Research seminar "Geometric and Harmonic Analysis"
This joint research seminar of the Universities of Aarhus and Paderborn is devoted to current research questions in the field of geometric and harmonic analysis.
The seminar takes place every Tuesday 2:15-3:30pm in room D 2 314 in hybrid form. If you are interested in participating online please contact Tobias Weich or Benjamin Delarue in order to receive the login details.
Organizers:
- Jan Frahm (Aarhus)
- Joachim Hilgert (Paderborn)
- Margit Rösler (Paderborn)
- Tobias Weich (Paderborn)
Program
15.10.2024 | Hamid Al Saqban (Paderborn) |
Titel: | On the Unique Ergodicity for Foliations of Generic Abelian Differentials |
Abstract: | In Teichmuller theory, a theorem due to H. Masur and W. Veech states that for generic Abelian differentials, the leaves of the horizontal foliation are uniquely ergodic. Said differently, the orbits of the horizontal straight-line flow on a generic translation surface are uniquely ergodic. Subsequently, G. Forni proved an effective form of this statement, establishing in particular precise power-laws for the deviations of ergodic averages of smooth functions from the power law (as predicted by the ergodic theorem). The main goal of this talk is to introduce and motivate translation surfaces, and to explain the main ideas behind a (microlocal) analytic approach to effective unique ergodicity for straight-line flows on a generic translation surface. This is a joint work in progress with D. Galli (University of Zurich). |
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22.10.2024 | Daniel Kahl (Paderborn) |
Titel: | Joint Spectrum of transfer operators on compact quotients of buildings |
Abstract: | We introduce an analog of the Weyl chamber flow for Euclidean buildings. We use this to define a semi-group of transfer operators and show that these have a joint spectrum of eigenvalues. |
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| 05.11.2024 |
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12.11.2024 | Tobias Simon (Erlangen) |
Titel: | Asymptotic behaviour of (Z(\g),K)-finite smooth functions of moderate growth at the boundary of the complex crown domain |
Abstract: | By results from Krötz and Schlichtkrull every K-finite smooth joint eigenfunction on a connected semisimple Lie group extends to a holomorphic function on the universal covering of the principal K_\C-fibre bundle over the complex crown domain. A natural way to see this is by writing the K-finite smooth joint eigenfunction as a vector-valued Poisson transform, i.e. applying a hyperfunction vector to the orbit map of a K-finite vector. If the hyperfunction vector is a distribution vector, i.e. the function is of moderate growth, we prove that the asymptotic at the boundary of the complex crown domain is at most polynomial. Using these growth rates, one obtains estimates for continuous seminorms in smooth Frechet globalizations of moderate growth of Harish-Chandra modules of finite length. |
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19.11.2024 | Yannick Bonthonneau (Université Sorbonne Paris Nord) |
Titel: | Regularity of the horospherical foliation and assumptions on pinching |
Abstract: | In the study of hyperbolic dynamics, it is an important observation that stable and unstable foliations are not as smooth as the flow, and only holder in general. It is widely believed that they are smooth only when the dynamics is of an algebraic nature. For example the diagonal flow on compact quotients of PSL_2(R). This has lead to investigations on precisely how much holder regularity one can expect. In the classical case of manifolds of variable negative curvature and their geodesic flow, the best results available are expressed in terms of pinching constants. I will recall the main events of that story and then explain why the pinching assumption can be made local instead of global. |
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26.11.2024 | Timon Weinmann (Imperial College London) |
Titel: | Lieb-Thirring Inequalities in Hyperbolic Space |
Abstract: | Consider a Schrödinger operator defined in the Euclidean space. In their celebrated 1976 paper, Lieb and Thirring proved an upper bound on the Riesz means of such an operator involving an integral of a power of the potential. Finding optimal constants in special cases of this inequality has been an active area of research for the past decades. In this talk we prove that these Lieb—Thirring inequalities in Euclidean space easily translate to corresponding inequalities for Schrödinger operators in Hyperbolic space and that the optimal constant can be retained. |
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03.12.2024 | N.N. |
Titel: | tba |
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10.12.2024 | Guendalina Palmirotta (Paderborn) |
Titel: | Patterson-Sullivan distributions on hyperbolic surfaces |
Abstract: | There is a curious relation between two kinds of phase space distributions associated to eigenfunctions of the Laplacian on a hyperbolic surface: Patterson-Sullivan distributions, which are geodesics-flow invariant distributions, and Wigner distributions, which arise in quantum chaos and are invariant under the wave group. In this talk, we will describe these two distributions and generalise them on convex-cocompact hyperbolic surfaces. Then, we will show how they are asymptotically intertwined. This is a joint work with Benjamin Delarue (Paderborn). |
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17.12.2024 | Dominik Brennecken (Paderborn) |
Title: | tba |
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07.01.2025 | Effie Papageorgiou (Paderborn) |
Titel: | tba |
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14.01.2025 | Stephan Stadler (MPI Bonn) |
Titel: | tba |
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21.01.2025 | N.N. |
Titel: | tba |
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28.01.2025 | James Reed Farre (MPI Leipzig) |
Titel: | tba |
Abstract: | tba |
Supported by Deutsche Forschungsgemeinschaft via CRC-TRR 358 "Integral Strucutres in Geometry and Representation Theory"
Seminar Program (Past)
16.04.2024 | Dominik Brennecken (Paderborn) |
Title: | 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. |
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23.04.2024 | Michael Heins (University of Würzburg) |
Title: | 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. |
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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 | Benjamin Delarue (Paderborn) |
Title: | Axiom A flows for projective Anosov subgroups |
Abstract: | Projective Anosov subgroups are higher rank generalizations of fundamental groups of convex cocompact hyperbolic manifolds. Key properties of these groups are closely related to the dynamical system given by the geodesic flow and in particular its restriction to its non-wandering set. In the projective Anosov case the latter dynamical system can be replaced by Sambarino's refraction flow (or a general Gromov geodesic flow). In my talk I will present recent joint work with Daniel Monclair and Andrew Sanders in which we show that every projective Anosov subgroup comes with an analytic contact Axiom A flow whose restriction to its non-wandering set becomes Sambarino's refraction flow. We deduce a general exponential mixing result and the existence of a discrete spectrum of Ruelle-Pollicott resonances with associated (co-)resonant states. If time allows, I will outline applications of our results to pseudo-Riemannian manifolds and Benoist subgroups. |
28.05.2024 | Joachim Hilgert (University of Paderborn) |
Title: | Quantum resonances and scattering poles |
Abstract: | tba |
04.06.2024 | Bartosz Trojan (TU Wroclaw) |
Title: | Compactifications of affine buildings |
Abstract: | In this talk I am going to introduce affine buildings and indicate what are the analytic problems one can be interested in. In particular, I explain several compactification procedures and motive the study of random walks. In the end I am going to describe Martin compactification. |
11.06.2024 | Pawel Plewa (Torino) |
Title: | Riesz transforms and spectral multiplier theorem for extensions of Bessel–Kingman hypergroups |
Abstract: | We consider a one parameter extension by dilations of the Bessel–Kingman hypergroup. We prove a sharp spectral multiplier theorem for the lifted Bessel operator. Moreover, we prove L^p-boundedness of the associated Riesz transforms in the full range of p. As an application, we show the L^p boundedness of Riesz transforms corresponding to a left-invariant sub-Laplacian on a solvable extension of stratified Lie groups. |
18.06.2024 | Colin Davalo (University of Heidelberg) |
Title: | Finite-sided Dirichlet domains for Anosov representations |
Abstract: | Dirichlet domains provide polyhedral fundamental domains in hyperbolic space for discrete subgroups of the isometry group. Selberg introduced a similar construction of a polyhedral fundamental domain for the action of discrete subgroups of the higher rank Lie group $\SL(n,\R)$ on the projective model of the associated symmetric space. His motivation was to study uniform lattices, for which these domains are finite-sided. However these domains can also be studied for smaller subgroups, and we will consider the following question asked by Kapovich: for which Anosov subgroups are these domains finite-sided ? Anosov subgroups are hyperbolic discrete subgroups satisfying strong dynamical properties, but are not lattices in higher rank. We will first consider an example of an Anosov subgroup for which the fundamental domain constructed by Selberg has infinitely many sides. We then provide a sufficient condition on a subgroup to ensure that the domain is finitely sided in a strong sense. This is joint work with Max Riestenberg. |
25.06.2024 | Elia Fioravanti (KIT) |
Title: | The pressure metric on quasi-Fuchsian spaces |
Abstract: | Thermodynamic formalism, as initially developed by Bowen and Ruelle, can be used to quantify the difference between two Anosov flows in terms of a "pressure semi-norm" on suitable spaces of Hölder functions. McMullen showed that the Weil-Petersson metric on Teichmüller space can be interpreted in these terms, after which "pressure metrics" were defined on broader representation varieties, for instance on quasi-Fuchsian spaces by Bridgeman and on Hitchin components by Bridgeman, Canary, Labourie and Sambarino. While the infinitesimal structure at the Fuchsian locus is well-understood in these varieties, almost nothing is known on the global behaviour of pressure metrics. In joint work with Ursula Hamenstädt, Frieder Jäckel and Yongquan Zhang, we show that the pressure metric on quasi-Fuchsian spaces has finite diameter. As a curious consequence, the Fuchsian locus is highly distorted. |
02.07.2024 | Robert Yuncken (Institut Élie Cartan de Lorraine) |
Title: | Hypoellipticity, pseudodifferential calculi and groupoids |
Abstract: | A differential operator P is hypoelliptic if it has the smoothness of solutions property: roughly, if Pf=g with g smooth then f is smooth. The best-known examples are elliptic operators, like the Laplacian, but there are many others, such as the heat operator, the Kolmogorov operator, or Hörmander’s sums-of-squares operators. In the late 70s, it was realized that hypoellipticity is linked to the representation theory of nilpotent Lie groups (Rockland, Beals-Greiner, Helffer-Nourrigat,...). Helffer and Nourrigat made a broad conjecture encompassing most of the known examples. We will recount this story and show how the Helffer-Nourrigat conjecture can be solved using ideas on the tangent groupoid from Connes and Debord-Skandalis. |
09.07.2024 | Konstantinos Tsouvalas (MPI Leipzig) |
Title: | Robust quasi-isometric embeddings of hyperbolic groups in SLn(C) |
Abstract: | Anosov representations, introduced by Labourie and further generalized by Guichard-Wienhard, form a large class of discrete and stable hyperbolic subgroups of semisimple Lie groups, generalizing classical convex cocompact subgroups of rank one Lie groups. In this talk, we will provide a construction of stable, non-rigid, quasi-isometrically embedded hyperbolic subgroups of SL(n,C), for sufficiently large n, which are not limits of Anosov representations. |
16.07.2024 | N.N. |
Title: | tba |
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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. |
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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. |
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31.10.2023 | N.N. |
Titel: | tba |
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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 magicangles 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). |
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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. |
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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. |
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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. |
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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). |
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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. |
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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. |
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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. |
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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. |
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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. |
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 |
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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 |
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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). |
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: | |
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. |
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. |
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 |
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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 |
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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. |
12.07.2022 | Norio Konno (Yokohama National University) -- zoom |
Titel: | tba |
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19.10.2021 | Kein Seminar |
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26.10.2021 | Lasse Wolf (Universität Paderborn) |
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02.11.2021 | Mohammad Talebi (Universität Oldenburg) |
Titel: | Analytic Torsion for manifolds with fibred boundary metrics
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| Abstract: | The Analytic Torsion was introduced by Daniel Burrill Ray and Isadore M.
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09.11.2021 | N.n. |
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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) |
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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 |
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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 P2N (g) (2N ≤n) of geometric differential operators of the form ∆gN + ··· 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. |
13.04.2021 | NN |
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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 |
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01.06.2021 | NN |
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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].
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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 This talk is based on the joint articles with J-Ph. Anker and J. Dziubanski.
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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. |
27.10.2020 | NN |
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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: |
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 |
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). |
07.04.2020 | N.N. |
Titel: |
tba |
Abstract: |
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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
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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: *
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: |
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16.06.2020 | Sven Möller (Rutgers University) |
Titel: | Dimension Formulae and Generalised Deep Holes of the Leech |
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. |
23.06.2020 | Michael Voit (TU Dortmund) |
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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. |
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: |
14.07.2020 | N.N. |
Titel: | tba |
Abstract: | tba |
08.10.2019 | Nguyen Thi Dang (Universität Heidelberg) |
Titel: |
Topological mixing of the Weyl chamber flow |
Abstract: | Let G be a semisimple Lie group without compact factors. Consider A a maximal split torus and a maximal compact subgroup K for which the Cartan decomposition holds. Denote by M the centralizer subgroup of A in K . Let Γ be a discrete subgroup of G, acting properly discontinuously on G/K . In the case of G = PSL(2, R), the right action of A on Γ\G identi es with the action of the geodesic ow on the unit tangent bundle of Γ\H2 . The latter has been well studied and satis es many chaotic properties such as topological mixing in its non-wandering set. My talk addresses the case when A is of higher dimension and Γ is not a lattice. First I will introduce the main topological property: topological mixing. Then I will state a joint result with Olivier Glorieux, a necessary and su cient condition for topological mixing of actions of one parameter subgroups φt of A on Γ\G/M . Then I will introduce a few key tools: the Benoist limit cone and Hopf coordinates of G/M . Finally, I will give the main ideas behind the proof of the topological mixing Theorem. Time permitting, I will present a generalization of this Theorem for the right action of φt on Γ\G when M is abelian and connected. |
15.10.2019 | Maxime Ingremeau (Université de Nice Sophia-Antipolis) |
Titel: | Around Berry’s random waves conjecture |
Abstract: | 40 years ago, the physicist Michael Berry suggested that eigenfunctions of the Laplacian on manifolds of negative curvature should be well described, in the high-frequency limit, by some random function, given by an isotropic monochromatic Gaussian field. After recalling various mathematical interpretations of this conjecture, we will discuss how Bourgain’s arithmetic « derandomization technique » allow to prove the conjecture for eigenfunctions on the two-dimensional torus. We will show that the conjecture holds in a weak form for some families of quasi-modes, namely, long-time evaluated Lagrangian distributions on manifolds of negative curvature. |
22.10.2019 | Lasse Wolf (Universität Paderborn) |
Titel | Spectral Asymptotics for kinetic Brownian Motion on hyperbolic surfaces |
Abstract: | The kinetic Brownian motion on the sphere bundle of a Riemannian manifold M is a stochastic process that models a random perturbation of the geodesic flow. If M is a orientable compact constant negatively curved surface, we show that in the limit of infinitely large perturbation the L2-spectrum of the infinitesimal generator of a time rescaled version of the process converges to the Laplace spectrum of the base manifold. In addition, we give explicit error estimates for the convergence to equilibrium. The proofs are based on noncommutative harmonic analysis of SL2(R). |
29.10.2019 | Fabian Januszewski |
Titel: | Rationalität von L-Werten |
Abstract: | Dieser Vortrag ist der erste in einer Serie von zwei Vorträgen über spezielle Werte von L-Funktionen. In diesem ersten Vortrag über Rationalität von L-Werten wird auf die konzeptionelle Motivation und Resultate eingegangen. Im Kontext letzterer spielen (g,K)-Moduln eine wichtige Rolle. |
05.11.2019 | N.N |
Titel: | tba |
Abstract: | tba
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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 |
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.
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28.01.2020 | N.N. |
Titel: | tba |
Abstract: | tba |
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). |
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 |
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.
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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.
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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 |
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. |
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 |
03.07.2019 | N.N. |
Titel: | tba |
Abstract: | tba |
10.07.2019 | Fällt aus wegen |