstructions coming from elements of order 5. Geometrically speaking we will describe log K3 surfaces which have a Brauer-Manin obstruction of order 5 to the existence of integral points.

*Adelina Mânzăţeanu (U Bristol) - Rational curves on cubic hypersurfaces over F_q*

Using a version of the Hardy – Littlewood circle method over F_q(t), one can count F_q(t)-points of bounded degree on a smooth cubic hypersurface X ⊂ P^{n−1} over F_q. Moreover, there is a correspondence between the number of F_q(t)-points of bounded height and the number of F_q-points on the moduli. space parametrising rational maps of degree d on X. In this talk I will give an asymptotic formula for the number of rational curves defined over F_q on X passing through two fixed points, one of which does not belong to the Hessian, for n ≥ 10, and q and d large enough. Further, I will explain how to deduce results regarding the geometry of the space of such curves.

*Erik Visse (UL) - Counting fibres with rational points*

We study a family of conics over projective space where we are interested in the number of points in the base up to bounded height whose fibres have arational point. In the early 1990s Serre found an upper bound when the base is the projective line. Since then there have been many results proving upper and lower bounds for more general fibrations. In recent joint work with Efthymios Sofos we return to the case of conics and we restrict ourselves to a family of a specific shape. We arrive at an asymptotic expression where we also give a description of the leading constant that is good enough to prove the Hasse principle. Our proofs make use of the circle method.

*Marieke van der Wegen (UU) - Recognizing hyperelliptic graphs in polynomial time*

Based on analogies between Riemann surfaces and graphs, Baker and Norine introduced a chip-firing game on multigraphs, leading to the introduction of a new graph parameter, so-called divisorial gonality. This parameter is related to treewidth, multigraph algorithms and number theory. We consider so-called hyperelliptic graphs (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigraphs, showing that we can recognize hyperelliptic graphs in O(n log n+m) time, where n is the number of vertices and m the number of edges of the multigraph. (Joint work with Jelco M. Bodewes, Hans L. Bodlaender and Gunther Cornelissen).