Geometrische und harmonische Analysis - Spektralanalysis

Winter 2023/24

17.10.2023	Carsten Peterson
Titel:	Quantum ergodicity on the Bruhat-Tits building for PGL(3, F) in the Benjamini-Schramm limit
Abstract:	Originally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which ``converge'' to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting. The proof uses tools from p-adic representation theory, polytopal geometry, and the geometry of affine buildings.

24.10.2023	Shi Wang (Shanghai) -- zoom talk
Titel:	Kleinian groups of small critical exponent
Abstract:	Given a finitely generated discrete subgroup G which acts isometrically on the hyperbolic space, the critical exponent of G measures the exponential growth rate of the G-orbits. In joint work with Beibei Liu, we show that if the critical exponent is small, then G must be convex cocompact and virtually free. This partly answers a conjecture of Kapovich.

31.10.2023	N.N.
Titel:	tba
Abstract:	tba

07.11.2023	Simon Becker (Zürich) -- joint with Complex Quantum Systems seminar
Titel:	Mathematics of magic angles
Abstract:	Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hörmander's bracket condition in a very simple setting). Recent mathematical progress also includes the proof of existence of generalized magic angles and computer assisted proofs of existence of real ones (Luskin--Watson, 2021). The results will be illustrated by colourful numerics which suggest many open problems (joint work with M Embree, J Wittsten, M Zworski 2020 and T Humbert, M Zworski 2022).

14.11.2023	Martin Hallnäs (Gothenburg)
Titel:	Integral equations for Heckman-Opdam hypergeometric functions and a quantum Calogero-Moser system
Abstract:	Motivated by spherical functions theory, Heckman and Opdam introduced in the late 1980s hypergeometric functions associated with root systems as joint eigenfunctions of a commutative algebra of Weyl group invariant PDOs, which are related by conjugation to quantum integrals of the hyperbolic Calogero-Moser system. In this talk, I will consider the type A case and show that the corresponding hypergeometric functions satisfy a one-parameter family of explicit integral equations. I will also discuss an interpretation of the result in terms of a so-called Q-operator for the hyperbolic Calogero-Moser system.

21.11.2023	Xiaocheng Li (Shandong University) -- zoom talk
Titel:	An estimate for spherical functions on SL(3,R)
Abstract:	We prove an estimate for spherical functions $\varphi_\lambda(a)$ on SL(3,R), establishing uniform decay in the spectral parameter $\lambda$ when the group variable $a$ is restricted to a compact subset of the abelian subgroup $A$. In the case of SL(3,R), it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when $\lambda$ and $a$ vary.

28.11.2023	Paul Nelson (Aarhus) -- zoom talk
Titel:	Microlocal lifts and quantum unique ergodicity for GL(2,Qp)
Abstract:	We will present the paper named in the title, which available at https://arxiv.org/abs/1601.02528. The paper constructs a representation-theoretic microlocal lift for principal series representations of GL(2,Qp), and uses this to prove a form of QUE for Hecke eigenfunctions on certain path spaces over finite arithmetic graphs. This provides an analogue at non-archimedean places of some results of Zelditch, Wolpert and Lindenstrauss at the archimedean place.

05.12.2023	Andrew Sanders (Cambridge, Masschusetts) -- zoom talk
Titel:	Uniformization of Locally Flag Manifolds
Abstract:	By the Poincare-Koebe uniformization theorem, every closed Riemann surface of negative Euler characteristic can be realized as a quotient of the upper half plane by a discrete subgroup of Mobius transformations. Moving from the Riemann sphere to other complex homogeneous flag varieties F, the theory of Anosov subgroups gives rise to a vast array of compact complex manifolds uniformized by open domains in F. In this talk, we aim to give an introduction through examples to the basic objects of the theory, discuss the (lack of a robust) function theory, and explain some results concerning the deformation theory of these objects. No prior knowledge of Anosov subgroups will be assumed. This is joint work with David Dumas (UIC).

12.12.2023	Quentin Labriet (Aarhus) -- zoom talk
Title:	Convolution formulas for Jacobi polynomials and representation theory of sl(2).
Abstract:	The goal of the talk is to present some convolutions formulas for Jacobi polynomials and how to obtain them using the representation theory of the Lie algebra sl(2). Doing so I will present some realizations of the so-called Hahn and Racah algebras. We will apply these formulas to prove identities involving the symmetry breaking operators involved in this problem called the Rankin-Cohen brackets. This is a joint work with L. Poulain d'Andecy from the university of Reims.

09.01.2024	Christian Arends (Aarhus) -- zoom talk
Titel:	Poisson transforms and a pairing formula for graphs
Abstract:	In this talk I will present key ideas from two recent papers (https://arxiv.org/abs/2312.09101, arxiv.org/abs/2312.10509). In doing so, I will focus on the case of regular graphs resp. its universal covering trees and show how to construct generalizations of scalar Poisson transforms using the automorphism group of the tree. We state some results on associated operator valued Hecke algebras and discuss how this construction gives rise to a spectral correspondence between eigenspaces of a Laplace operator acting on functions on the edges and eigendistributions (so-called resonances) of a transfer operator associated to the shift on one-sided infinite non-backtracking paths. Moreover, we discuss a pairing formula of resonant states and their negative time analogues, the coresonant states. This is joint work with Jan Frahm and Joachim Hilgert.

16.01.2024	Mostafa Sabri (NYU Abu Dhabi) -- zoom talk
Titel:	Ergodic Theorems for Continuous-time Quantum Walks on Crystal Lattices and the Torus
Abstract:	We give several quantum dynamical analogs of the classical Kronecker-Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk exp(−itΔ)ψ starting from a localized initial state ψ. Then the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.

23.01.2024	Jean-Philippe Anker (Orleans)
Titel:	Spectral projections on hyperbolic surfaces
Abstract:	In an ongoing collaboration with Pierre Germain (Imperial College) and Tristan Léger (NYU and Princeton University), we study $L^2 \to L^p$ estimates (p>2) for spectral projections in a small window on (locally) symmetric spaces. For hyperbolic surfaces of infinite area and with no cusps, we have recently obtained almost optimal results [arXiv:2306.12827]. In this talk, I will give a brief survey of the problem, which goes back to the restriction theorem of Stein-Tomas in the Euclidean setting, comment on our result for hyperbolic surfaces and present the main steps of its proof.

30.01.2024	Daniele Galli (Universität Zürich)
Titel:	A cohomological approach to Ruelle-Pollicott resonances of Anosov diffeomorphisms
Abstract:	Given a transitive Anosov diffeomorphism on a closed connected manifold, it is known that, for enough smooth observables, the system is mixing w.r.t. the measure of maximal entropy. Accordingly, it makes sense to investigate the speed of decay of correlations and to look for the so-called Ruelle-Pollicott resonances, in order to determine an asymptotic for the correlation limit. In this talk I will describe some recent ideas to tackle these questions. In particular, I will point out some surprising connections between the spectrum of a particular transfer operator acting on suitable Anisotropic Banach spaces of currents and the spectrum of the action induced by the Anosov map on the De Rham cohomology. As a corollary, we obtain an upper bound for the speed of mixing, w.r.t. the measure of maximal entropy. This talk is based on my PhD thesis that I defended last June at the University of Bologna.

Sommer 2023

04.04.2023	Joachim Hilgert (Paderborn University)
Titel:	Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem
Abstract:	For Lie groups and symmetric spaces one has Fourier transformations between spaces on which the symmetry group acts. The Fourier transforms are intertwining the (highly reducible) representations, thus showing their equivalence. We analyze the question whether the intertwining operator can be obtained from a change of polarization in the sense of geometric quantization and give a positive answer fro the case of compact groups. In this case the change of polarization yields at the same time the Peter-Weyl decomposition of L^2(G) and the Borel-Weil theorem for the irreducible components. The key tool is a connection on the space of Kähler polarizations on the cotangent bundle whose parallel transport provides the intertwiner. This is joint work with T. Baier, O. Kaya, J. Mourao and J. Nunes.
11.04.2023	No seminar due to CIRM Workshop
18.04.2023	N.N.
Titel:	tba
Abstract:	tba
25.04.2023	Hongwei Zhang (Ghent)
Titel:	Dispersive equations on noncompact Riemannian symmetric spaces
Abstract:	In this presentation, we provide an overview of space-time mixed norm estimates for dispersive equations on noncompact symmetric spaces. We focus on two key estimates: the Strichart inequality and the Smoothing property. The former is based on the pointwise kernel estimate, while the latter relies on the Stein-Weiss inequality, also known as the Hardy-Littlewood-Sobolev inequality with double weights. These results differ from the classical ones in the Euclidean setting due to the particular geometry at infinity. Additionally, we discuss how the geometric properties of the discrete group affect these results on locally symmetric spaces.
02.05.2023	no seminar due to parallel Complex Quantum systems seminar
Titel:	tba
Abstract:	tba
09.05.2023	Karl-Hermann Neeb (Uni Erlangen)
Titel:	Causal symmetric spaces and nets of operator algebras
Abstract:	In the theory of local observables in Algebraic Quantum Field Theory (AQFT) modular theory creates a one-parameter group of modular automorphisms from a single state and this group often has a geometric implementation. If modular groups are contained in finite-dimensional Lie groups, they naturally lead to 3-gradings of the Lie algebra and further to causal symmetric spaces. Conversely, we explain how nets of local observables (resp. of standard subspaces) on causal symmetric spaces can be constructed for all irreducible unitary representations of simple Lie groups which are either linear or locally isomorphic to SL(2,R). This is joint work with Jan Frahm (Aarhus), Gestur Olafsson (Baton Rouge) and Vincenzo Morinelli (Rome)
16.05.2023	no seminar due to parallel Complex Quantum systems seminar
Titel:	tba
Abstract:	tba
23.05.2023	Thibault Lefeuvre (Paris Jussieu)
Titel:	Marked length spectrum rigidity for Anosov surfaces
Abstract:	I will show that on a closed Anosov surface (e.g. surface with negative sectional curvature), the marked length spectrum, that is the length of closed geodesics marked by the free homotopy of the surface, determines the metric up to isometry. The proof combines microlocal analysis with the geometry of complex curves. If time permits, I will also discuss the case of surfaces with boundary. Joint works with A. Erchenko, C. Guillarmou and G. Paternain.
30.05.2023	Zhicheng Han (Uni Göttingen)
Titel:	Various spectra of the universal cover of SL2(R)
Abstract:	In this talk we will discuss the universal cover of SL_2(R), treating them as Riemannian manifolds, and study its associated differential form spectrum and Dirac operator spectrum. We will also discuss its application to computing certain topological invariants (Novikov-Shubin invariants) If time permits, we will discuss some of the ongoing attempts to generalize this to more general semisimple Lie groups (due to Herb-Wolf) such as the universal cover of Sp(2n, R).
06.06.2023	Nikhil Savale (Uni Köln)
Titel:	Spectral Theory of the Sub-Riemannian Laplacian
Abstract:	Sub-Riemannian (sR) geometry is the geometry of bracket-generating metric distributions on a manifold. Peculiar phenomena in sR geometry include the exotic Hausdorff dimension describing the growth rate of the volumes of geodesic balls. As well as abnormal geodesics that do not satisfy any variational equation. In this talk I will survey my results which show how both these phenomena are reflected in the spectral theory of the hypoelliptic Laplacian in sR geometry.
13.06.2023	Effie Papageorgiou (Uni Paderborn)
Titel:	Asymptotic behaviour of solutions to the heat equation on noncompact symmetric spaces
Abstract:	Link to abstract
20.06.2023	Sebastien Gouezel (University Rennes) (zoom)
Titel:	Ruelle Resonances for Geodesic Flows on Noncompact Manifolds
Abstract:	Ruelle resonances are complex numbers associated to a dynamical system that describe the precise asymptotics of the correlations for large times. It is well known that this notion makes sense for smooth uniformly hyperbolic dynamics on compact manifolds. In this talk, I will consider the case of the geodesic flow on some noncompact manifolds. In a class of such manifolds (called SPR), I will explain that one can define Ruelle resonances in a half-plane delimited by a critical exponent at infinity. Joint work with Barbara Schapira and Samuel Tapie.
27.06.2023	Giovanni Forni (University of Maryland)
Titel:	Weakly mixing billiards in polygons
Abstract:	According to numerical simulations (Artuso,Casati, Guarneri, Prosen, Wang, ...) the billiard in the random polygon is ergodic and mixing. The mathematical theory of billiards in polygons is rather well-developed for billiards in rational polygons (with limitations that will be explained), but there are few results on the ergodic theory of typical polygons. Kerckhoff-Masur-Smillie proved in 1986 that ergodic polygons are dense (in the space of polygons), by fast approximation based on their ergodicity result for rational polygons. In this talk we will present a joint result with Jon Chaika that weakly mixing polygons are also dense.
04.07.2023	Yann Chaubet (Cambridge) (zoom)
Titel:	Dynamical zeta functions counting triangles and Betti numbers
Abstract:	In this talk, I will present recent results concerning special values of certain combinatorial zeta functions counting geodesic paths in triangulations. I will show that those values are related to some topological invariants. As such, we recover the first Betti number or L^{^2}-Betti number of a compact manifold, as well as the linking number of knots in a 3-manifold. This is a joint work with Léo Bénard, Viet Dang and Thomas Schick.
11.07.2023	Chris Lutsko (Ruttgers University)
Titel:	An abstract spectral approach to orbit counting
Abstract:	One of Selberg's many contributions to hyperbolic geometry is an effective count for the number of (hyperbolic) lattice points lying in a ball of radius T. To do this he used the pretrace formula and estimates on the different spectral components. If, instead of a lattice, one considers a thin group, there is no spectral decomposition, and so one must use softer methods to approach the same problem. In this talk I will present this soft approach, and discuss an application to counting circles in Apollonian packings. I will present joint work with Alex Kontorovich and Dubi Kelmer.

Winter 2022/23

11.10.2022 20:30 zoom	Zhongkai Tao (UC Berkeley)
Titel:	Spectral asymptotics for kinetic Brownian motion on locally symmetric spaces
Abstract:	We prove the strong convergence of the spectrum of the kinetic Brownian motion to the spectrum of base Laplacian for locally symmetric spaces. This generalizes recent work of Kolb--Weich--Wolf on constant curvature surfaces in two aspects. First we give a construction of Casimir operators for a large family of locally homogeneous spaces, including all locally symmetric spaces. Second, we analyze the spectrum carefully using the Schur complement formula and prove a strong convergence by studying projections to the spherical harmonics. This is joint work with Qiuyu Ren.
18.10.2022	Allan Merino (University of Ottawa, Canada) -- online
Titel:	Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup
Abstract:	In his wonderful paper “Remarks on classical invariant theory”, Roger Howe suggested that his classical duality should be extendable to superalgebras/ supergroups. Roughly speaking, by restricting the spinor-oscillator representation (w, H) of the complex orthosymplectic Lie superalgebra spo(V) to some particular super dual pairs (g, g’) (or (G, g’)), he proved that the action of g’ on every G-isotypic components is irreducible. Some results have been obtained for other dual pairs (Howe, Nishiyama, Sergeev, Cheng-Wang, Howe-Lu, Davidson-Kujawa-Muth ...) but a general theory for a real or complex orthosymplectic Lie superalgebra (or supergroup) is not known yet. In a recent work with Hadi Salmasian, we obtained a classification of irreducible reductive dual pairs in a real or complex orthosymplectic Lie supergroup SpO(V). Moreover, we proved a “double commutant theorem” for all dual pairs in a real or complex orthosymplectic Lie supergroup. Time permitting, I will explain other questions we are currently working on related to the extension of Howe duality to super dual pairs.
25.10.2022	Clemens Weiske -- online
Titel:	Heisenberg parabolically induced representations of Hermitian Lie groups
Abstract:	Let G be a hermitian Lie group. Then G naturally contains a maximal parabolic subgroup MAN whose unipotent radical is a Heisenberg group. Consider the corresponding spherical principal series representations of G, realised on functions on the radical opposite to N, which is also a Heisenberg group. Under the Heisenberg group Fourier transform, this space transforms into operators on Fock spaces. We show that these Fock spaces decompose into a multiplicity free direct sum under the action of M, which is in general non-compact. We find an explicit expression for the Knapp-Stein intertwining operator on the Fourier transformed side, generalising classical results by Cowling for the case G=SU(1,n), where M is compact. This gives a new construction of the complementary series and of certain interesting unitarizable subrepresentatio. The presented results are joint work with Jan Frahm and Genkai Zhang.
08.11.2022	no seminar due to parallel Complex Quantum Systems seminar
15.11.2022	no seminar due to parallel Complex Quantum Systems seminar
22.11.2022	no seminar due to parallel Complex Quantum Systems seminar
29.11.2022	Hamid Al Saqban (Universität Paderborn) -- hybrid
Titel:	Limit theorems and the Kontsevich-Zorich cocycle
Abstract:	The Kontsevich-Zorich (KZ) cocycle is a dynamical system that is closely related to the derivative cocycle of the Teichmuller geodesic flow. While these dynamical systems are interesting in and of themselves, they also act as renormalizing dynamical systems for straight-line flows on translation surfaces. The main goal of these two talks is to introduce and motivate translation surfaces and their moduli spaces, and to state and sketch the proofs of recent results, parts of which are joint with G. Forni, that establish the existence of large fluctuations for the asymptotic growth of the norm of the KZ cocycle.
06.12.2022	Hamid Al Saqban(Universität Paderborn) -- hybrid
Titel:	Limit theorems and the Kontsevich-Zorich cocycle
Abstract:	The Kontsevich-Zorich (KZ) cocycle is a dynamical system that is closely related to the derivative cocycle of the Teichmuller geodesic flow. While these dynamical systems are interesting in and of themselves, they also act as renormalizing dynamical systems for straight-line flows on translation surfaces. The main goal of these two talks is to introduce and motivate translation surfaces and their moduli spaces, and to state and sketch the proofs of recent results, parts of which are joint with G. Forni, that establish the existence of large fluctuations for the asymptotic growth of the norm of the KZ cocycle.
13.12.2022	Ali Suri (Universität Paderborn) -- hybrid
Titel:	Curvature and stability of quasi-geostrophic motion
Abstract:	In this talk, first we restate the necessary background about quantomorphism group and central extension of the corresponding Lie algebra. Then we derive the quasi-geostrophic equation as an Euler-Arnold equation of the L^2 metric on this central extension. Afterwards, using the Lie algebra structure constants, we study the curvature and its impact on the stability of geodesics which are solutions of the quasi-geostrophic equation.
20.12.2022	Pritam Ganguly (Universität Paderborn) -- hybrid
Titel:	An uncertainty principle and its connection with quasi-analyticity
Abstract:	An Uncertainty principle due to Ingham provides the best possible decay of the Fourier transform of a function on \mathbb{R} which vanishes on a nonempty open set. To prove this result Ingham used the classical Denjoy-Carleman theorem for quasi-analytic functions on the real line. In this talk, we plan to discuss similar results in a more general context. To be precise, we will discuss an $L^2$-version of the Denjoy-Carelman theorem (due to Chernoff) and use this to prove the Ingham-type uncertainty principle for the (generalized) spectral projections associated with the Laplacian. To life less complicated, we will limit ourselves mainly to Euclidean spaces.
17.01.2023	Frédéric Naud (Sorbonne Université, France) -- online
Titel:	GOE and GUE statistics on random hyperbolic surfaces.
Abstract:	We will describe some models of Random compact hyperbolic surfaces on which it is possible to study, in a large genus regime, the smooth linear spectral statistics for the (twisted) Laplace spectrum. We will show that the variance converges, after taking adhoc limits, to the one given by the random matrix models of GOE and GUE. This is, in a fairly weak sense, a rigourous justification of some popular conjectures of quantum chaos from the 80s.
31.01.2023	William Hide -- hybrid
Titel:	Short geodesics and small eigenvalues on random hyperbolic punctured spheres
Abstract:	We study the geometry and spectral theory of random genus 0 hyperbolic surfaces with n cusps as n tends to infinity. In particular, we are interested in the number of "short" closed geodesics and "small" Laplacian eigenvalues for surfaces sampled with Weil-Petersson probablity. Inspired by the work of Mirzakhani and Petri (in the case of large genus compact surfaces), we demonstrate Poisson statistics for the number of closed geodesics on surfaces with n cusps whose lengths are on scales 1/sqrt(n). Using similar ideas we show that with high probability, a random genus 0 surface with n cusps has polynomially (in n) many small eigenvalues as n tends to infinity. This is joint work with Joe Thomas (Durham).
07.02.2023	Henrik Gustafsson (Umeå University, Sweden) -- online
Titel:	Automorphic forms and their Fourier coefficients
Abstract:	I will give a basic introduction to automorphic forms on reductive groups and their Fourier coefficients with respect to different unipotent subgroups. I will review a series of papers joint with Dmitry Gourevitch, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi for how these Fourier coefficients can be computed by a reduction principle. In particular, we have shown that they simplify drastically for automorphic forms in small automorphic representations, and I will give a brief overview of the applications of these results to scattering amplitudes in string theory.

Sommer 2022

12.04.2022	Moritz Doll (Bremen)
Titel:	Divisor of the Selberg Zeta Function for Hyperbolic Surfaces with Twists
Abstract:	We consider a finitely generated co-compact Fuchsian group $\Gamma$. The quotient $X = \mathbb{H} / \Gamma$ is a geometrically finite hyperbolic surface with infinite area that might have conical singularities. Given a unitary representation $\chi$ of $\Gamma$, we show that the associated Selberg zeta function admits a factorization, which gives a spectral interpretation of its zeros in terms of the resonances of a suitable Laplacian. This generalizes a theorem by Borthwick-Judge-Perry, where a similar factorization was shown for untwisted hyperbolic surfaces without conical singularities. This is based on joint work with Ksenia Fedosova and Anke Pohl.
19.04.2022	Benjamin Hinrichs (Jena)
Titel:	Infrared-Criticality and the Qubit
Abstract:	In models from quantum physics describing the interaction of a quantum mechanical particle with a quantum field of bosons, one observes the phenomenon of infrared-criticality if the bosons are massless. From the physical perspective, this is attributed to the fact that even tiny energy fluctuations can lead to the creation of an infinite cloud of low-energy bosons. Mathematically, infrared-criticality means that the selfadjoint Hamilton operator describing the system does not have a ground state, i.e., the infimum of its spectrum is not an eigenvalue. In this talk, we will first demonstrate this phenomenon on a toy quantum field theory. After that, we consider the spin boson model, in which the particle interacting with the massless bosonic field is a qubit. We discuss a recent proof that the spin boson model does have a ground state, even in the infrared-critical situation, if the coupling of qubit and field is below a critical value. Heuristically, this is explained by the fact that symmetries of the model can lead to cancellations of infrared-divergences. Our proof combines three ingredients: a compactness argument, a Feynman-Kac-Nelson type formula and a correlation bound for one-dimensional Ising models. The talk is based on joint work with David Hasler and Oliver Siebert.
10.05.2022	Renato Renner (ETH Zürich)
Titel:	From Quantum Cryptography to Quantum Gravity
Abstract:	What statements can we make about information that is inaccessible to us? This question arises both in quantum cryptography and in quantum gravity. In the former, we would like to put bounds on the information that an eavesdropper may possess. In the latter, we are interested, for instance, in the information content of a black hole. In this talk I will show how quantum information theory allows us to characterise such inaccessible information. As applications, I will discuss some implications to quantum cryptography and quantum gravity.
17.05.2022	Josh Maglione (Uni Bielefeld)
Titel:	Igusa zeta functions and flag Hilbert-Poincaré series of hyperplane arrangements
Abstract:	We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert-Poincaré series. We show how these rational functions are connected to local Igusa zeta functions and class counting zeta functions for certain graphical group schemes studied by Rossmann and Voll. We report on a general self-reciprocity result and explore other connections within algebraic combinatorics. This is joint work with Christopher Voll and with Lukas Kühne.
24.05.2022	Javier Echevarria Cuesta (Ecole Polytechnique, Paris)
Titel:	An algorithm to compute the spectrum of the Laplace-Beltrami operator on hyperbolic surfaces
Abstract:	The eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a given manifold encode a wealth of information; they help us solve inverse problems, the heat equation, the wave equation, and the Schrödinger equation, to name a few applications. It is hence a classical problem to compute them. Without having to look any further, the case of two-dimensional oriented surfaces of constant curvature is already very rich. Indeed, while we fully understand the spectral theory for these manifolds when the curvature is non-negative, the hyperbolic case eludes explicit computations. Here we propose a new algorithm for their rigorous approximation. This is joint work with Alexander Strohmaier.
31.05.2022	NN
Titel:
Abstract:
07.06.2022	Krystal Guo (University of Amsterdam) -- Hörsaal D1
Titel:	Quantum walks on graphs
Abstract:	The interplay between the properties of graphs and the eigenvalues of their adjacency matrices is well-studied. Important graph invariants, such as diameter and chromatic number, can be understood using these eigenvalue techniques. In this talk, we use classical techniques in algebraic graph theory to study quantum walks. A system of interacting quantum qubits can be modelled by a graph. The evolution of the quantum system can be completely encoded as a quantum walk in a graph, which can be seen, in some sense, as a quantum analogue of random walk. This gives rise to a rich connection between algebraic graph theory, linear algebra and quantum computing. In this talk, I will present recent results on the average mixing matrix of a graph; a quantum walk has a transition matrix which is a unitary matrix with complex values and thus will not converge, but we may speak of an average distribution over time, which is modelled by the average mixing matrix.
14.06.2022	NN
Titel:
Abstract:
21.06.2022	Samuel Edwards (University of Bristol) -- Hörsaal D1
Titel:	Spectral theory and dynamics on infinite-volume quotients Γ\G
Abstract:	Connections between representation and spectral theory, and dynamical systems on homogeneous spaces related to semisimple Lie groups G have a long history of study, going back to Gelfand-Fomin. In the most classical setting, one considers dynamics on quotient spaces Γ\G, where Γ is a lattice in G. The other main setting that has been studied are infinite-volume quotients of rank-one Lie groups by geometrically finite discrete subgroups Γ. Results in this setting (initiated by Patterson and Sullivan) relate the critical exponent of the group Γ, the spectrum of the Laplace operator on the locally symmetric space Γ\G/K (K being a maximal compact subgroup of G), and dynamics of the geodesic flow on the unit tangent bundle of Γ\G/K. Recently, there has been some interest in extending these results to infinite volume quotients Γ\G, where G is a higher-rank group, and Γ is an Anosov subgroup of G. I will discuss join work with Hee Oh in this setting, where we study relations between the eigenfunctions of the invariant differential operators on Γ\G/K, the growth indicator function of Γ (which is the higher-rank analogue of the critical exponent), and the conformal measures that govern dynamics on Γ\G.
28.06.2022	Guenda Palmirotta (Luxembourg) -- now ONLINE!
Titel:	Solvability of invariant systems of differential equations on the hyperbolic plane
Abstract:	In the Euclidean case, it is well-known, by Malgrange and Ehrenpreis, that linear differential operators with constant coefficients are solvable. However, what happens, if we genuinely extend this situation and consider systems of linear invariant differential operators, is still solvable? In case of $\mathbb{R}^n$ (for some positive integer $n$), the question has been proved mainly by Hörmander. We will show that this remains still true for Riemannian symmetric spaces of non-compact type $X=G/K$. More precisely, we will present a possible strategy to solve this problem by using the Fourier transform and its Paley-Wiener(-Schwartz) theorem for (distributional) sections of vector bundles over $X$. We will get complete solvability for the hyperbolic plane $\mathbb{H}^2=SL(2, \mathbb{R})/SO(2)$ and beyond. This work was part of my doctoral dissertation supervised by Martin Olbrich.
05.07.2022	Viet Dang (Institut Math. Jussieu) -- zoom
Titel:	Length Orthospectrum between convex bodies on flat tori.
Abstract:	I will describe joint work with Gabriel Rivière and Matthieu Léautaud where we construct new anisotropic Sobolev spaces on cotangent of tori. These spaces are adapted to the study of the transport by the geodesic flow. As an application, we show a new summation formula of Meyer-Guinand type relating the orthogeodesic spectrum between two convex bodies and the spectrum of the magnetic Laplacian on the torus.
06.07.2022 11:15-12:45	Job Kuit (Paderborn) -- Raum J3.330
Titel:	The most continuous part of the Plancherel decomposition for a real spherical space
Abstract:	Let Z be a homogeneous space of a real reductive group G. The Plancherel decomposition of Z is the decomposition of the space L^2(Z) of square integrable functions into a direct integral of irreducible unitary representations of G. In general this decomposition has a mixed discrete and continuous nature. In this talk we consider real spherical homogeneous spaces Z. In particular, we will focus on the most continuous part of L^2(Z), i.e., the closed subspace of L^2(Z) that decomposes in the largest continuous families. Recently, Eitan Sayag and I finished an article in which we give a precise description of the Plancherel decomposition of the most continuous part. I will discuss some of the ingredients and key ideas that went into this article.
12.07.2022	Norio Konno (Yokohama National University) -- zoom
Titel:	tba
Abstract:	tba

Winter 2021/22

19.10.2021	Kein Seminar
Titel:	-
Abstract:	-
26.10.2021	Lasse Wolf (Universität Paderborn)
Titel:
Abstract:
02.11.2021	Mohammad Talebi (Universität Oldenburg)
Titel:	Analytic Torsion for manifolds with fibred boundary metrics
Abstract:	The Analytic Torsion was introduced by Daniel Burrill Ray and Isadore M. Singer as analytic counterpart of R-Torsion from Topology. In this talk we define the Analytic Torsion for Manifolds with Geometry at infinity using structure results of Heat kernel in short and long time regimes. We determine the structure of heat kernel by means of Geometric Microlocal Analysis in the sense of Melrose et al.
09.11.2021	N.n.
Titel:
Abstract:
16.11.2021	Simon Roby (Tsinghua University)
Titel:	Resonances of the Laplace operator on homogeneous vector bundles over the real hyperbolic space
Abstract:	We study the resonances of the Laplacian acting on the compactly supported sections of the homogeneous vector bundle over the real hyperbolic space $H^n(\mathbb{R})=G/K$ determined by a representation $\tau$ of the maximal compact $K$ of $G$. We will see how to determine the resonances. Under the additional assumption that $\tau$ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We will see how to find their Langlands parameters and their wave front sets. If additional time is available, we will see what are the differences if we replace the real hyperbolic space by others hyperbolic spaces.
23.11.2021	Christian Arends (Universität Paderborn)
Titel:
Abstract:
30.11.2021	Damian Osajda (University of Wroclaw)
Titel:	Helly graphs and their automorphism groups
Abstract:	A graph is Helly if each family of pairwise intersecting (combinatorial) balls has non-empty intersection. Groups acting geometrically on such graphs are themselves called Helly. The family of such groups is vast, it contains: Gromov hyperbolic groups, CAT(0) cubical groups, Garside groups, FC type Artin groups, and others. On the other hand, being Helly implies many important algorithmic and geometric features of the group. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a kind of CAT(0)-like structure. I will present basic properties and examples of Helly groups. The talk is based on joint works with Jeremie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, Jingyin Huang, Motiejus Valiunas, Thomas Haettel.
07.12.2021	Kein Seminar wegen Oberwolfach Workshop
Titel:
Abstract:
14.12.2021	Andreas Juhl (Humboldt-Universität Berlin)
Titel:	Singular Yamabe Problems and extrinsic conformal Laplacians
Abstract:	For any even-dimensional Riemannian manifold (Mⁿ, g), there is a distin- guished sequence P_2N (g) (2N ≤n) of geometric differential operators of the form ∆_g^N + ··· which are conformally invariant (GJMS-operators). Here P₂(g) is the Yamabe operator (or conformal Laplacian) and P₄(g) is the Paneitz operator (discovered 1983). The definition and analysis of these operators rest on the ambient metric or the Poincaré-Einstein metric in the sense of Fefferman and Graham. In recent years, it was found that there is a far-reaching generalization of these constructions for hypersurfaces M ↪ X in any Riemannian manifold X. The resulting conformally invariant operators still live on M but their definition depends on the embedding. I will describe the basic ideas of this theory which among other things leads to higher-order analogs of the Willmore functional. The talk rests on work of Gover-Waldron and Juhl-Orsted.
21.12.2021	Antoine Meddane (Université de Nantes)
Titel:	Morse complex and resonances for Axiom A flows.
Abstract:	Axiom A flows were flows introduced by Smale in the 70' to generalize the Morse gradient flows and the geodesic flows on negatively curved manifolds. I will explain how tools from microlocal analysis and from the theory of semiclassical resonances of Helffer-Sjöstrand can be used to construct a Morse complex for Axiom A flows. In particular, it implies a generalization of Morse-inequalities in this context which was something not known yet.
11.01.2022	Dominik Brennecken (Universität Paderborn)
Titel:	Dunkl theory in line with the analysis on symmetric cones
Abstract:	Radial analysis on symmetric cones is closely related to the Dunkl theory associated to root systems of type A. Spherical functions, Laplace transform and hypergeometric series have analogues in Dunkl theory which allow to extend many concepts and results to this setting.
18.01.2022	Philipp Schütte (Universität Paderborn)
Titel:	Weighted Zeta Functions and Invariant Ruelle Distributions on Open Hyperbolic Systems
Abstract:	Ruelle resonances constitute important invariants for chaotic (hyperbolic) dynamical systems and their theory has progressed greatly in the last couple of decades. Building on the work of Dyatlov/Guillarmou (2016) in this subject area we define and discuss a notion of weighted zeta function for open hyperbolic systems. First we sketch a proof of their meromorphic continuation and the fact that their poles encode the resonances. Then we show how their residues can be identified with so-called invariant Ruelle distributions. On the one hand this yields a residue interpretation of Patterson-Sullivan distributions, on the other hand this enables their numerical calculation for example systems like geodesic flows on Schottky surfaces and 3-disk obstacle scattering.
25.01.2022	Anders Karlsson (University of Geneva)
Titel:	Spectral zeta functions of graphs and analytic number theory
Abstract:	From the spectrum of a Laplace operator one can form a zeta function. In the case of the circle it gives Riemann’s zeta function. We study such functions for certain finite and infinite graphs. These functions appear (incognito) in several areas of mathematics and physics. In particular we study rather precise asymptotic of discrete tori as is done in statistical physics, and recover spectral zeta functions of real tori which are of interest in number theory. As it turns out, the Riemann hypothesis is equivalent to an approximate functional equation of the spectral zeta functions of cyclic graphs (whose special values incidentally appear in the Verlinde formulas in algebraic geometry / mathematical physics). Joint work with F. Friedli, G. Chinta and J. Jorgenson.
01.02.2022	Khalid Koufany (Université de Lorraine)
Titel:	Symmetry breaking differential operators for tensor products of spinorial representations
Abstract:	Joint work with Jean-Louis Clerc.

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

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
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 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.

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

07.01.2020	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.

14.01.2020	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.

28.01.2020	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

Oberseminar "Geometrische und harmonische Analysis"

Programm

Programm (Historie)

16.04.2024	Dominik Brennecken (Paderborn)
Titel:	Convolution and Elliptic Operators in Dunkl Theory
Abstract:	We consider generalized convolutions in the rational Dunkl setting and give new results on the (singular) support of these convolutions. Furthermore, elliptic Dunkl operators are introduced, which are motivated by radial parts of elliptic operators on Riemannian symmetric spaces. The statements about the generalized convolution make it possible to prove a theorem on hypoellipticity for elliptic Dunkl operators, as well as a theorem on elliptic regularity on generalized Sobolev spaces. Depending on time, an outlook on the multitemporal wave equation in the Dunkl setting is given.

23.04.2024	Michael Heins (Universität Würzburg)
Titel:	Spectral Theory of the invariant Laplacian on the hyperbolic Disk and the Riemann Sphere
Abstract:	We discuss a unifying complex analytic framework for the eigenvalue theory of the Laplace operators of the hyperbolic disk and the Riemann sphere. The principal strategy is to pass to two complex coordinates to work with holomorphic objects instead of real-analytic ones. Our main theorem consists in an intrinsic classification of the Möbius invariant eigenspaces of the Laplacians.

30.04.2024	no seminar due to hiring committee talks
07.05.2024	no seminar due to hiring committee talks
14.05.2023	no seminar due to hiring committee talks
21.05.2024	Elia Fioravanti (KIT)
Titel:	tba
Abstract:	tba

28.05.2024	Joachim Hilgert (Universität Paderborn)
Titel:	Quantum resonances and scattering poles
Abstract:	tba

04.06.2024	N.N.
Titel:	tba
Abstract:	tba

06.06.2024 9-11 Uhr!	Bartosz Trojan (TU Wroclaw)
Titel:	tba
Abstract:	tba

11.06.2024	Pawel Plewa (Torino)
Title:	tba
Abstract:	tba

18.06.2024	Colin Davalo (Universität Heidelberg)
Titel:	tba
Abstract:	tba

25.06.2024	N.N.
Titel:	tba
Abstract:	tba

02.07.2024	Robert Yuncken (Institut Élie Cartan de Lorraine)
Titel:	tba
Abstract:	tba

09.07.2024	Konstantinos Tsouvalas (MPI Leipzig)
Titel:	Robust quasi-isometric embeddings of hyperbolic groups in SL_n(C)
Abstract:	tba

16.07.2024	N.N.
Titel:	tba
Abstract:	tba