Organizers: Anders Södergren, Christian Johansson.

## Past seminars

**11.12.2018 - Arash Ardehali (Uppsala):**Microscopic structure of black holes from asymptotic analysis of special functions.

**21.11.2018 - Brandon Williams (TU Darmstadt):**Hilbert modular forms and Borcherds products.

Abstract: I will talk about how Borcherds products can be used to compute graded rings of Hilbert modular forms.

**14.11.2018 - Pär Kurlberg (KTH):**Class numbers and class groups for definite binary quadratic forms.

Abstract: Gauss made the remarkable discovery that the set of integral binary quadratic forms of fixed discriminant carries a composition law, i.e., two forms can be "glued together" into a third form. Moreover, as two quadratic forms related to each other via an integral linear change of variables can be viewed as equivalent, it is natural to consider equivalence classes of quadratic forms. Amazingly, Gauss' composition law makes these equivalence classes into a finite abelian group - in a sense it is the first abstract group "found in nature". Extensive calculations led Gauss and others to conjecture that the number h(d) of equivalence classes of such forms of negative discriminant d tends to infinity with |d|, and that the class number is h(d)=1 in exactly 13 cases: d is in { -3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163 }. While this was known assuming the Generalized Riemann Hypothesis, it was only in the 1960's that the problem was solved by Alan Baker and by Harold Stark. We will outline the resolution of Gauss' class number one problem and survey some known results regarding the growth of h(d). We will also consider some recent conjectures regarding how often a fixed abelian group occur as a class group, and how often an integer occurs as a class number. In particular: do all abelian groups occur, or are there "missing" class groups?

**31.10.2018 - Martín Sombra (ICREA & Universitat de Barcelona):**

**The zero set of the independence polynomial of a graph.**

**10.10.2018 - Jonathan Nilsson (Chalmers/GU):**Representation theory for Lie algebras of vector fields.

**26.09.2018 - Christian Johansson (Chalmers/GU):**Mod p completed cohomology of locally symmetric varieties.

Abstract: This talk is about singular cohomology of (certain) manifolds, with coefficients in F_p (p some fixed prime). More precisely, the manifolds in question are certain locally symmetric varieties. I will discuss how one can prove that the direct limit of such cohomology groups vanish above the middle degree as one goes up a certain tower of locally symmetric varieties. This confirms many cases of a conjecture of Calegari and Emerton, which is of central importance in the p-adic part of Langlands program. This is joint work in progress with David Hansen.

**Rainer Dietmann (Royal Holloway, University of London)**: Lines on cubic hypersurfaces.

**31.05.2018 - Chunhui Liu (Kyoto University):**Counting rational points in arithmetic varieties by the determinant method.

Abstract: By the slope method in Arakelov geometry, we can construct a family of hypersurfaces which cover the rational points of bounded height on an arithmetic variety but don't contain the generic point of this variety. By estimating some invariants of Arakelov geometry, we can control the number and the maximal degree of this family of auxiliary hypersurfaces explicitly. In this talk, I will explain the method of studying the problem of counting rational points by the approach of Arakelov geometry.

**Metaplectic covers of Kac-Moody groups and Whittaker functions.**

29.05.2018 - Manish M. Patnaik (University of Alberta):

29.05.2018 - Manish M. Patnaik (University of Alberta):

**- Will Sawin (ETH-ITS): The circle method and free rational curves on hypersurfaces.**

18.05.2018

18.05.2018

Abstract: In joint work in progress with Tim Browning, we study a system of two of Diophantine equations over F_q[t], and use the circle method to show a relationship between their numbers of solutions. As a consequence, we bound the dimension of the singular locus of the moduli space of rational curves on a smooth projective hypersurface. I will explain how these problems are related and what techniques we use to get the best bound.

**26.04.2018 - Sofia Tirabassi (University of Bergen):**Fourier--Mukai partners of Enriques and bielliptic surfaces in positive characteristic.

Abstract: We show that Enriques and bielliptic surfaces in positive characteristic do not have any non-trivial Fourier--Mukai partners. This is joint work with K. Honigs and M. Lieblich.

24.04.2018 - Wanmin Liu (IBS Center for Geometry and Physics): Classification of full exceptional collections of line bundles on three blow-ups of P^3.

Abstract: A fullness conjecture of Kuznetsov says that if a smooth projective variety X admits a full exceptional collection of line bundles of length l, then any exceptional collection of line bundles of length l is full. In this talk, we show that this conjecture holds for X as the blow-up of P^3 at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on X is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such X. This is a joint work with Song Yang and Xun Yu at Tianjin University.

**12.04.2018 - Dennis Eriksson (Chalmers/GU):** Transfinite diameters in the Berkovich setting.

Abstract: The classical transfinite diameter is a notion of size of a compact set in the complex plane, given by a well-defined limiting process. It generalizes the notion of diameter of a disc. The definition and existence of the limit were further generalized and studied on more general complex manifolds by Berman-Boucksom. In this talk, I will explain how these results can be extended to the non-archimedean/Berkovich setting. A key ingredient is given by Knudsen-Mumfords version of the Grothendieck-Riemann-Roch theorem. This is joint work with Sébastien Boucksom.

**08.02.2018**

**-**

**Håkan Granath (Stockholms universitet):**Heun functions and quaternionic modular forms.

02.02.2018 - Martin Raum (Chalmers/GU): Unifying relaxed notions of modular forms.