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Resonance chains: On Schottky surfaces the resonances of the Laplacian often form interesting chains (See Borthwick-Weich J. Spec. Theor 6(2) (2016) or Weich Comm.Math.Phys. 337(2) (2015))

# Research seminar "Geometric Analysis and Number Theory"

This joint research seminar of the Universities of Aarhus, Bielefeld and Paderborn is devoted to current research questions in the field of geometric analysis and number theory.

The seminar currently takes place as a virtual research seminar via zoom (usually Tuesdays 2pm-3:15pm). If you are interested to participate please contact  Tobias Weich or Benjamin Küster in order to receive the login details.

Organizers:

• Claudia Alfes-Neumann (Bielefeld)
• Michael Baake (Bielefeld)
• Jan Frahm (Aarhus)
• Polyxeni Spilioti (Aarhus)

# Programm

Sommer 2021
 13.04.2021 NN Titel: tba Abstract: tba 20.04.2021 Corina Ciobotaru (IHÉS) Titel: Chabauty limits of subgroups of SL(n,Q_p) Abstract: For a locally compact group G (e.g. G=SL(n,Q_p)) the set S(G) of all closed subgroups of G is a compact topological space with respect to the Chabauty topology. Given a subset X of S(G) it is natural to ask what is the closure of X in S(G) with respect to the Chabauty topology. In a joint work with Arielle Leitner and Alain Valette we study such closure when X is the set of all SL(n,Q_p)-conjugates of the diagonal Cartan subgroup C of SL(n,Q_p). By using the action of SL(n,Q_p) on its associated Bruhat—Tits building and an explicit, well-behaved, replacement of a Lie functor on the closure of our set X, we are able to give a classification of all the limits of X and to compute all of them (up to conjugacy) when n < 5. 27.04.2021 Yann Chaubet (Université Paris-Saclay) Titel: Closed geodesics with prescribed intersection numbers Abstract: On a closed negatively curved surface, Margulis gave the asymptotic growth of the number of closed geodesics of bounded length, when the bound goes to infinity. In this talk, we will investigate such a counting result for closed geodesics of which certain intersection numbers (with a given family of pairwise disjoint simple closed geodesics) are prescribed. 04.05.2021 Léo Bénard (Universität Göttingen) Titel: Asymptotics of twisted Alexander polynomials and hyperbolic volume Abstract: Given a hyperbolic 3-manifold of finite volume M, we compute the asymptotics of the family of twisted Alexander polynomials on the unit circle. We show it grows exponentially as the volume times the square of the dimension of the representation. Joint work with J. Dubois, M. Heusener and J. Porti. The proof goes through the study of the analytic torsion of some compact hyperbolic manifolds obtained by Dehn surgery on M. 11.05.2021 Quentin Labriet (Université de Reims Champagne-Ardenne) Titel: Symmetry breaking operators and orthogonal polynomials Abstract: Symmetry breaking operators are intertwinning operators for the restriction of an irreducible representation. In some cases, these are given by differential operators whose symbol is related some classical orthogonal polynomials. First, I will  describe the example of the Rankin-Cohen brackets which are symmetry breaking operators for the tensor product of two representation of the holomorphic discrete series of SL2(R). I will explain how they are related to Jacobi polynomials, and to the classical Jacobi transform. In a second part, I will discuss some ongoing work on the restriction for representation of the holomorphic discrete series for the conformal group of a tube domain. 18.05.2021 NN Titel: tba Abstract: tba 01.06.2021 NN Titel: tba Abstract: tba 08.06.2021 Benjamin Küster (Universität Paderborn) Titel: Oscillatory integrals in equivariant cohomology Abstract: When a symplectic manifold carries a Hamiltonian action of a compact Lie group, it is an elementary question how the topology of the symplectic quotient can be described in terms of the original Hamiltonian group action. Some tools that allow to partially achieve this goal are the Cartan model of equivariant cohomology and the Kirwan map, which I will briefly introduce in my talk. I will then explain how oscillatory integrals arise in that context and what the difficulties and new features are when one considers singular symplectic quotients. 15.06.2021 Jean-Philippe Anker (Universite d'Orleans) Titel: Bottom of the $L^2$ spectrum of the Laplacian on locally symmetric spaces Abstract: Let $X=G/K$ be a Riemannian symmetric space of noncompact type, let $\Gamma$ be a discrete torsion free subgroup of $G$, let $Y\=\Gamma\backslash G/K$ be the associated locally symmetric space and let $\Delta$ be the  Laplace-Beltrami operator on $Y$. In rank one, a celebrated result, due to Elstrodt, Patterson, Sullivan and Corlette, expresses the bottom of the $L^2$ spectrum of $-\Delta$ in terms of the critical exponent of the Poincar\'e series of $\Gamma$ on $X$. A less precise result was obtained later on by Leuzinger in higher rank. We shall discuss in this talk higher rank analogs of the rank one result, which are obtained by considering suitable Poincar\'e series. This is joint work with Hong-Wei Zhang [arXiv:2006.06473]. 22.06.2021 Gerhard Keller (Universität Erlangen) Titel: B-free systems and their automorphisms Abstract: I plan to start with a brief introduction to B-free dynamical systems (derived from B-free numbers like e.g. the square-free ones). After a short description of the range of dynamical possibilities of these systems, I will concentrate on the two extreme ends of this range, namely on those B-free systems which are "as chaotic as possible" (like the square-free system), and those which are "as close to being periodic as possible" (regular Toeplitz systems). For these two  classes of topological dynamical systems I will report about recent and ongoing progress in understanding the automorphism group of the systems. 29.06.2021 Genkai Zhang (Chalmers University of Technology) Titel: Branching of metaplectic representation of Sp(n, R) under its principal SL(2, R) subgroup Abstract: Any simple split Lie group $G$  has a unique  principal S=SL(2, R) subgroup, by the work of Kostant. We study the branching problem for holomorphic representations  of G=Sp(n, R) under S. A complete decomposition is found for the metaplectic representation of G=Sp(2, R). We use both some classical method of orthogonal polynomials and the geometric tool of Chern connection. 06.07.2021 Agnieszka Hejna (University of Wroclaw) Titel: Harmonic analysis in the rational Dunkl setting Abstract: Dunkl theory is a generalization of Fourier analysis and special function theory related to root systems and reflections groups. The Dunkl operators T_j, which were introduced by C. F. Dunkl in 1989, are deformations of directional derivatives by difference operators related to the reflection group. The goal of this talk will be to study harmonic analysis and Hardy spaces in the rational Dunkl setting. The first part will be devoted to two results: improved estimates of the heat kernel h_t(x,y) of the Dunkl heat semigroup generated by the Dunkl--Laplace operator Delta_k=\sum_{j=1}^NT_j^2, and a theorem regarding the support of Dunkl translations  of L^2 compactly supported functions  (not necessarily radial). This kind of results turn out to be useful tools in studying harmonic analysis in Dunkl setting. We will discuss this kind of applications in the second part of the talk. We will discuss how our tools can be used to for studying singular integrals of convolution type or Littlewood-Paley square functions in the Dunkl setting. This talk is based on the joint articles with J-Ph. Anker and J. Dziubanski. 13.07.2021 Malte Behr (Universität Oldenburg) Titel: Quasihomogeneous Blow-Ups and Pseudodifferential Calculus on SL(n,R) Abstract: We consider the quasihomogeneous blow-up of a submanifold Y in a surrounding manifold with corners X. It generalizes the concept of radial blow-up and revolves around the idea of assigning different weights to functions vanishing at the submanifold Y. In the second part, we consider the hd-compactification of SL(n,R), introduced by Albin, Dimakis, Melrose and Vogan. . We introduce a resolution of this compactification, on which right-invariant differential operators have simple degeneracies at the boundary. We construct an algebra of pseudodifferential operators on X. It is constructed using a resolution of X^2 by a series of quasihomogeneous blow-ups. 20.07.2021 Jorge Vargas (Universidad Nacional de Córdoba) Titel: Restriction of square-integrable representations Abstract: Let $G$ be a semisimple Lie group, and $(\pi,V)$ a irreducible square integrable representation for $G$. Thus, a model for $V$ is the $L^2$-kernel of a elliptic operator  on a fiber bundle over the symmetric space $G/K$ attached to $G$. Let $H$ be a closed reductive subgroup for $G$. We say $\pi$ is $H$-discretely decomposable ( $H$-admissible) if the sum of the closed  $H$-irreducible subspaces in $V$ is dense in $V$, ($H$-admissible if it is $H$-discretely decomposable and the multiplicity of each irreducible factor is finite). We will give criteria for being $H$-$\cdots$ in language of spherical functions as well as in terms of structure of  intertwining operators. We will present   some aspects of branching problems and results in Orsted-Vargas, Branching problems in reproducing kernel spaces, Duke mathematical journal, Vol. 169, 3478-3537, 2020 and some  consequences.
Winter 2020/21
 27.10.2020 NN Titel: tba Abstract: tba 03.11.2020 Beatrice Pozzetti (Universität Heidelberg) Titel: Orbit growth rate and Hausdorff dimensions for Anosov representations Abstract: Anosov representations are a robust and well studied class of discrete subgroups of non-compact simple Lie groups, which generalize many geometric features of hyperbolic manifolds to higher rank locally symmetric spaces. After a gentle introduction motivating the study of this class of groups, I will discuss joint work with Andres Sambarino and Anna Wienhard in which we establish a relation between a suitably chosen growth rate in the symmetric space and the Riemannian Hausdorff dimension of the minimal invariant subset in a flag manifold.
 10.11.2020 Clemens Weiske (Aarhus University) Titel: Symmetry breaking operators and unitary branching laws Abstract: Let $\pi$ be a unitary irreducible representation of a real reductive Lie group $G$. Naturally the restriction $\pi|_H$ to a reductive subgroup $H$ defines a unitary representation of $H$ and decomposes into a direct integral of unitary irreducible $H$-representations. We introduce a method to prove such a direct integral decomposition for unitary representations inside principal series representations, which expands classical Plancherel theorems for homogenous $H$-spaces to branching laws. This is done by an analytic continuation procedure in meromorphic parameters of certain $G$ and $H$-intertwining operators, namely symmetry breaking operators and Knapp--Stein operators. We will study examples where $G$ and $H$ are of real rank one.
 17.11.2020 Frederic Naud (Institut Mathematique Jussieu, Paris) Titel: Spectral gaps of random hyperbolic surfaces Abstract: We will introduce and review the notion of spectral gap related to the Laplace operator, both for compact and geometrically finite hyperbolic surfaces. We will then define a notion of random covers above a given hyperbolic surface and state some new results pertaining to the spectral gap in the large degree regime.
 24.11.2020 Stefan Hante (Universität Halle) Titel: Geometric time integration of a flexible Cosserat beam model Abstract: Cosserat beam models are used in industry as well as in academia to describe long and slender flexible structures like cables, hoses, rotor blades, etc. In my talk I will show how we can exploit the Lie group structure of the configuration space in order to discretize the beam model in space. Furthermore, I will introduce a numerical algorithm that can be used to approximately solve the remaining differential equation on a Lie group as well as touch on its numerical analysis. I will show computational tests as well as some industrial applications.
 01.12.2020 Nils Matthes (University of Oxford) Titel: Meromorphic modular forms and their iterated integrals Abstract: Meromorphic modular forms are generalizations of modular forms which are allowed to have poles. Part of the motivation for their study comes from recent work of Li–Neururer, Pasol–Zudilin, and others, which shows that integrals of certain meromorphic modular forms have integer Fourier coefficients – an arithmetic phenomenon which does not seem to exist for holomorphic modular forms. In this talk we will study iterated integrals of meromorphic modular forms and describe some general algebraic independence results, generalizing results of Pasol–Zudilin. If time permits we will also mention an algebraic geometric interpretation of meromorphic modular forms which generalizes the classical fact that modular forms are sections of a certain line bundles.
 08.12.2020 Mihajlo Cekic (Universite Paris-Sud, Orsay) Titel: First Band of Pollicott-Ruelle resonances in dimension 3 Abstract: If a vector field X generates an Anosov flow, using the theory of anisotropic Sobolev spaces it is possible to define a discrete spectrum of X (as a differential operator), called Pollicott-Ruelle resonances. In this talk we show that if X additionally preserves a contact form (eg. X is the geodesic flow), this spectrum has structure, i.e. there exists a vertical band of resonances. In particular, it follows that the flow is exponentially mixing. Our proof uses semiclassical measures, slightly exotic pseudodifferential calculus and propagation estimates, and a careful analysis near the trapped set. Joint work with Colin Guillarmou.
 15.12.2020 Jan Frahm (Aarhus University) Titel: Conformally invariant differential operators on Heisenberg groups and minimal representations Abstract: On Euclidean space, the Fourier transform intertwines partial derivatives and coordinate multiplications. As a consequence, solutions to a constant coefficient PDE $p(D)u=0$ are mapped to distributions supported on the variety $\{p(x)=0\}$. In the context of unitary representation theory of semisimple Lie groups, so-called minimal representations are often realized on Hilbert spaces of solutions to systems of constant coefficient PDEs whose inner product is difficult to describe (the non-compact picture of a degenerate principal series). The Euclidean Fourier transform provides a new realization on a space of distributions supported on a variety where the invariant inner product is simply an $L^2$-inner product on the variety. Recently, similar systems of differential operators have been constructed on Heisenberg groups. In this talk I will explain how to use the Heisenberg group Fourier transform to obtain a similar picture in this context.
 12.01.2021 Fabian Januszewski (Universität Paderborn) Titel: Tensor-structures in non-commutative harmonic analysis Abstract: Motivated by arithmetic questions, I will discuss approaches involving the symmetric monoidal structure of (g,K)-modules in the representation theory of real reductive groups.
 19.01.2021 Paul Nelson (ETH Zürich) Titel: The orbit method, microlocal analysis and applications to L-functions Abstract: I will describe how the orbit method can be developed in a quantitative form, along the lines of microlocal analysis, and applied to local problems in representation theory and global problems concerning automorphic forms.  The local applications include asymptotic expansions of relative characters.  The global applications include moment estimates and subconvex bounds for L-functions.  These results are the subject of two papers, the first joint with Akshay Venkatesh: https://arxiv.org/abs/1805.07750
 26.01.2021 Jasmin Matz (University of Copenhagen) Titel: Quantum ergodicity of compact quotients of SL(n,R)/SO(n) in the level aspect Abstract: Suppose M is a closed Riemannian manifold with an orthonormal basis B of L2(M) consisting of Laplace eigenfunctions. Berry's Random Wave Conjecture tells us that under suitable conditions on M, in the high energy limit (ie, large Laplace eigenvalue) elements of B should roughly behave like random waves of corresponding wave number. A classical result of Shnirelman and others that M is quantum ergodic if the geodesic flow on the cotangent bundle of M is ergodic, can then be viewed as a special case of this conjecture. We now want to look at a level aspect, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds together with Laplace eigenfunctions f whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of SL(n,R)/SO(n), n>2.
 02.02.2021 Andreas Mono (Universität zu Köln) Titel: On small divisor functions and a construction of polar harmonic mass forms Abstract: Recently, Mertens, Ono, and Rolen studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, and have shadows given by classical Shimura theta functions. Here, we construct a second class of small divisor functions, and prove that these generate the holomorphic part of polar harmonic (weak) Maaß forms of weight 3/2. Specializing to a certain choice of parameters, we obtain an identity between our small divisor function and Hurwitz class numbers. Lastly, we present p-adic congruences of our polar (weak) harmonic Maaß form, when p is an odd prime. This is joint work with Joshua Males and Larry Rolen.
 09.02.2021 Luz Roncal (BCAM- Basque Center for Applied Mathematics) Titel: Hardy's inequalities and an extension problem on $NA$ groups Abstract: We will introduce Hardy's inequality from several points of view and we will turn into its fractional version. One of the approaches to prove fractional Hardy's inequality leads to the study of solutions of the extension problem (that has received much attention in the last few years, especially among the PDE community). It happens that these solutions are related with the eigenfunctions of Laplace-Beltrami operators, which motivates the problem of characterising such eigenfunctions. The aim of this talk is to provide an overview of the above topics and to report recent progress, putting an emphasis on the context of $NA$ groups. Joint work with Sundaram Thangavelu (Indian Institute of Sciences in Bangalore, India).

Sommer 2020
 07.04.2020 N.N. Titel: tba Abstract:
 14.04.2020 Tobias Weich (Paderborn) Titel: Ruelle-Taylor Resonanzen für Anosov Wirkungen höheren Rangs Abstract: In diesem Vortrag soll erklärt werden wie man mittels der Theorie des Taylor Spektrums kommutierender Operatoren sowie mikrolokaler Analysis einer Anosov Wirkung höheren Rangs ein intrinsisches diskretes Resonanzsspektrum zuordnen kann.
 21.04.2020 Lasse Wolf (Paderborn) Titel Quantum-Classical correspondance in higher rank Abstract: tba
 28.04.2020 Beatrice Pozzetti (Heidelberg) Titel: tba Abstract: canceled due to Corona restrictions
 05.05.2020 Margit Rösler (Paderborn) Titel: Riesz-Distributionen im Dunkl-Setting vom Typ A Abstract: Nach einem klassischen Resultat von Gindikin ist eine Riesz-Distribution auf einem symmetrischen Kegel genau dann ein positivies Maß, wenn ihr Index zur Wallachmenge gehört. Gegenstand dieses Vortrags ist ein Analogon dieser Aussage im Rahmen der rationalen Dunkl-Theorie zu Wurzelsystemen vom Typ A. Die verallgemeinerte Wallachmenge ist dabei durch die Multiplizität der Dunkl-Operatoren parametrisiert. Wir erläutern auch die Rolle dieser verallgemeinerten Wallachmenge im Zusammenhang mit der Existenz positiver Vertauschungsoperatoren.
 12.05.2020 Kai-Uwe Bux (Bielefeld) Titel Coarse Topological Group Invariants Abstract: It is hopeless to classify infinite groups up to isomorphism. There are several invariants one can use to chart the vast area inhabited by such groups. I shall discuss several numerical group invariants coming from topology, homology, and geometry: * finiteness properties  (co)homologicial and geometric dimensions  isoperimetric inequalities I shall illustrate these concepts (with a focus on finiteness properites). Groups of matrices provide a good source of examples.
 19.05.2020 Jungwon Lee (Sorbonne Université) Titel Dynamics of continued fractions and conjecture of Mazur-Rubin Abstract: Mazur and Rubin established several conjectural statistics for modular symbols. We show that the conjecture holds on average. We plan to introduce the approach based on spectral analysis of transfer operator associated to a certain skew-product Gauss map and consequent result on mod p non-vanishing of modular L-values with Dirichlet twists (joint with Hae-Sang Sun).
 26.05.2020 Thomas Mettler (Frankfurt) Titel Lagrangian minimal surfaces, hyperbolicity and dynamics Abstract: The Beltrami—Klein model leads to a natural generalisation of hyperbolic surfaces in terms of so-called properly convex projective surfaces. In my talk I will relate these properly convex projective surfaces to certain Lagrangian minimal surfaces. This gives rise to a new class of dynamical systems and I will discuss some of their properties. In parts, this talk is based on joint work with Maciej Dunajski and Gabriel Paternain.
 02.06.2020 Gabriel Rivière (Nantes) Titel Poincaré series and linking of Legendrian knots Abstract: Given two points on a compact Riemannian surface with variable negative curvature, one can consider the lengths of all the geodesic arcs joining these two points and form a natural zeta function associated with these lengths (the so-called Poincaré series). I will explain that this Poincaré series has a meromorphic continuation to the whole complex plane. Then, I will show that the value at 0 is given by the inverse of the Euler characteristic by interpreting this value at 0 as the linking of two Legendrian knots. If time permits, I will explain how the results can be extended when one consider the geodesic arcs orthogonal to two closed geodesics. (joint work with Nguyen Viet Dang)
 09.06.2020 N.N. Titel: tba Abstract:
 16.06.2020 Sven Möller (Rutgers University) Titel: Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra Abstract: Conway, Parker and Sloane (and Borcherds) showed that there is a natural bijection between the Niemeier lattices (the 24 positive-definite, even, unimodular lattices of rank 24) and the deep holes of the Leech lattice, the unique Niemeier lattice without roots. We generalise this statement to vertex operator algebras (VOAs), i.e. we show that all 71 holomorphic VOAs (or meromorphic 2-dimensional conformal field theories) of central charge 24 correspond to generalised deep holes of the Leech lattice VOA. The notion of generalised deep hole occurs naturally as an upper bound in a dimension formula we obtain by pairing the character of the VOA with a certain vector-valued Eisenstein series of weight 2. (This is joint work with Nils Scheithauer.)
 23.06.2020 Michael Voit (TU Dortmund) Titel: Limit theorems for Calogero-Moser-Sutherland particle models in the freezing regime Abstract: Calogero-Moser-Sutherland models are described by some root system and some coupling constants. They are closely related with several random matrix models and, for some cases, with Brownian motions on Grassmann manifolds. In this talk we discuss some limit results when the the coupling constants tend to infinity. Here the limits can be described  in terms of deterministic limit dynamical systems and zeroes of classical orthogonal polynomials. We also present some results when the number of particles tends to infinity.
 30.06.2020 Lennart Gehrmann (Essen) Titel: Big principal series and L-invariants Abstract: By a result of Bertolini, Darmon and Iovita the Orton L-invariant of a modular form equals the derivative of the U_p-eigenvalue of a p-adic family passing through it. In this talk I will give a new, more conceptual proof of the result. One advatange of the method is that it can be generalized to automorphic forms on higher rank groups. This is joint work with Giovanni Rosso. 07.07.2020 Elmar Schrohe (Hannover) Titel: Degenerate Elliptic Boundary Value Problems with Non-smooth Coefficients Abstract: Abstract (as pdf file)
 14.07.2020 N.N. Titel: tba Abstract: tba
Winter 2019/20
 08.10.2019 Nguyen Thi Dang (Universität Heidelberg) Titel: Topological mixing of the Weyl chamber flow Abstract: Let G be a semisimple Lie group without compact factors. Consider A a maximal split torus and a maximal compact subgroup K for which the Cartan decomposition holds. Denote by M the centralizer subgroup of A in K . Let Γ be a discrete subgroup of G, acting properly discontinuously on G/K . In the case of G = PSL(2, R), the right action of A on Γ\G identies with the action of the geodesic ow on the unit tangent bundle of Γ\H2 . The latter has been well studied and satises many chaotic properties such as topological mixing in its non-wandering set. My talk addresses the case when A is of higher dimension and Γ is not a lattice. First I will introduce the main topological property: topological mixing. Then I will state a joint result with Olivier Glorieux, a necessary and sucient condition for topological mixing of actions of one parameter subgroups φt of A on Γ\G/M . Then I will introduce a few key tools: the Benoist limit cone and Hopf coordinates of G/M . Finally, I will give the main ideas behind the proof of the topological mixing Theorem. Time permitting, I will present a generalization of this Theorem for the right action of φt on Γ\G when M is abelian and connected.
 15.10.2019 Maxime Ingremeau (Université de Nice Sophia-Antipolis) Titel: Around Berry’s random waves conjecture Abstract: 40 years ago, the physicist Michael Berry suggested that eigenfunctions of the Laplacian on manifolds of negative curvature should be well described, in the high-frequency limit, by some random function, given by an isotropic monochromatic Gaussian field. After recalling various mathematical interpretations of this conjecture, we will discuss how Bourgain’s arithmetic « derandomization technique » allow to prove the conjecture for eigenfunctions on the two-dimensional torus. We will show that the conjecture holds in a weak form for some families of quasi-modes, namely, long-time evaluated Lagrangian distributions on manifolds of negative curvature.
 22.10.2019 Lasse Wolf (Universität Paderborn) Titel Spectral Asymptotics for kinetic Brownian Motion on hyperbolic surfaces Abstract: The kinetic Brownian motion on the sphere bundle of a Riemannian manifold M is a stochastic process that models a random perturbation of the geodesic flow. If M is a orientable compact constant negatively curved surface, we show that in the limit of infinitely large perturbation the L2-spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold. In addition, we give explicit error estimates for the convergence to equilibrium. The proofs are based on noncommutative harmonic analysis of SL2(R).
 29.10.2019 Fabian Januszewski Titel: Rationalität von L-Werten Abstract: Dieser Vortrag ist der erste in einer Serie von zwei Vorträgen über spezielle Werte von L-Funktionen. In diesem ersten Vortrag über Rationalität von L-Werten wird auf die konzeptionelle Motivation und Resultate eingegangen. Im Kontext letzterer spielen (g,K)-Moduln eine wichtige Rolle.
 05.11.2019 N.N Titel: tba Abstract: tba
 12.11.2019 N.N. Titel tba Abstract: tba
 19.11.2019 N.N. Titel tba Abstract: tba
 26.11.2019 N.N. Titel tba Abstract: tba
 03.12.2019 Markus Schwagenscheidt, Universität zu Köln Titel Generating series involving meromorphic modular forms Abstract: To each non-zero discriminant D one can associate a modular form f_k,D of weight 2k, which is a cusp form if D > 0 and a meromorphic modular form if D < 0. It is well known that the two-variable generating series obtained by summing up the cusp forms f_k,D with D > 0 is modular in both variables. It yields the Kohnen-Zagier kernel function for the Shimura correspondence. Furthermore, the generating series of traces of geodesic cycle integrals of f_k,D for fixed D > 0 is a cusp form of half-integral weight 1/2+k. In this talk we explain how the two-variable generating series of the meromorphic modular forms f_k,D for D < 0 as well as the generating series of traces of cycle integrals of f_k,D for fixed D < 0 can be completed to real-analytic modular forms. Furthermore, we explain some rationality results for the traces of cycle integrals of the meromorphic modular forms f_k,D for D < 0.
 10.12.2019 Julia Budde, Universität Paderborn Titel: Wellenfrontmengen unitärer Darstellungen Abstract: Wir geben eine Einführung in Definition und elementare Eigenschaften von Wellenfrontmengen unitärer Darstellungen von Lie Gruppen.
 17.12.2019 N.N. Titel: tba Abstract: tba
 07.01.2019 Martin Raum, Chalmers University of Technology Titel: Congruences of modular forms on arithmetic progressions Abstract: One purpose of modular forms, and more generally, weakly holomorphic modular forms is to aid or even enable the analysis of certain generating series, namely modular generating series. A common question asked for coefficients $c(n)$ of a generating series is which patterns of divisibility they satisfy. Among the most accessible patterns, there is divisibility on arithmetic progressions: $c(an + b)$ is divisible by a given positive integer $\ell$ for all $n$. The theory of modular forms modulo primes yields a good handle on such question as long as $a$ and $b$ are remain fixed and $\ell$ is prime. In this talk, we showcase a new technique that allows us to answer in the affirmative the question of whether there is a connection among congruences for varying $b$ and fixed $a$. As a result, we discover surprisingly strong restrictions on maximal arithmetic progressions that admit a congruence. We primarily build upon results by Deligne-Rapoport on the arithmetic compactification of the moduli of elliptic curves. We reduce the original problem to a very concrete one in the modular representation theory of finite groups of Lie type and their covers, which in our motivating setting can be solved by a calculation.
 14.01.2019 Julia Budde Titel: Wellenfrontmengen von Darstellungen Nilpotenter Lie Gruppen Abstract: tba 21.01.2019 Jens Funke, Durham University Titel: Theta series and (singular) theta lifts Abstract: In this talk we give an introduction to theta series and theta lifts and its representation-theoretic background. We then explain singular theta lifts of Borcherds type and employ the reductive dual pair U(p, q) × U(1, 1) to construct two different kinds of Green forms for codimension q-cycles in Shimura varieties associated to unitary groups. In particular, we establish an adjointness result between the singular theta lift and the Kudla-Millson theta lift and discuss further applications in the context of the Kudla Program. This is joint work with Eric Hofmann.
 28.01.2019 N.N. Titel: tba Abstract: tba

Sommer 2019
 10.04.2019 Claudia Alfes-Neumann -- Universität Paderborn Titel: Harmonic weak Maass forms and Harish-Chandra modules Abstract: In this talk we review results of Bringmann and Kudla on the classification of harmonic Maass forms. In their paper they gave a classification of the Harish-Chandra modules generated by the pullback (to SL_2(R)) of harmonic weak Maass forms for congruence subgroups of SL_2(Z). ACHTUNG:  Gemeinsames Oberseminar mit AG Algebra Vortrag findet in  D1 320 statt und startet schon um 14h.
 17.04.2019 Jan Frahm (geb Möllers) -- Universität Erlangen-Nürnberg Titel: Periodenintegrale, L-Funktionen und Multiplizität Eins Abstract: Einer automorphen Form auf der oberen Halbebene kann man durch Ihre Fourierkoeffizienten eine L-Funktion zuordnen. Allgemeiner kann man zwei automorphen Formen auf der oberen Halbebene die sogenannte Rankin-Selberg L-Funktion zuordnen, die durch Faltung der Fourierkoeffizienten gegeben ist. Diese Konstruktion lässt sich auf Paare automorpher Formen auf GL(m) und GL(n) verallgemeinern und hängt eng zusammen mit der Restriktion automorpher Formen von  GL(m) auf GL(n) (m\geq n) und Periodenintegralen. Im Vortrag wird dieser Zusammenhang erklärt und die Beziehung zur Restriktion (unendlich-dimensionaler) Darstellungen von GL(m) auf GL(n) hergestellt. Dadurch wird es möglich Abschätzungen für Rankin-Selberg L-Funktionen mit darstellungstheoretischen Methoden zu erreichen, insbesondere mit der Multiplizität Eins Eigenschaft.
 24.04.2019 Polyxeni Spilioti -- Universität Tübingen Titel Dynamical zeta functions, trace formulae and applications Abstract: The dynamical zeta functions of Ruelle and Selberg are functions of a complex variable s and are associated with the geodesic flow on the unit sphere bundle of a compact hyperbolic manifold. Their representation  by Euler-type products traces back  to the Riemann zeta function. In this talk, we will present  trace formulae and Lefschetz formulae, and the machinery that they provide to study the analytic properties of the dynamical zeta functions and their relation to spectral invariants. In addition, we will present other applications of the Lefschetz formula, such as the prime geodesic theorem for locally symmetric spaces of higher rank.
 01.05.2019 Feiertag
 08.05.2019 Joachim Hilgert -- Universität Paderborn Titel: Satake-Kompaktifizierung Abstract: tba
 15.05.2019 Valentin Blomer -- Universität Göttingen Titel: Spurformeln in der Analytischen Zahlentheorie Abstract: Mit der Poissonschen Summationsformel als Ausgangspunkt werden spektrale Summationsformeln auf lokal-symmetrischen Räumen vorgestellt zusammen mit einer Reihe arithmetischer und analytischer Anwendungen.
 22.05.2019 Anna von Pippich -- TU Darmstadt Titel The special value Z'(1) of the Selberg zeta function Abstract: In this talk, we report on an explicit formula for the special value at s=1 of the derivative of the Selberg zeta function for the modular group Gamma=PSL_2 (Z). The formula is a consequence of a generalization of the arithmetic Riemann--Roch theorem of Deligne and Gillet-Soule to the case of the trivial sheaf on Gamma\H, equipped with the hyperbolic metric. The proof uses methods of zeta regularisation and relies on Mayer-Vietoris type formulas  for determinants of Laplacian. This is joint work with Gerard Freixas.   Achtung: Der Vortrag findet von 12:30-14h im Raum N 3.211 statt.
 29.05.2019 Michael Baake -- Universität Bielefeld Titel Spectral aspects of point sets and their dynamical systems Abstract: The plan of this talk is to recall some properties around the diffraction and dynamical spectra of cut and project and inflation point sets, with some emphasis on cases with mixed spectrum and some examples of number-theoretic origin.
 05.06.2019 Anna Wienhard -- Universität Heidelberg Titel Vortrag entfällt Abstract: tba
 12.06.2019 Christopher Voll Titel Zeta functions of groups and rings -- uniformity at the edge of the wilderness Abstract: In asymptotic group theory, zeta functions have become important tools to study the asymptotic and finer arithmetic properties of the distribution of finitary invariants of infinite groups. Defined as Dirichlet generating series, they encode, for instance, the numbers of finite-index subgroups or finite-dimensionsional representations of a given infinite group. Zeta functions associated to arithmetic groups often enjoy Euler products, indexed by the (Archimedean and non-Archimedean) places of number fields. The non-Archimedean factors tend to be rational functions. To understand how these functions vary with the place is among the fundamental challenging questions in the field. I will report on recent work with Angela Carnevale and Michael Schein: we prove a conjecture of Grunewald, Segal, and Smith on the variation of local normal subgroup zeta functions of finitely generated free class-2-nilpotent groups under base extension with number rings. Our result establishes that, in this setup, the variation is "uniform on primes of fixed decomposition type" in the relevant number field. ACHTUNG:  Es finden an diesem Tag aufgrund der Lesewoche zwei Vorträge 11:00-12:30 (in J 2.213) und 14:00-15:30 (E 2.304) statt.
 19.06.2019 Kein Seminar
 26.06.2019 Jasmin Matz Titel: Asymptotics of traces of Hecke operators Abstract: The distribution of spectral parameters in families of automorphic representations has many applications, such as density estimates for exceptional eigenvalues or low-lying zeros in families of L-functions. I want to talk about joint work with T. Finis in which we prove an effective equidistribution result for Satake parameters of spherical automorphic forms on many split reductive groups with growing Laplace eigenvalue. Compared to previously known results for GL(n), we can improve the bounds for the remainder terms. As a special case we also obtain the Weyl law on the associated locally symmetric space together with an upper bound for the remainder.
 03.07.2019 N.N. Titel: tba Abstract: tba
 10.07.2019 Fällt aus wegen PBMath Sommerschule

Lie Theory

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### Prof. Dr. Fabian Januszewski

Algebra and Number Theory

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### Dr. Benjamin Küster

Spectral Analysis

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### Prof. Dr. Margit Rösler

Harmonic Analysis

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### Prof. Dr. Tobias Weich

Spectral Analysis

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