20.12.2017, 10.00-11.00 hrs. MV:L14 - Henrik Gustafsson (Stanford University): Fourier coefficients of automorphic forms attached to small representations.

Abstract: In this talk I will discuss Fourier coefficients of automorphic forms on adelic groups with respect to different unipotent subgroups, how they can be classified by nilpotent orbits and how these orbits are related to automorphic representations.

Based on joint work with Daniel Persson, Axel Kleinschmidt, Baiying Liu and Olof Ahlén, I will present theorems for evaluating Fourier coefficients of automorphic forms attached to small automorphic representations of SL(n) with respect to unipotent radicals of maximal parabolic subgroups by reduction to the Casselman-Shalika formula. I will also briefly discuss recent progress for other semi-simple groups based on joint work with Dmitry Gourevitch and Siddhartha Sahi.

06.12.2017 - Daniel Persson (Chalmers/GU): Small automorphic representations.

Abstract: I will start with an introduction to certain aspects of the theory of automorphic representations, with emphasis on the Fourier-Whittaker coefficients of automorphic forms. I will then explain the notion of “small representation” and discuss recent results on minimal and next-to-minimal representations of SL(n). If time permits, I will explain some key ingredients in the proofs, involving wavefront sets and nilpotent coadjoint orbits.

29.11.2017 - Sho Tanimoto (Københavns Universitet): Zero loci of Brauer elements on semi-simple algebraic groups.

Abstract: Dan Loughran recently studied the problem of counting the number of varieties containing a rational point in a family using harmonic analysis on tori. We explore an analogous question for semi-simple groups using automorphic representation theory of semi-simple groups. This is joint work with Dan Loughran and Ramin Takloo-Bighash.

29.11.2017 - Tuomas Sahlsten (University of Manchester): Equidistribution of Laplacian eigenfunctions without number theory.

Abstract: We will give a gentle introduction to the topic of “quantum ergodicity” and review the history and current challenges of the problem. The quantum ergodicity theorem states that on Riemannian surfaces with a “chaotic” geodesic flow, most eigenfunctions of the Laplacian equidistribute spatially in the large eigenvalue limit (semiclassical limit). In this talk, we will present an alternative equidistribution theorem for eigenfunctions where the eigenvalues stay bounded and we take instead sequences of compact hyperbolic surfaces that become large/coarse (thermodynamic limit).

This result is motivated by the recent works of Anantharaman, Brooks, Le Masson, and Lindenstrauss on eigenvectors of the discrete Laplacian on regular graphs, limit multiplicity theory in the geometry of manifolds by DeGeorge-Wallach and more recently by Abert et al., and the level aspect Quantum Unique Ergodicity for Maass forms by Nelson-Pitale-Saha under Grand Riemann Hypothesis. Our approach relies more on dynamical systems, in particular a quantitative ergodic theorem established by Nevo using the spectral transfer principle from representation theory.

This is a joint work with Etienne Le Masson (Bristol).

22.11.2017 - Lars Halvard Halle (Københavns Universitet): Degenerations of Hilbert schemes of points.

Abstract: I will present a 'good' compactification of the relative Hilbert scheme of points associated to the smooth locus of a (simple) degeneration of varieties. This compactification is obtained by GIT-methods, and yields a new, refined, approach to earlier constructions of J. Li and B. Wu.

I will also discuss applications of this to degenerations of irreducible holomorphic symplectic varieties.

This is joint work with M. Gulbrandsen, K. Hulek and Z. Zhang.

15.11.2017 - Lisa Carbone (Rutgers University): A Lie group analog for the Monster Lie algebra.

Abstract: The Monster Lie algebra m, which admits an action of the Monster finite simple group M, was constructed by Borcherds as part of his program to solve the Conway-Norton conjecture about the representation theory of M. We associate the analog of a Lie group G(m) to the Monster Lie algebra m. We give generators for large free subgroups and we describe relations in G(m).

14.11.2017 - Paloma Bengoechea (ETH): Hyperbolic periods of the j-invariant around Markov geodesics.

Abstract: The hyperbolic periods of the j-modular function are integrals of j along geodesics in the hyperbolic plane joining a real irrational quadratic number with its Galois conjugate. They have been the object of various recent works of Duke, Imamoglu and Toth and have been treated as the analog of singular moduli for real quadratic fields. Kaneko conjectured, based on numerical evidence, some very specific behaviours of the periods of j around geodesics that correspond to Markov quadratics. Markov quadratics are those which can be worse approximated by rationals, they give the beginning of the Lagrange spectrum in Diophantine approximation. In this talk, we will explain and prove Kaneko's conjectures.

31.10.2017 - Sebastián Herrero (Chalmers/GU): A p-adic Linnik problem with applications.

Abstract: We will present a p-adic analogue of the classical Linnik problem on the distribution of lattice points with norm tending to infinity. Our result has applications to the distribution of CM elliptic curves over the p-adic complex numbers with supersingular reduction, and to the finiteness of singular moduli that are S-units, for any given finite set of rational primes S. This is joint work with Juan Rivera-Letelier and Ricardo Menares.

10.10.2017 - Olga Balkanova (Chalmers/GU): Moments of L-functions and applications.

Abstract: We shall discuss several problems in number theory related to asymptotic evaluation of moments of L-functions. Particular attention will be paid to the families of Hecke L-functions and symmetric square L-functions associated to primitive cusp forms of large weight.

03.10.2017 - Oleksiy Klurman (KTH): Rigidity theorems for multiplicative functions.

Abstract: In this talk, we describe how one can combine recent breakthroughs by Matomaki, Radziwill and Tao, Szemeredi theorem for long arithmetic progressions together with some new correlation formulas for multiplicative functions developed earlier by the speaker to settle several old open problems about multiplicative functions among which the solution of the Erdo-Coons-Tao conjecture (1957), conjecture of Chudakov (1971) and folklore conjecture on the consecutive values of unimodular multiplicative functions. This is joint work with A. Mangerel.

19.09.2017 - Per Salberger (Chalmers/GU): Diophantine approximation and Diophantine equations.

Abstract: We explain how Chow forms can be used to give precise reformulations of theorems on Diophantine approximation by Faltings and Wüstholz and of theorems on Diophantine equations by Broberg and myself.

09.06.2017 - Dan Petersen (Københavns Universitet): A spectral sequence associated to a stratification, and a conjecture of Vakil-Wood.

Abstract: Given a space with a stratification, there is a well known spectral sequence computing the Borel-Moore homology of the total space in terms of the BM homology of the open strata. I will describe a "dual" spectral sequence, which seems not to have been considered before, that takes as input the BM homology of closed strata and calculates the BM homology of an open stratum. Many familiar spectral sequences arise as special cases. As an application we prove a very general representation stability theorem for configuration spaces of points. This result generalizes a theorem of Church and specializes to prove a conjecture of Vakil and Wood (previously proven by Kupers-Miller-Tran).

08.06.2017 - John Shareshian (Washington University): Noncontractibility of coset posets, and an elementary problem on binomial coefficients.

Abstract: The order complex of a partially ordered set P is the simplicial complex with vertex set P whose k-dimensial faces are totally ordered subsets of P having size k+1. Ken Brown asked whether there is a finite group G such that the order complex of the poset of all cosets of all proper subgroups of G (ordered by conclusion) is contractible. In joint work with Russ Woodroofe, we gave a negative answer to Brown's question. Our proof requires the classification of finite simple groups. In particular, we observed that for every finite simple group of Lie type G, there exists a prime p such that <S,P>=G whenever S is a Sylow 2-subgroup and P is a Sylow p-subgroup of G. This last claim is not true for alternating groups. However, a weaker claim might be true, and this leads to an elementary but apparently challenging problem on prime factors of binomial coefficients. I will discuss the motivation for Brown's question, the structure of our proof, and what we know so far about the binomial coefficient problem.

31.05.2017 - Stefan Lemurell (Chalmers/GU): First zero of L-functions.

Abstract: The talk will discuss the possibility of enumerating all L-functions according to their first critical zero, and whether this enumeration would start with most people's favourite the Riemann zeta-function.

24.05.2017 - James Parks (KTH): The Lang-Trotter Conjecture for two elliptic curves.

Abstract: Fix an integer t and let E be an elliptic curve defined over the rationals without complex multiplication. In 1976, Lang and Trotter gave an asymptotic conjecture for the number of primes up to x such that for a prime p, the trace of the Frobenius automorphism a_p(E)=t with an explicit conjectural constant. In this talk I will discuss the conjectural constant of the Lang-Trotter Conjecture and give a version of this conjecture for two elliptic curves. This is joint work with Amir Akbary.

17.05.2017 - Dennis Eriksson (Chalmers/GU): Theta functions, discriminants and Riemann-Roch.

Abstract: Mumford has shown, using a version of the Riemann–Roch theorem, that certain line bundles associated to a family of curves are isomorphic. The equality of line bundle classes can be seen as a special form of a tautological relation, and the isomorphism is fundamental in arithmetic intersection theory. The isomorphism is, however, abstract, in the sense that it is deduced from general considerations on the geometry of the moduli space of curves.

In this talk, I will give the background on the above material, and show that this relation can be described concretely by classical theta functions (nullwerte) and discriminants of polynomials, and, if time permits, offer some applications. This is joint work with Gerard Freixas i Montplet.

03.05.2017 - Kristian Ranestad (Universitet i Oslo): Divisors in the moduli space of cubic fourfolds.

Abstract: In the talk I shall describe divisors in the moduli space of cubic fourfolds that are not Noether Lefschetz divisors, and how they relate to hyperkähler 4-folds that parametrize the decompositions of the defining cubic form as a sum of ten cubes of linear forms. This is joint work with Claire Voisin.

26.04.2017 - Yiannis Petridis (University College London): Arithmetic Statistics of modular symbols.

Abstract: Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. In joint work with Morten S. Risager we prove these on average using analytic properties of Eisenstein series twisted by modular symbols. We also prove another conjecture predicting the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps.

12.04.2017 - Fabien Pazuki (U. de Bordeaux/Københavns Universitet): Elliptic curves and isogenies.

Abstract: Two elliptic curves E and E' defined over a number field K are isomorphic over the algebraic closure of K if and only if they have the same j-invariant. A natural question is: how is this invariant transformed by general isogenies? We prove a new height bound on the difference of heights of the j-invariants of isogenous elliptic curves, and derive several consequences, for instance bounds for the height of modular polynomials and for Vélu's formulas. If time permits, we will add a remark on Mordell-Weil ranks of elliptic curves.

05.04.2017 - Georgios Dimitroglou Rizell (Uppsala Universitet): Classification of Lagrangian tori inside the complex two-dimensional plane.

Abstract: In a joint work with E. Goodman and A. Ivrii we show that, up to smooth isotopy through Lagrangian tori, there is a unique Lagrangian torus inside the complex two-dimensional plane endowed with its linear symplectic form. The goal is producing a suitable filling of the torus by pseudoholomorphic discs, whose construction involves considering limits of pseudoholomorphic foliations while "stretching of the neck" around the torus.

**22.03.2017 - Anders Södergren (Chalmers/GU):** Sato-Tate equidistribution and zeros of Artin L-functions.

Abstract: In this talk I will survey various aspects of the relation between zeros of L-functions and eigenvalues of random matrices. The focus will be on recent refinements of the Katz-Sarnak heuristics for low-lying zeros in families of L-functions. The main goal of the talk is then to describe results verifying the Katz-Sarnak heuristics for families of Artin L-functions attached to geometric parametrizations of number fields. In particular I will describe the fundamental role played by the Sato-Tate equidistribution in these families.

**17.03.2017 - Eric Mortenson (Max Planck Institute für Mathematik):** Kronecker-type identities and formulas for sums of squares and sums of triangular numbers.

Abstract: We recall Kronecker's identity and review how limiting cases give the representations of a number as a sum of four squares and the representations of a number as a sum of two squares. The two formulas imply respectively Lagrange's theorem that every number can be written as a sum of four squares and Fermat's theorem that an odd prime can be written as the sum of two squares if and only if it is congruent to 1 modulo 4. By considering a limiting case of a higher-dimensional Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number as a sum of three squares. From the Kronecker-type identity, we also deduce Gauss's theorem that every positive integer is representable as a sum of three triangular numbers.

**08.03.2017 - Lionel Lang (Uppsala Universitet):** Tropical degenerations of curves and Jacobians.

Abstract: A family of planar curves C_t is said to converge to a tropical curve C in R^2 if the corresponding family of amoebas A(C_t) converges in Hausdorff distance to C (after some rescaling). Our aim is to understand this convergence abstractly, in terms of the moduli of the underling family of Riemann surfaces. Doing so, we come to an abstract notion of tropical convergence of families in M_g to abstract tropical curves.

This notion allows to keep track of the periods of the Riemann surfaces of the family, unifying two already existing concepts: the Jacobians of algebraic curves and the Jacobians of Tropical curves. This approach has potential applications in classical problems: compactification of moduli spaces, Riemann-Schottky, Brill-Noether.

We will try to introduce every object carefully. In particular, no background in tropical geometry should be required.

**22.02.2017 - Orsola Tommasi (Chalmers/GU):** Structure of tautological rings.

Abstract: A main theme in the study of the cohomology of moduli spaces of curves is the study of the tautological ring, a subring generated by certain geometrically natural classes. For a long time, one of the open questions was whether the tautological ring is a Gorenstein ring, as conjectured by Carel Faber in the case of smooth curves without marked points. In this talk we discuss the definition of the tautological rings, their properties and an approach that allows to detect the existence of non-tautological classes in the cohomology ring of the moduli space of stable curves of genus 2 with sufficiently many marked points. We use this to prove that the Gorenstein conjecture does not hold for these spaces. This is joint work with Dan Petersen (KTH, Stockholm).

**08.02.2017 - Jan Stevens (Chalmers/GU):** Tom and Jerry.

Abstract: Projection from a point of a variety in projective space often gives a birational map to a variety in a smaller space. Unprojection takes a graded ring with a codimension 1 ideal and produces a graded ring of higher embedding dimension. This strategy has been advocated by Miles Reid, in order to study explicit descriptions of Gorenstein codimension 4 rings, such as anticanonical rings of certain Fano 3-folds. For codimension 3 there is a nice structure by the Buchsbaum-Eisenbud theorem, but for higher codimension it is still elusive. I illustrate unprojection methods with the example of cones over (singular) Del Pezzo surfaces of degree 6 and their deformations.

**18.01.2017 - Ulf Persson (Chalmers/GU): **Surfaces in P^3 with curve singularities.

Abstract: 'Curves were given by God, while surfaces where invented by the Devil'. This is a paraphrase of a saying of an Italian geometer - Enriques - struggling with the classification of surfaces. True every one needs to know about curves and the essentials can be succinctly summarized. Not so with surfaces. The surface geometer meets an embarrassment of riches, not unlike the naturalist who studies the diversity of the world of organism. This talk will be given in that spirit. In the past, I did together with Jan Stevens and Stephen Endrass, systematically look at sextics with triple points, which provided an unexpected variety of surfaces in your backyard so to speak, some which I did not expect to encounter, such as Kodairas fake K-3 surfaces (making multiple fibers of elliptic K-3 surfaces). In this talk I will look more specifically at surfaces singular along configurations of lines. As the above simile suggests, the motivation is to find 'new' surfaces unknown to science.