27.10.2020

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Titel:

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03.11.2020

Beatrice Pozzetti (Universität Heidelberg)

Titel:

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

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:

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

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

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:

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

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

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:

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

https://arxiv.org/abs/2012.02187                                                                                                                                                                                                                       

26.01.2021

Jasmin Matz (University of Copenhagen)

Titel:

Quantum ergodicity of compact quotients of SL(n,R)/SO(n) in the level aspect

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

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

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).                                                           

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Tobias Weich (Paderborn)

Titel:

Ruelle-Taylor Resonanzen für Anosov Wirkungen höheren Rangs

 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.                                                                                                         

Lasse Wolf (Paderborn)

Quantum-Classical correspondance in higher rank

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Beatrice Pozzetti (Heidelberg)                                  

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canceled due to Corona restrictions                                                                                                                               

Margit Rösler (Paderborn)

Riesz-Distributionen im Dunkl-Setting vom Typ A

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.

Kai-Uwe Bux (Bielefeld)

Coarse Topological Group Invariants

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.                                                                                                                              

Jungwon Lee (Sorbonne Université)

Dynamics of continued fractions and conjecture of Mazur-Rubin

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).
                                                                                                                               

Thomas Mettler (Frankfurt)

Lagrangian minimal surfaces, hyperbolicity and dynamics

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.                                                                                                                                  

Gabriel Rivière (Nantes)

Poincaré series and linking of Legendrian knots

 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)                                                                                                    

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Sven Möller (Rutgers University)

Dimension Formulae and Generalised Deep Holes of the Leech
Lattice Vertex Operator Algebra

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.)                                                                                                                                  

Michael Voit (TU Dortmund)

Limit theorems for Calogero-Moser-Sutherland particle models in the freezing regime

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.                                                                                                                         

Lennart Gehrmann (Essen)

Big principal series and L-invariants

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.                                                                                                                            

Elmar Schrohe (Hannover)

Abstract (as pdf file)

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Nguyen Thi Dang (Universität Heidelberg)

Topological mixing of the Weyl chamber flow

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 identies with the action of the geodesic ow on the unit tangent bundle of Γ\H2 . The latter has been well studied and satises 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 sucient 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.                                                                                                                     

Maxime Ingremeau (Université de Nice Sophia-Antipolis)

Titel:

Around Berry’s random waves conjecture

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.                                                                                                                   

Lasse Wolf (Universität Paderborn)

Spectral Asymptotics for kinetic Brownian Motion on hyperbolic surfaces

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).                                                                                                                              

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.                                                                                                                                    

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N.N.

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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.                                                                                                        

tba                                                                                                                                   

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.                                                                                                                              

tba                                                                                                                                   

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.

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Claudia Alfes-Neumann -- Universität Paderborn

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.

Jan Frahm (geb Möllers) -- Universität Erlangen-Nürnberg

Titel:

Periodenintegrale, L-Funktionen und Multiplizität Eins

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.

Polyxeni Spilioti -- Universität Tübingen

Dynamical zeta functions, trace formulae
and applications

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.                                                                                                                                 

Feiertag

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Valentin Blomer -- Universität Göttingen

Mit der Poissonschen Summationsformel als Ausgangspunkt werden spektrale Summationsformeln auf lokal-symmetrischen Räumen vorgestellt zusammen mit einer Reihe arithmetischer und analytischer Anwendungen.

Anna von Pippich -- TU Darmstadt

The special value Z'(1) of the Selberg zeta function

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.

Michael Baake -- Universität Bielefeld

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.
                                                                                                                               

Anna Wienhard -- Universität Heidelberg

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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.                                                                         

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PBMath Sommerschule