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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))

# Oberseminar "Geometrische Analysis und Zahlentheorie"

In dem Oberseminar  "Geometrische Analysis und Zahlentheorie" werden aktuelle Forschungsresultate auf den Gebieten der geometrischen Analysis und der Zahlentheorie präsentiert und diskutiert.

Das Seminar findet

Dienstags 14:15-15:45
im Raum E2-304

statt.

# Seminarprogramm

Wintersemester 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 N. N. Titel: tba 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
Sommersemester 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

### Jun.-Prof. Dr. Claudia Alfes-Neumann

Zahlentheorie und automorphe Formen

Telefon:
+49 5251 60-2663
Büro:
D2.241
Web:

Sprechzeiten:

Nach Vereinbarung.

### Prof. Dr. Fabian Januszewski

Algebra und Zahlentheorie

Telefon:
+49 5251 60-2610
Büro:
D2.231

### Prof. Dr. Joachim Hilgert

Lie-Theorie

Telefon:
+49 5251 60-2630
Büro:
D2.234
Web:

Sprechzeiten:

Nach Vereinbarung

### Jun.-Prof. Dr. Tobias Weich

Spektral Analysis

Leiter der AG Spektralanalysis

Telefon:
+49 5251 60-2621
Büro:
D2.207
Web:

Sprechzeiten:

Nach Terminvereinbarung per Mail

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