Oberseminar Algebra, Zahlentheorie und Diskrete Mathematik

Simplicial arrangements and the geometry of planar cubic curves (online)

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Grünbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points. This is a joint work with Dmitri Panov.

Certain families of K3 surfaces and their modularity

Abstract: We start with a double sextic family of K3 surfaces with four parameters with Picard number 16. Then by geometric reduction (top-to-bottom) processes, we obtain three, two and one parameter families of K3 surfaces of Picard number 17, 18 and 19 respectively. All these families turn out to be of hypergeometric type in the sense that their Picard--Fuch differential equations are given by hypergeometric or Heun functions. We will study the geometry of two parameter families in detail.

 We will then prove, after suitable specializations of parameters, these K3 surfaces will have CM (complex multiplication), and will become modular. This is done starting with one-parameter family establishing the modularity at special fibers, and then applying arithmetic induction (bottom-to-top) processes to multi-parameter families. This is a joint work with A. Clingher, A. Malmendier, and N. Yui.

Terminality of moduli spaces of curves on hypersurfaces via the circle method

Abstract: I will explain how one can use tools from analytic number theory to study moduli spaces of curves on Fano varieties. In particular, I will report on joint work with Matthew Hase-Liu that shows that the moduli space of genus g curves of degree e on a smooth hypersurface of low degree only has terminal singularities, provided e is sufficiently large with respect to g. Using a spreading argument together with a result of Mustata, we reduce the problem to counting points over finite fields on the jet schemes of these moduli spaces. We solve this counting problem by developing a suitable version of the circle method.

A nonabelian circle method

Abstract: We will present a new form of the circle method targeted to counting solutions to equations with variables in non-commutative division algebras, and apply it to the question of how often a sum of squares of quaternions equals zero.

Frieze patterns with constant boundary

Abstract: Frieze patterns with constant boundary arise as a natural intermediate step between ordinary frieze patterns and frieze patterns with coefficients.
Thus one can ask what parts of the classical theory developed by Conway and Coxeter can be generalized to this scenario. By introducing the notion of primitive CC-k-frieze patterns, we are able to obtain a variant of their bijection. With the help of computer generated lists of CC-k-frieze patterns of small heights, we could find a transformation that explains a phenomenon visible in those lists. This transformation leads to the definition of revolving door equivalence. It turns out that all ordinary frieze patterns of the same height over the integers are revolving door equivalent.

Orthogonality of restricted primes with nilsequences

The randomness of arithmetic functions with respect to linear correlations can be measured by Gowers uniformity norms. We show that the von Mangoldt function of primes restricted to a fixed Chebotarev class varies randomly around its average, up to structure arising from congruences to small moduli. By the inverse theory of Green-Tao-Ziegler, we can achieve this by studying correlations with nilsequences. Under GRH, we get analogous results for primes with a prescribed primitive root. This is joint work with Magdaléna Tinková.

High-dimensional cohomology of arithmetic groups

Group cohomology allows one to associate to a group a sequence of algebraic invariants. Even for classical groups like SL_n(Z), computing these invariants leads to very challenging algebraic and combinatorial problems, especially in high dimensions/ cohomological degrees. In recent years, duality approaches have yielded new results about this high-dimensional cohomology for arithmetic groups such as SL_n(Z) or Sp_2n(Z). I will talk about the challenges of these computations, explain the strategy of the new approaches in the example of SL_n(Z) and mention generalisations of these for other types of groups.

Göttingen-Hannover Number Theory Workshop

15:00: Zhizhong Huang (CAS, Beijing)

16:30: Fabian Gundlach (Paderborn)