17.01.2023

Severin Barmeier (Koln),
Strict quantization of polynomial Poisson structures
Deformation quantization is a general framework for obtaining quantum observable algebras from a classical mechanical system through algebraic deformations of its (classical) observable algebra, where the reduced Planck constant hbar plays the role of the deformation parameter. A breakthrough was achieved through Kontsevich's Formality Theorem which implies that any Poisson manifold can be formally quantized. This general positive result motivates one to continue pursuing the deformation quantization programme and look for the existence of convergent deformations, where the deformation parameter can be evaluated to the physical value of hbar, giving a "strict quantization". This next step is rather nontrivial even for constant or linear Poisson structures on Rᵈ and widely open for general Poisson manifolds. In this talk I will present a combinatorial approach to the deformation quantization problem of (nonlinear) polynomial Poisson structures on Rᵈ for which convergence and continuity results can be shown directly, yielding strict quantizations. In simple examples these strict quantizations can even be represented as *algebras of adjointable operators on a Hilbert space and thus can be viewed as quantum observable algebras as in the standard formulation of quantum mechanics. This talk is based on https://arxiv.org/abs/2201.03249 joint with Philipp Schmitt and https://arxiv.org/abs/2002.10001 joint with Zhengfang Wang. 


16.12.2022
(10:30) 
Jihun Yum (Daejeon),
Limit of Bergman kernels on a tower of coverings of compact Kaehler
manifolds
The Bergman kernel $B_X$, which is by the definition the reproducing kernel of the space of $L^2$ holomorphic $n$forms on a $n$dimensional complex manifold $X$, is one of the important objects in complex geometry. In this talk, we observe the asymptotics of the Bergman kernels, as well as the Bergman metric, on a tower of coverings. More precisely, we show that, for a tower of finite Galois coverings $\{ \phi_j : X_j \rightarrow X\}$ of compact K{\"a}hler manifold $X$ converging to an infinite Galois covering $\phi : \widetilde{X} \rightarrow X$, the sequence of pushforward Bergman kernels $\phi_{j*} B_{X_j}$ locally uniformly converges to $\phi_* B_{\widetilde{X}}$. Also, as an application, we show that sections of canonical line bundle $K_{X_j}$ for sufficiently large $j$ give rise to an immersion into some projective space, if so do sections of $K_{\widetilde{X}}$. This is a joint work with S. Yoo in IBSCCG. The talk takes place in Seminarraum 2 at 10:30 am. 


13.12.2022

Pau Mir Garcia (Barcelona),
BohrSommerfeld quantization of bsymplectic toric manifolds
BohrSommerfeld quantization uses techniques from symplectic geometry to understand the relation between classical physics and quantum physics. It has been successfully applied in the symplectic case for several compact systems and has been extended to some noncompact ones. For the class of $b$symplectic manifolds, the quantization procedure has to be redefined in order to obtain results in finite dimensions. In this talk, I will define the BohrSommerfeld quantization for $b$symplectic toric manifolds and prove that its dimension is given by a signed count of the integral points in the moment polytope of the torus action on the manifold. This is joint work with Eva Miranda and Jonathan Weitsman. 


29.11.2022

Eva Miranda (Barcelona),
From Symplectic to Poisson manifolds and back: Applications to
quantization
bStructures and other generalizations (such as Esymplectic structures) are ubiquitous and sometimes hidden, unexpectedly, in a number of problems including the space of pseudoRiemannian geodesics and regularization transformations of the threebody problem. Esymplectic manifolds include symplectic manifolds with boundary, manifolds with corners, compactified cotangent bundles and regular symplectic foliations. Their deformation quantization was studied à la Fedosov by Nest and Tsygan. How general can such structures be? Can we use this perspective to quantize Poisson manifolds using Fedosov recipe? In this talk, I first explain how to associate an Esymplectic structure to a Poisson structure with transverse structure of semisimple type (joint work with Ryszard Nest) and I will connect this to a result by Cahen, Gutt and Rawnsley on tangential star products. This result illustrates how Esymplectic manifolds serve as a trampoline to the investigation of the geometry of Poisson manifolds and the different facets of their quantization. This should let us address a number of open questions in Poisson Geometry and the study of its quantization from a brandnew perspective. 

12.7.2021

Workshop on Complex Analytic Geometry



7.7.2021 (18:00)

Fabrizio Bianchi (Lille),
A Spectral Gap for the Transfer Operator on Complex Projective Spaces
We study the transfer (PerronFrobenius) operator on $P^k(C)$ induced by a generic holomorphic endomorphism and a suitable continuous weight. We prove the existence of a unique equilibrium state and we introduce various new invariant functional spaces, including a dynamical Sobolev space, on which the action of f admits a spectral gap. This is one of the most desired properties in dynamics. It allows us to obtain a list of statistical properties for the equilibrium states. Most of our results are new even in dimension 1 and in the case of constant weight function, i.e., for the operator $f_*$. Our construction of the invariant functional spaces uses ideas from pluripotential theory and interpolation between Banach spaces. This is a joint work with TienCuong Dinh. 


31.5.2021

Tamas Darvas (Minnesota),
The volume of pseudoeffective line bundles and partial equilibrium
Let $L$ be a line bundle with positive singular Hermitian metric $he^{−u}$, on an ndimensional compact K ̈ahler manifold X. Let $d_k$ be the dimension of the space of global sections of $L^k$ that are $L^2$integrable with respect to the weight $e^{−ku}$. We show that the limit of $d_k/k^n$ exists, and equals the nonpluripolar volume of the Imodel potential associated to u. We give applications to the quantization of partial equilibrium measures. Joint work with Mingchen Xia. 


24.5.2022

Dan Coman (Syracuse),
Extension problems for plurisubharmonic functions
Let $V$ be a complex manifold and $X$ be an analytic subvariety of $V$. We discuss results about the following two extension problems: (a) plurisubharmonic functions on $X$ extend to plurisubharmonic functions on $V$, under the assumption that $V$ is Stein. (b) $\omega$plurisubharmonic functions on $X$ extend to $\omega$plurisubharmonic functions on $V$, under the assumption that $(V,\omega)$ is a compact K\"ahler manifold. 


10.5.2022

Slawomir Kolodziej (Jagiellonian University),
Complex MongeAmpere and Hessian equations and their applications
I would like to give an overview of existence and stability results for complex MongeAmp`ere and Hessian equations. With the focus on weak (nonsmooth) solutions obtained by means of pluripotential theory meth ods. Several geometric applications will be discussed. 


3.5.2022

Ood Shabtai (Koln) ,
Pairs of spectral projections of spin operators
We study the semiclassical behavior of an arbitrary bivariate polynomial, evaluated on certain spectral projections of spin operators, and contrast it with the behavior of the polynomial when evaluated on random pairs of projections. 

5.4.2022

Turgay Bayraktar (Sabanci) & Norman Levenberg (Bloomington),
Zeros of random polynomial mappings in several complex variables
We discuss some results on random polynomials with an eye towards obtaining universality results under the most general assumptions on the random coefficients. In particular, we generalize and strengthen some previous results on asymptotic distribution of normalized zero measures and currents associated to random poly nomials and random polynomial mappings in several complex variables. The talk is based on joint work with Tom Bloom. 


22.3.2022

Abdellah Laaroussi (Hannover),
Heat invariants and geometry of quaternionic contact manifolds
We consider the small time asymptotics for the heat kernel associated to the intrinsic sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold. More precisely, we explicitly compute the first two coefficients $c_0$ and $c_1$ appearing in the small time asymptotics expansion of the heat kernel on the diagonal. We show that the second coefficient $c_1$ equals the scalar curvature $\kappa$ (up to a constant multiple) associated to the canonical connection defined on such a manifold. 


1.2.2022

Ved Datar (Indian Institute of Science),
Some new rigidity results in Kahler geometry
Two basic comparison theorems from classical Riemannian geometry are BishopGromov volume comparison theorem and the BonnetMyers' diameter comparison theorem. Loosely speaking these theorems tell us that compact Riemannian manifolds with positive Ricci curvature have their volume and diameter bounded by that of a suitably scaled round sphere. Moreover, we have equality if and only if the Riemannian manifold is isometric to a round sphere. The rigidity in volume comparison follows immediately from the proof, while the diameter rigidity was proved by Cheng about 20 years after Myers' diameter comparison first appeared. In the volume case, one also has an almost rigidity theorem due to Colding, namely if a compact Riemannian manifold with positive curvature has almost maximal volume, then it is GromovHausdorff close to a round sphere. In my talk, I will speak about some recent results on the Kahler analogues of the above theorems. This is joint work with Harish Seshadri and Jian Song. 


1.12.2021

Hoang Chinh Lu (Paris Saclay),
MongeAmpere volumes on compact complex manifolds
We investigate in depth the behaviour of MongeAmpere volumes of quasipsh functions on a given compact hermitian manifold. We shall prove that the property for these MongeAmpere volumes to stay bounded away from zero or infinity is a bimeromorphic invariant. We shall show in particular that a conjecture of BoucksomDemaillyPaunPeternell holds true if and only if such MongeAmpere volumes stay bounded away from infinity. 


28.09.2021

Mercator Lecture Leonid Polterovich (Tel Aviv), Approximate representations of symplectomorphisms via quantization
We show that for a special class of geometric quantizations with "small" quantum errors, the quantum classical correspondence gives rise to an approximate projective representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups. Joint work with Laurent Charles.


20.07.2021

Ksenia Fedosova (AlbertLudwigsUniversitat Freiburg),
Spectral theory of infinitevolume hyperbolic manifolds
In this talk, we define a twisted Laplacian on an orbibundle over a hyperbolic surface (that might be of infinite volume). We prove the meromorphic continuation of the resolvent to the entire complex plane and prove an upper bound on the number of resonances. Additionally, we introduce the corresponding scattering matrix and prove an explicit formula for its determinant in terms of the Weierstrass product over the resonances. This is a joint work with M. Doll and A. Pohl. 


13.07.2021

Bingxiao Liu (Koln),
Asymptotic real analytic torsions for compact locally symmetric orbifolds
In this talk, I would like to explain a result on the full asymptotics of real analytic torsions for a certain sequence of flat vector bundles on a given locally symmetric orbifold. The basic idea is applying the Selberg's trace formula and then computing the semisimple orbital integrals. The key part is to evaluate explicitly the elliptic orbital integrals of heat kernels on the symmetric space. In particular, the identity orbital integrals correspond to the $L_2$torsion, which has been wellstudied. Here, we deduce a geometric localization formula, so that an elliptic orbital integral can be written as a sum of several identity orbital integrals for the corresponding centralizer, a reductive Lie subgroup. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations. 


06.07.2021

Michela Egidi (RuhrUniversitat Bochum),
The observability problem for generalized rectangles
We will introduce the (internal) observability problem for generalized rectangles in R^d, i.e. domains which are exhausted by hyperrectangles in R^d, and present a geometric condition on the observability set for its solvability. This condition is borred from complex analysis and it allows to show a spectral inequality which, in turn, gives the observability property. The technique used are also of complex analytic nature. Moreover, we will show that, if the domain is unbounded, the introduced geometric condition is both sufficient and necessary for observability. The talk is based on joint works with Ivan Veselic and Albrecht Seelmann. 


29.06.2021

Yihan Li (Nankai University),
An Index Theorem for EndPeriodic Toeplitz Operators
In this talk, I will present a recent result on the index theorem for EndPeriodic Toeplitz operators. This result can be viewed as a generalization of the theorem by Dai and Zhang for Toeplitz operators on manifolds with boundary and also an odddimensional analogue of the index theorem for endperiodic Dirac operators by MrowkaRubermanSaveliev. In particular, we find a new etatype invariant in the result and we will show its relation with the etatype invariant introduced by DaiZhang. The approach follows mainly the heat kernel method with a bcalculuslike modification. In the proof, we also introduce a etatype invariant for endperiodic Dirac operators and a variation formula for it. This is a joint work with professor Guangxiang Su. 


22.06.2021

Weixia Zhu (Faculty of Mathematics, University of Vienna),
Spectral Stability of the $\bar{\partial}$Neumann Laplacian
In this talk, we study the spectral stability of the complex Laplacians when the underlying structures are slightly perturbed. Our focus is on the stability of the variational eigenvalues of the $\bar{\partial}$Neumann Laplacian on a bounded (pseudoconvex) domain. This talk is based on joint work with Siqi Fu. 


15.06.2021

Alix Deleporte (Universite ParisSaclay),
Determinantal point processes and semiclassical spectral projectors
Determinantal point processes (DPPs) form a family of probabilistic models which capture the statistical properties of free fermions. The study of DPPs is further motivated by natural mathematical instances, such as random matrix theory or random representations of finite groups. To each (sequence of) locally finite rank projections is naturally associated a (sequence of) DPPs; this provides a supplementary motivation for the study of the semiclassical limit of natural spectral projectors. In this talk, I will discuss first the DPPs associated with Bergman/Szego projectors on holomorphic sections of a large positive curvature line bundle, whose study was initiated by Berman. Then, I will present an ongoing work with G. Lambert (UZH) on the semiclassical limit of DPPs associated with Schrodinger operators. 


08.06.2021

Hendrik Herrmann (Bergische University of Wuppertal),
Generic Features in the Spectral Decomposition of Correlation Matrices
We show that correlation matrices with particular average and variance of the correlation coefficients have a notably restricted spectral structure. Applying geometric methods, we derive lower bounds for the largest eigenvalue and the alignment of the corresponding eigenvector. We explain how and to which extent, a distinctly large eigenvalue and an approximately diagonal eigenvector generically occur for specific correlation matrices independently of the correlation matrix dimension.



01.06.2021

Cyril Letrouit (Laboratoire JacquesLouis Lions, Paris),
Subelliptic PDEs: observability and propagation of singularities
We present two results about subelliptic PDEs. The first one
(joint with Chenmin Sun) concerns observability for a family of
subelliptic Schrodinger equations: using tools of semiclassical
analysis, we prove a resolvent estimate which shows that energy
propagates slowly in directions which need brackets to be generated. The
second one (joint with Yves Colin de Verdiere) is a general result about
the propagation of singularities for subelliptic PDEs, formulated with
the help of subRiemannian geometry.



16.03.2021 (18:00)

Ming Xiao (University of California, San Diego),
Normal Stein spaces with BergmanEinstein metric and finite ball quotients.
In this talk, we will start with a conjecture posed by Cheng,
which states that the Bergman metric of a bounded, strongly pseudoconvex
domain in $\mathbb{C}^n$ with smooth boundary is KahlerEinstein if and only if the
domain is biholomorphic to the unit ball $\mathbb{B}^n$. Then we will discuss the recent
developments on solving and generalizing Cheng's conjecture. The talk is
based on a joint paper with Huang, and a recent preprint with Ebenfelt and
Xu.



02.02.2021

Zuoqin Wang (University of Science and Technology of China),
Semiclassical oscillating functions of isotropic type and their applications
Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semiclassical analysis. In this talk I will describe how to associate classes of semiclassical oscillating functions to isotropic submanifolds in phase space, and show that these classes are invariant under the action of arbitrary Fourier integral operators (modulo the usual clean intersection condition). Some special classes and applications will also be discussed. This is based on joint works with V. Guillemin (MIT) and A. Uribe (U. Michigan).



26.01.2021

Guokuan Shao (Sun Yatsen University),
Gequivariant Szego kernel asymptotics on CR manifolds
Let $X$ be a compact connected orientable CR manifold of dimension $2n+1$ with nondegenerate Levi curvature, which admits a connected compact Lie group $G$ action. In this talk, we will discuss a
Boutet de MonvelSjostrand type theorem for $G$equivariant Szego kernels $S_k^{(q)}$. When $X$ admits also a transversal CR $S^1$ action, we study the asymptotics of Fourier components of $S_k^{(q)}$. If time permits, we will show the coefficients of lower order terms in the asymptotic expansion when $X$ is strongly pseudoconvex.



08.12.2020

Masanori Adachi (Shizuoka University),
On Levi flat hypersurfaces with transversely affine foliation
In this talk, we discuss the classification problem of Levi flat hypersurfaces
in complex surfaces by restricting ourselves to the case that the Levi foliation
is transversely affine. After presenting known examples, we give a proof for
the nonexistence of real analytic Levi flat hypersurface whose complement
is 1convex and Levi foliation is transversely affine in a compact Kaehler surface.
This is a joint work with Severine Biard (arXiv:2011.06379).



01.12.2020

Xiaoshan Li (Wuhan University),
BergmanEinstein metric on a Stein space with strongly
pseudoconvex boundary
In this talk, I will give a discussion when a Stein space with
compact strongly pseudoconvex boundary is biholomorphic to a ball
under the assumption the Bergman metric is K ahlerEinstein.
This talk is based on a joint work with Professor Xiaojun Huang.



24.11.2020

Purvi Gupta (Indian Institute of Science),
Polynomially convex embeddings of compact real manifolds
A compact subset of $\mathbb{C^n}$ is polynomially convex if it is defined by a family of polynomial inequalities. In this talk, we will elaborate on this definition and discuss some questions and recent results regarding the minimum embedding (complex) dimension of abstract compact (real) manifolds subject to this convexity constraint. The primary challenge arises from the CRsingularities of a generic embedding. We will explain why this is the case, and discuss the main techniques that have been used in this problem so far. This is joint work with R. Shafikov.



07.07.2020

Wei Guo Foo (Chinese Academy of Sciences),
Equivalence problem of 5dimensional real hypersurfaces of type $C_{2,1}$
We consider the CR equivalence problem of 5dimensional
real hypersurfaces in C3 that are Levi degenerate of constant rank 1,
and are 2 nondegenerate in the sense of Freeman; and we will review some
of the progress made in this area. A wellknown homogeneous model is
the tube over the future light cone, whose Lie algebra of
infinitesimal CR automorphism is isomorphic to so(2,3).
In recent joint works with Zhangchi Chen, Joel Merker, and TheAnh Ta,
we study such manifolds in the rigid setting,
giving rise to 7dimensional Lie subalgebra of so(2,3), with 2 primary
invariants.



30.06.2020

Huan Wang (Academia Sinica),
Semipositive Line Bundles and Growth of Dimension of Cohomology
Firstly, we recall Siu and Demailly's proof of GrauertRiemenschneider conjecture, and Berndtsson's refined estimates on the cohomology of semipositive line bundles over compact complex manifolds.
Secondly, on possibly noncompact complex manifolds, we present the refined estimate for the dimension of the space of harmonic (0,q)forms with values in high tensor powers of semipositive line bundle when the fundamental estimate holds. In the end, we discuss two questions about nef and pseudoeffective line bundles given by Demailly and Matsumura analogue to Berndtsson's estimates. Both the background and the recent progresses of this topic would be introduced.



23.06.2020

Bo Liu (East China Normal University),
Differential Ktheory and localization formula for $\eta$ invariants
We obtain a localization formula in differential Ktheory for S1action. Then by combining an extension of Goette's result on the comparison of two types of equivariant $\eta$ invariants, we establish a version of localization formula for equivariant $\eta$ invariants. An important step of our approach is to construct a pre$\lambda$ring structure in differential Ktheory which is interesting in its own right.



16.06.2020

WeiChuan Shen (Köln),
Asymptotics of torus equivariant Szego kernel on a compact CR manifold
The study of the Szego
kernel is a classical subject in several complex variables and CauchyRiemann geometry. For example, when $X$ is the boundary of a strongly pseudoconvex domain, Boutet de Monvel and Sjostrand proved
that locally the Szego kernel $\Pi^{(0)}(x,y)$ on $(0,0)$forms is a Fourier integral operator with complexvalued phase
function. This kind of description of kernel function has profound impact in many aspects, such
as spectral theory for Toeplitz operator, geometric quantization and Kahler geometry.
In this talk, we let $(X,T^{1,0}X)$ be a compact CR manifold
of real dimension $2n+1$, $n\geq 2$, admitting a $S^1\times T^d$ action, $d\geq 1$. We consider a lattice point $(p_1,\cdots,p_d)\in Z^d$, where
$(p_1,\cdots,p_d)$ is a regular value of the associate CR
moment map $\mu$. Our goal is to
study the asymptotic expansion of the corresponding torus equivariant Szego kernel $\Pi^{(0)}_{m,mp_1,\cdots,mp_d}(x,y)$ as $m\to\infty$ under certain assumptions on $Y:=\mu^{1}(p_1,\cdots,p_d)$. As a corollary, we find a condition when the space of Requivariant CR functions on an irregular Sasakian manifold is nontrivial in semiclassical limit.


27.02.2020

Maxim Braverman (Northeastern University), Geometric quantization of noncompact and bsymplectic manifolds
We introduce a method of geometric quantization for of
noncompact symplectic manifolds in terms of the index of an
AtiyahPatodiSinger (APS) boundary value problem. We then apply it to a
class of compact manifolds with singular symplectic structure, called
bsymplectic manifolds. We show further that bsymplectic manifolds have
canonical Spinc structures in the usual sense, and that the APS index
above coincides with the index of the Spinc Dirac operator. We show that
if the manifold is endowed with a Hamiltonian action of a compact connected
Lie group with nonzero modular weights, then this method satisfies the
GuilleminSternberg ``quantization commutes with reduction'' property. In
particular our quantization coincides with the formal quantization defined
by Guillemin, Miranda and Weitsman, providing a positive answer to a
question posed in their paper. (joint with Yiannis Loizides and Yanli Song)



04.02.2020

Benjamin Kuster (Orsay), Resonances of the frame flow generator on hyperbolic 3manifolds
The frame flow on hyperbolic 3manifolds is an example of a
flow that is only partially hyperbolic (Anosov) due to the presence of
an additional neutral direction besides the flow direction. I will
present some very recent results, obtained together with Colin
Guillarmou, on the existence of a spectral gap in the resonance spectrum
of the generator of the frame flow on hyperbolic 3manifolds, based on
Fourier analysis and a semiclassical calculus for powers of line bundles.



21.01.2020

Son Duong (Universitat Wien), Semiisometric CR immersions of CR manifolds into Kahler manifolds
In this talk, we discuss a notion of semiisometric immersions of
strictly pseudoconvex CR manifolds into a Kahler manifold. This is
applied in particular to estimate the first positive eigenvalue of the
Kohn Laplacian on compact manifolds, to determine the CR umbilical
points, and to characterize the totally umbilic pseudohermitian
submanifolds of the complex euclidean space.



10.12.2019

N. Savale (Köln), Szego kernel and embedding for weakly pseudoconvex 3D CR manifolds of finite type
We construct a pointwise parametrix of Boutet de Monvel Sjostrand type for weakly pseudoconvex three dimensional CR manifolds of finite type assuming the range of the tangential CR operator to be closed. This improves the earlier analysis of Christ and consequently generalizes an embedding theorem of Lempert. Joint work with CY. Hsiao.



3.12.2019

Mihajlo Cekic (MPIM Bonn/Orsay), Resonant spaces for volumepreserving Anosov flows
Recently Dyatlov and Zworski proved that the order of vanishing of the Ruelle zeta function at zero, for the geodesic flow of a negatively curved surface, is equal to the negative Euler characteristic. They more generally considered contact Anosov flows on 3manifolds. In this talk, I will discuss an extension of this result to volumepreserving Anosov flows, where new features appear: the winding cycle and the helicity of a vector field. A key question is the (non)existence of Jordan blocks for one forms and I will give an example where Jordan blocks do appear, as well as describe a resonance splitting phenomenon near contact flows. This is joint work with Gabriel Paternain.


24.07.2019

R.T. Huang (National Central University), $S^1$equivariant Index Theorems and Morse Inequalities on
complex manifolds with boundary
Let $M$ be a complex manifold of dimension $n$ with smooth
connected boundary $X$. Assume that $\overline{M}$ admits a holomorphic
$S^1$action preserving the boundary $X$ and the $S^1$action is
transversal and CR on $X$. We show that the $m$th Fourier component of
the $q$th Dolbeault cohomology group $H^q_m(\overline{M})$ is finite
dimensional, for every $m\in\mathbb{Z}$ and every $q=0,1,\ldots,n$. This
enables us to define$\sum^{n}_{j=0}(1)^j{\rm dim \,}
H^j_m(\overline{M})$ the $m$th Fourier component of the Euler
characteristic on $M$ and to study large $m$behavior of
$H^q_m(\overline{M})$.In this talk, I will present an indexformula for
$\sum^{n}_{j=0}(1)^j{\rm dim\,}H^j_m(\overline{M})$ and Morse
inequalities for $H^q_m(\overline{M})$. This is based on a joint work
with ChinYu Hsiao, Xiaoshan Li and Guokuan Shao.



11.07.2019 (5:00 pm)

Joint seminar with "Interactions between symplectic geometry, combinatorics and number theory" M. Hamilton (Mount Allison University), Integral integral affine geometry, quantization, and RiemannRoch Let $B$ be a compact integral affine manifold. If there is an atlas whose coordinate changes are not only affine but also preserve the lattice $\mathbb{Z}^n$, then there is a welldefined notion of "integral points" in $B$, and we call $B$ an \emph{integral integral affine manifold.} I will discuss the relation of integral integral affine structures to quantization as well as some associated results, in particular the fact that for a regular Lagrangian fibration $M \to B$, the RiemannRoch number of $M$ is equal to the number of "integral points" in $B$. Along the way we encounter the fact that the volume of $B$ is equal to the number of integral points, a simple claim from "integral integral affine geometry" whose proof turns out to be surprisingly tricky. This is joint work with Yael Karshon and Takahiko Yoshida.



18.06.2019

U. Ludwig (Essen),
An Extension of a Theorem by Cheeger and Mueller to Spaces with Isolated Conical
Singularities
An important comparison theorem in global analysis is the comparison of analytic and
topological torsion for smooth compact manifolds equipped with a unitary flat vector bundle. It has
been conjectured by Ray and Singer and has been independently proved by Cheeger and Mueller in the
70ies. Bismut and Zhang combined the Witten deformation and local index techniques to generalise
the result of Cheeger and Mueller to arbitrary flat vector bundles with arbitrary Hermitian metrics.
The aim of this talk is to present an extension of the CheegerMueller theorem to spaces with isolated
conical singularities by generalising the proof of Bismut and Zhang to the singular setting.



28.05.2019

B. Guneysu (Bonn),
The geometry of semiclassical limits on regular Dirichlet spaces.
In this talk, I will first explain how one can reformulate the
known semiclassical limit results for the heat trace of Schrodinger
operators on Riemannian manifolds in a form which makes sense for abstract
Schrodinger type operators on locally compact spaces. These are results of
the form ``the quantum partition function converges to the classical
partition function as the Planck parameter tends to zero''. Then I will
give a probabilistic proof of this reformulation in case the ``free
operator'' stems from a regular Dirichlet form which satisfies a principle
of not feeling the boundary. This abstract result leads to completely new
results for Schrodinger operators on arbitrary Riemannian manifolds, and
simultaniously allows to recover also some results for weighted infinite
graphs.



21.05.2019

J. Kellendonk (Lyon),
The bulk boundary correspondence for quasiperiodic chains
Almost periodic chains (models for incommensurate phases) and quasiperiodic chains (models for
quasicrystals) have a richer topological phase structure than periodic chains, due to the existence of phasons.
This effects the bulk boundary correspondence for these models. We investigate in particular quasiperiodic chains
where the phason degree of freedom lives in a totally disconnected space. We show how we can nevertheless
define a winding number for the boundary resonances and thus obtain an equation which relates the integrated
density of states at a gap to a winding number which can be interpreted as the work the phason motion exhibits
on the edge states of the system.



30.04.2019

S. Finski (IMJ, Paris 7),
RiemannRochGrothendieck theorem for families of surfaces with hyperbolic cusps and its applications to the moduli space of curves
We generalize RiemannRochGrothendieck theorem on the level of differential forms for families of Riemann surfaces with hyperbolic cusps. The study of the spectral properties of the Kodaira Laplacian lies in the heart of our approach. When applied directly to the moduli space of punctured stable curves, our main result gives a formula for the WeilPetersson form in terms of the first Chern form of the Hodge line bundle, which generalizes the result of TakhtajanZograf. Our result gives also some nontrivial consequences on the growth of the WeilPetersson form near the compactifying divisor of the moduli space, which permits us to give a new approach to some wellknown results of Wolpert on the WeilPetersson geometry of the moduli space of curves.



13.03.2019

Y. Kordyukov (Ufa),
Asymptotic spectral analysis of Toeplitz operators on symplectic manifolds


22.1.2019

D. T. Huynh (MPI Bonn), Decreasing the truncation level in Cartan's Second Main Theorem
Abstract: Let $f:\mathbb{C}\rightarrow\mathbb{P}^2(\mathbb{C})$ be an entire holomorphic curve and let $\{L_i\}_{1\leq i\leq q}$ be a family of $q\geq 4 $ lines in general position in projective plane. If $f$ is linearly nondegenerate, i.e. its image is not contained in any line, then the classical Second Main Theorem of Cartan states that the following inequality holds true outside a subset of $(0,\infty)$ of finite Lebesgue measure:
\[
(q3)\,T_f(r)
\leq
\sum_{i=1}^qN_f^{[2]}(r,L_i)
+
o(T_f(r)).
\]
Here $T_f(r)$ and $N_f^{[2]}(r,L_i)$ stand for the characteristic function and the $2$truncated counting functions in Nevanlinna theory. It is conjectured that in the above estimate, the truncation level of the counting functions can be decreased to $1$, provided that that $f$ is algebraically nondegenerate (i.e. its image is not contained in any algebraic curve). In this talk, we will provide a partial answer to this conjecture in the special case where $f$ clusters to some algebraic curve. We also want to propose a strategy to achieve the full proof.



15.1.2019

F. Javier Torres de Lizaur (MPI Bonn), Geometric structures in the nodal sets of eigenfunctions of the Dirac operator
Abstract: Let $S_1,...,S_N$ be a collection of codimension 2 smooth submanifolds, of arbitrarily complicated topology, in the round sphere $\mathbb{S}^n$, $n \geq 3$, ($N$ being the complex dimension of the spinor bundle). In this talk I will show that there is always an eigenfunction $\psi:=(\psi_1,..., \psi_N)$ of the Dirac operator for which each submanifold $S_j$ is (modulo ambient diffeomorphism) a structurally stable nodal set of the spinor component $\psi_j$. These structures appear at small scales and sufficiently high energies. The result holds for any choice of trivialization of the spinor bundle.



18.12.2018

N. Savale (Köln), Bochner Laplacian and Bergman kernel expansion of semipositive line bundles on a Riemann surface
Abstract: We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semipositive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the subRiemannian (sR) Laplacian.



27.11.2018

N. Raymond (Angers), On the semiclassical spectrum of the Pauli operator with
Dirichlet boundary conditions
Abstract: This talk is devoted to the spectrum of the electromagnetic
Laplacian $(ih\nablaA)^2hB$ on a bounded, regular, and
simplyconnected open set of the plane.
Here $B$ is the magnetic field associated with $A$. We will see that,
when the magnetic field is positive (and under generic conditions), we
can accurately describe the lowlying eigenvalues in the semiclassical
limit $h\to 0$. We will show the crucial role of the magnetic
CauchyRiemann operators (and of their ellipticity), of the
BergmanHardy spaces, and of the Riemann mapping theorem in the
description of the spectrum.
This is a joint work with JeanMarie Barbaroux, Loïc Le Treust, and
Edgardo Stockmeyer.



20.11.2018

B. Liu (Bonn), Asymptotics of equivariant
analytic torsion on compact locally symmetric spaces
Abstract: In many cases, the size of torsion subgroups
in the cohomology of a closed manifold can be studied by computing
the RaySinger analytic torsion, which can be related
to the topological torsion by the CheegerMüller theorem.
Müller initiated the study of asymptotic analytic torsions associated
with a family of flat vector bundles on compact locally symmetric spaces.
Bismut, Ma and Zhang considered the analytic torsion forms in the
more general context and they introduced the Winvariants to denote
the leading terms of asymptotic analytic torsion forms.
Here we consider the leading term of the asymptotics of
equivariant analytic torsions on compact locally symmetric spaces,
which suggests an extension of BismutMaZhang's results to the equivariant case.



6.11.2018

V.D. Vu (Cologne), Equidistribution of Fekete points of large order II 


30.10.2018

N. Romao (Augsburg),
Quillen metrics and geometric quantization of vortex moduli
Abstract: The vortex equations provide an equivariant generalization
of GromovWitten theory for Kähler manifolds X equipped with a holomorphic
Hamiltonian action of a compact Lie group. Their moduli spaces support Kähler
structures which are invaluable to understand certain gauge theories
(for example gauged sigmamodels, but not only) at both classical and quantum level.
In my talk, I shall describe the geometric quantization of the moduli spaces of vortices
in line bundles (i.e when X=C with usual circle action) on a compact Riemann surface
\Sigma with fixed compatible area form
\omega_\Sigma. As complex manifolds, the moduli spaces identify with symmetric powers of \Sigma. A crucial ingredient of our construction is the Deligne pairing of line bundles over a familiy of curves, which carries a metric defined in terms of Quillen's metric on a determinant of cohomology. In a natural complex polarization, the resulting quantum Hilbert spaces are finitedimensional, and they can be interpreted as spaces of multispinors on \Sigma valued in a prequantization of an integral rescaling of
\omega_\Sigma. I will also address the issue of relating Hilbert spaces corresponding to different quantization data geometrically. Joint work with Dennis Eriksson.



23.10.2018

V.D. Vu (Cologne), Equidistribution of Fekete points of large order I
Abstract: Let F be a compact with piecewise smooth boundary
in n dimensional Euclidean space and N_k the dimension of the vector
space of the restrictions of real polynomials of n variables to K.
A Fekete point of order k is a point in F^{N_k} maximising the
determinant of a certain matrix of Vandermonde type.
These points are important in the interpolation problem
of continuous functions on F by polynomials.
By using deep tools from complex geometry and pluripotential theory,
we prove that Fekete points of large order are equidistributed toward
a canonical measure associated to F with an explicit speed of convergence.
Such equidistribution is crucial for approximation of Fekete points in practice.


19.06.2018

K. Fritsch (Bochum),
On equivariant embeddings for nonproper group actions
Abstract: There are several wellknown embedding theorems for real and
complex manifolds, the most prominent one arguably being the embedding
theorem of Whitney. If one adds an additional structure to the manifold
in the form of a Liegroup action, one may ask about the existence of
embeddings that respect the group action, which are called equivariant
embeddings.
I will give an introduction into this topic and talk about my results
for nonproper group actions. I will also give a small introduction into
the theory of moment maps and their connection to CR manifolds.



29.05.2018

R. Teodorescu (South Florida),
Projective connections and extremal domains
for analytic content
Abstract: An unexpected outcome of the recent proof of the 30 year old
conjecture that disks and annuli are the only domains where analytic content
 the uniform distance from z bar to analytic functions  achieves its lower
bound, is a new insight into projective connections and the
classification of quadratic differential spaces.
In particular, we reveal a new relation between
the symmetry constraints characterizing extremal domains
(in the approximation theory sense) and invariance groups for projective connections
in the case of finitegenus Riemann surfaces.



10.04.2018

M. Braverman (Northeastern), The spectral Flow of a family of Toeplitz operators
Abstract: We show that the (graded) spectral flow of
a family of Toeplitz operators on a complete Riemannian manifold
is equal to the index of a certain Calliastype operator.
When the dimension of the manifold is even this leads
to a cohomological formula for the spectral flow.
As an application, we compute the spectral flow of
a family of Toeplitz operators on a strongly pseudoconvex domain.
This result is similar to the Boutet de Monvel's computation
of the index of a single Toeplitz operator on a strongly pseudoconvex domain.
Finally, we show that the bulkboundary correspondence
in the GrafPorta model of topological insulators
is a special case of our result.


30.01.2018

A. Kokotov (Concordia/MPI Bonn), Surfaces of constant positive curvature with conical singularities and spectral determinants
Abstract: Let $f: X\to P^1$ be a meromorphic
function of degree N with simple poles and simple
critical points on a compact Riemann surface $X$ of genus $g$
and let $m$ be the standard round metric of curvature $1$ on the
Riemann sphere $P^1$. Then the pullback $f^*m$ of $m$ under $f$
is a metric of curvature $1$ with conical singularities of conical
angles $4\pi$ at the critical points of $f$. We study the $\zeta$regularized
determinant of the Laplace operator on $X$ corresponding to the metric $f^*m$
as a functional on the moduli space of the pairs $(X,f)$
(i.e. on the Hurwitz space $H_{g,N}(1,...,1)$) and derive an explicit formula for the functional.
Using closely related methods, we find an explicit expression
for the determinant of (the Friedrichs extension) of the Laplacian
on a compact Riemann surface of genus one with conformal metric of
curvature 1 having a single conical singularity of angle $4\pi$.
The talk is based on the joint works with V. Kalvin (Concordia University).



23.01.2018

G. Borot (MPI Bonn),
Matrix models, topological and geometric recursion
Abstract: I will first review the structure of the large N expansion in 1d loggases/matrix models, which is for a large class of models governed by a "topological recursion". Then, I will present a refinement of the topological recursion, called "geometric recursion", which is a general construction of functorial assignments for surfaces, by means of successive excisions of embedded pairs of pants. This leads to some (very preliminary) thoughts about the relevance of the possible relevance of the geometric recursion in the study of matrix models.
Based on joint works with GuionnetKozlowski, and AndersenOrantin.



19.12.2017

N. Savale (Köln), The GaussBonnetChern theorem: a probabilistic perspective
Abstract: We prove a probabilistic refinement of the GaussBonnetChern theorem
at the level of differential forms in the spirit of local index theory. Namely, for a real oriented vector
bundle with metric connection, we show that its Euler form may be identified with the expectation
of the current defined by the zerolocus of an appropriate random section of the bundle.



5.12.2017

L. Ioos (Paris), Lagrangian states in BerezinToeplitz
quantization
Abstract: A quantization is a process which, given a classical dynamical system, produces
the underlying quantum dynamics. In the case of BerezinToeplitz quantization, to a
symplectic manifold with some additional structures, we associate a sequence of Hilbert
spaces parametrized by the integers. Asymptotic results when this parameter tends to
infinity are supposed to describe the socalled semiclassical limit, when the scale gets
so large that we recover the laws of classical mechanics.
In the case of geometric quantization associated to a real polarization, quantum
states are represented by Lagrangian submanifolds satisfying the socalled BohrSommerfeld
condition. In this talk, I will construct these "Lagrangian states" in the framework
of BerezinToeplitz quantization and study their semiclassical properties. I will
then give an application to the problem of relative Poincaré series in the theory of automorphic
forms, and if time allows, I will present some links with geometric quantization
of ChernSimons theory.



21.11.2017

G. Marinescu (Köln), Universality results for zeros of random polynomials
Abstract: We investigate the distribution of roots of polynomials of high degree
with random coefficients
which, among others, appear naturally in the context of "quantum chaotic dynamics".
If the coefficients are i.i.d. Gaussian random variables, then the
roots tend to concentrate near the unit circle in the
complex plane. In contrast to this singular distribution, the zeros of
SU(2) polynomials spread uniformly over the Riemann sphere.
While these results are consistent with random matrix
theory predictions, they provide a new and different insight
into the problem of quantum ergodicity.
They admit higherdimensional generalizations and form a field called
"stochastic Kähler geometry".



07.11.2017

A. Drewitz (Köln),
Recent developments in some percolation models with longrange correlations
Abstract: We will introduce the Gaussian free field and the model of
random interlacements as prototypical examples for percolation models
with longrange correlations. After a review some of the developments in
these fields during the last decades, a recently established isomorphism
theorem will be introduced that leads to a deeper understanding of
the connection between these two models.
In particular, we will then outline how this isomorphism theorem can be used
in order to infer new interesting properties from one of the models via known
properties of the other.




19.07.2016

S. Klevtsov (Köln), Integer quantum Hall effect and Quillen metric



05.07.2016

A. Alldridge (Köln), Noncommutative geometry and the quantum Hall effectIV



21.06.2016

A. Alldridge (Köln), Noncommutative geometry and the quantum Hall effectIII



14.06.2016

I. Burban (Köln), Mathematical aspects of the quantum Hall effect on a torusII



07.06.2016

I. Burban (Köln), Mathematical aspects of the quantum Hall effect on a torusI



31.05.2016

L. Galinat (Köln), Landau problem on Riemann surfaces of higher genus



24.05.2016

G. Marinescu (Köln), Geometric quantization



10.05.2016

G. Marinescu (Köln), Laplacians on Hermitian manifoldsII



03.05.2016

F. Lapp (Köln), Laplacians and Dirac operatorsII



26.04.2016

F. Lapp (Köln), Laplacians and Dirac operatorsI



01.02.2016

S. Jansen (Bochum), Symmetry breaking in Laughlin's state on a cylinder



25.01.2016

G. Marinescu (Köln), Laplacians on Hermitian manifolds



18.01.2016

S. Klevtsov (Köln), Quantum Hall effect on Riemann surfaces: aims and achievements (II)



11.01.2016

A. Alldridge (Köln), Noncommutative geometry and quantum Hall effect (III)



07.12.2015

S. Klevtsov (Köln), Quantum Hall effect on Riemann surfaces: aims and achievements (I)



30.11.2015

I. Burban (Köln), Landau problem on a torus, line bundles and thetafunctions



23.11.2015

A. Alldridge (Köln), Noncommutative geometry and quantum Hall effect (II)



16.11.2015

A. Alldridge (Köln), Noncommutative geometry and quantum Hall effect (I)



09.11.2015

M. Zirnbauer (Köln), Quantum Hall Effect (III): quantum mechanical foundations



02.11.2015

M. Zirnbauer (Köln), Quantum Hall Effect (II): quantum mechanical foundations



26.10.2015

M. Zirnbauer (Köln), Quantum Hall Effect (I): electromagnetic response


14.07.2015

A. Hochenegger (Köln), Poincaré bundles on compex tori



07.07.2015

I. Burban (Köln), Derived categories and FourierMukai transforms II



30.06.2015

I. Burban (Köln), Derived categories and FourierMukai transforms I



19.05.2015

B. Liu (Köln), Analytic torsion and Quillen metrics II



12.05.2015

B. Liu (Köln), Analytic torsion and Quillen metrics I



29.04.2015

X. Ma (Paris/Köln), Introduction to geometric quantization
Vorsicht! Zeit und Raumänderung: Mittwoch, 16:00 Uhr im Hörsaal des MI



28.04.2015

G. Marinescu (Köln), Operators, kernels and Laplacians II



21.04.2015

G. Marinescu (Köln), Operators, kernels and Laplacians I


20.01.2015

H. Geiges (Köln), Integrable Systeme und Kontaktgeometrie



13.01.2015

I. Burban (Köln), LaxPaare und algebrogeometrische Methoden



09.12.2014

L. Galinat (Köln), AdlerKostant Schema und die klassische YangBaxter Gleichung



02.12.2014

C. Lange (Köln), Geodätischer Fluss



25.11.2014

A. Huckleberry (Bochum), LiouvilleArnold Integrabilität und WinkelWirkung KoordinatenII



18.11.2014

A. Huckleberry (Bochum), LiouvilleArnold Integrabilität und WinkelWirkung KoordinatenI



11.11.2014

A. Hochenegger (Köln), Symplektische Blätter von g^{*} und koadjungierte Wirkung



04.11.2014

T. Quella (Köln), Impulsabbildung und NoetherSätzeII



28.10.2014

A. Alldridge (Köln), Impulsabbildung und NoetherSätzeI



21.10.2014

G. Marinescu (Köln), Übersicht über symplektische Geometrie



14.10.2014

AG Kunze (Köln), HamiltonJacobi Gleichung



07.10.2014

M. Kunze (Köln), Übersicht über Hamiltonsche Systeme


08.07.2014

George Marinescu, Metric aspects of Okounkov bodies



01.07.2014

Vorbesprechung WS 14/15



17.06.2014

W. Kirwin (Köln), Complextime flows in toric geometry



03.06.2014

I. Burban (Köln), Survey of the theory of the classical YangBaxter equation



27.05.2014

D. Greb (Bochum), Completely integrable systems and Okounkov bodies



20.05.2014

Th. Bachlechner (Heidelberg), Inflation in UV theories
Abstract: In light of the discoveries that the universe went through a phase of accelerated expansion and is in a state with extremely small vacuum energy, physicists are struggling to produce theoretical models that are consistent with these observations. I will discuss the potential implications of the recent detection of B mode polarization by the BICEP2 experiment. I will discuss two theoretical approaches giving rise to (1) large field inflation and (2) small field inflation. The first realizes Nflation via kinematic alignment in the axion kinetic term while the second approach aims towards an understanding inflation within random supergravity theories. Both approaches crucially rely on a detailed understanding of the geometry on which the underlying string theory is compactified.



13.05.2014

P. Littelmann (Köln), Introduction to NewtonOkounkov bodies II



06.05.2014

X. Ma (Paris und Köln), AtiyahSinger Index Theorem VI



29.04.2014

P. Littelmann (Köln), Introduction to NewtonOkounkov bodies I



22.04.2014

X. Ma (Paris und Köln), AtiyahSinger Index Theorem III



15.04.2014

M. Hien (Köln), Partner orbits and action differences on compact factors of the hyperbolic plane (Thesis defense)

10.12.2013

J. Weyman (Essen), Local cohomology supported in determinantal varieties
Abstract: Let K be a field of characteristic zero.
Consider the polynomial ring S=K[X_{i,j}]_{1\le i\le m,1\le j\le n} on the entries of a generic m\times n matrix X=(X_{i,j}).
Let I_p be the ideal in S generated by p\times p minors of X.
I explain how to calculate completely the local cohomology modules H^i_{I_p}(S). I will also explain why the problem is interesting. It turns put the result allows to classify the maximal CohenMacaulay modules of covariants for the action of SL(n) on the set of m nvectors.
It also allows to describe the equivariant simple Dmodules, where D is the Weyl algebra of differential operators on the space of m\times n matrices.
This is a joint work with Claudiu Raicu and Emily Witt. The relevant references are arXiv 1305.1719 and arXiv 1309.0617.



03.12.2013

A. Kahle (Bochum), An elementary theorem for Field Theories
Abstract: Field theories, as axiomatised by Atiyah and Segal, have a
natural 'multiplication' coming from the monoidal structure on the
source and target categories. We show that when the target category is
permutative (roughly meaning that it has two monoidal structures that
interact as the addition and multiplication on a ring), that the field
theories may also be 'added', and that the 'multiplication'
distributes over the addition.



26.11.2013

Dr. A. Jarosz (Holon Institute of Technology, Israel), Quantum dots and Jack polynomials
Abstract: The talk will discuss a randommatrix approach to quantum transport in chaotic quantum dots with one nonideal lead and Dyson's symmetry parameter 1, 2 and 4. The reflection eigenvalues (the fundamental quantities of the theory) are shown to form a novel probability ensemble, described in terms of Jack polynomials, which are objects appearing in various settings in mathematics and physics; an introduction to this subject is given. This ensemble reveals links to various challenging mathematical questions.



22.10.2013

D. Ostermayr (Bonn), BottPeriodizität in KRTheorie via äquivarianten GammaRäumen
Abstract: Atiyah's Reelle KTheorie, oder KRTheorie, ist eine Verallgemeinerung
topologischer KTheorie auf Räume mit Involution, die reelle und komplexe
topologische
KTheorie als Spezialfälle enthält. Nach Einführung der grundlegenden
Definitionen,
werde ich ein Modell für das konnektive Cover kr als C_2Spektrum
vorstellen, welches auf
Segal und Suslin zurückgeht. Anschließend werde ich, hierbei Suslin
folgend, zeigen, wie aus Segal's
Arbeit zu GammaRäumen eine Version der (1, 1)Periodizität für
KRTheorie folgt.


25.02.2013

N. Orantin (Lisabon), From random matrix theory to enumerative geometry, a journey into integrable systems
Abstract: Random matrix theory has recently raised a lot of interest both in mathematics and physics.
It is one of the few solvable models, said to be integrable, which at the same time can be explicitly solved and has many applications ranging from biology to highenergy physics. This makes it not only a formidable toy model but also an important tool for studying modern complex systems as well as dualities in highenergy physics and mathematics. In particular, it was recently understood that, in some regime, a large class of matrix models can be solved by a universal inductive method called topological recursion. In this elementary talk, I will review some of the main applications of this new method in problems such as statistical physics on a random lattice, combinatorics, GromovWitten theories, Givental theory, knot theory or integrable systems in a larger sense. I will show how combinatorics allows very often to fill the gap between a solvable system and its solution.



14.12.2012

W. Kirwin (Köln), Polarisierung in geometrischer Quantisierung



27.11.2012

S. Klevtsov (Köln), Introduction to random metrics



13.11.2012

W. Kirwin (Köln), Complextime evolution in geometric
quantization


12.06.2012

S. Garnier (Bochum), Flows of supervector fields and local actions



22.05.2012

H.C. Herbig (Aarhus), On orbifold criteria for singular symplectic toric quotients



08.05.2012

A. Alldridge (Köln), Twisted equivariant matter



02.05.2012

A. Alldridge (Köln), Twisted equivariant matter
Vorsicht! Zeit und Raumänderung: Mittwoch, 10:15 Uhr im Seminarraum B der Chemie



24.04.2012

M. Zirnbauer (Köln), Twisted equivariant matter



17.04.2012

G. Schwarz (Brandeis), Automorphisms of quotients
Abstract: Let G be a complex reductive group and V a representation space of G. Then there is a quotient space Z and a canonical map pi: V>Z.
The quotient space Z has a natural stratification which reflects properties of the Gaction on V.
Let phi: V>Z be an automorphism. Then one can ask the following questions.
(1) Does phi automatically preserve the stratification?
(2) Is there an automorphism Phi: V>Z which lifts phi? This is, can we have pi(Phi(v))=phi(pi(v)) for all v in V. If so, can we choose Phi to be equivariant, i.e., can we have that Phi(gv)=g Phi(v) for all v in V and g in G?
We give conditions for positive responses to these questions, expanding upon work of Kuttler and Reichstein.


6.12.2011

S. Klevtsov (Brussels), 2D gravity and random Kahler metrics
Abstract: We propose a new approach to define theories of random metrics in two and higher dimensions, based on recent methods in Kahler geometry. The main idea is to use finite dimensional spaces of Bergman metrics, parameterized by large N hermitian matrices, as an approximation to the full space of Kahler metrics. This approach suggests the relevance of a new type of gravitational effective actions, corresponding to the energy functionals in Kahler geometry. These actions appear when a nonconformal field theory is coupled to gravity, and generalize the standard Liouville model in two dimensions.



22.11.2011

S. Schmittner (Köln), Spherical representations of the Lie super algebra gl(qr+s)
Abstract: Spherical representations are interesting because their matrix
coefficients are eigenfunctions of invariant differential operators on
symmetric spaces.
A classical theorem proven by Helgason in his '84 book classifies all
spherical representations of a noncompact semisimple Lie group with
Iwasawa decomposition G=KAN. It states that a given finite
dimensional irreducible representation (which is necessarily a highest
weight representation) contains a nonzero Kinvariant vector if and
only if the highest weight vector is Minvariant, where M is the
centralizer of A in K. This immediately translates into a condition on
the highest weight.
For the simplest example, the spherical representations of su(2)
containing a u(1) invariant vector are exactly those with even highest
weight.
(For physicists: The multiplets with integer total spin are exactly
those containing a state with vanishing magnetic quantum number.)
In my Diploma thesis we could fully generalize this results to the case
of gl(qr+1) (for r>q or high enough highest weight).
A necessary condition for a representation to be spherical is given for
any gl(qr+s) and in fact also for any strongly reductive symmetric pair.
The method of prove is similar to the one used by Schlichtkrull ('84).



15.11.2011

E. Vishnyakova (MPI Bonn), Locally free sheaves on complex supermanifolds
Abstract: The main results of our study are the following ones: the classification of locally free sheaves of modules which have a given retract in terms of nonabelian 1cohomology; the study of such sheaves on projective superspaces, in particular, generalization of the Barth  Van de Ven  Tyurin Theorem for supercase; a spectral sequence connecting the cohomology with values in a locally free sheaf of modules with the cohomology with values in its retract. In the case of split supermanifold the necessary and sufficient conditions for triviality of cohomology class which corresponds to the tangent sheaf are given. 


25.10.2011

M. Schulte (Bochum), Elementary aspects of the
topology of compact complex symmetric spaces: A discussion via examples (2)



18.10.2011

M. Schulte (Bochum), Elementary aspects of the
topology of compact complex symmetric spaces: A discussion via examples (1)


06.07.2011

M. Huruguen (Institut Fourier, Grenoble), Toric varieties and spherical embeddings over an arbitrary field



01.06.2011

D. Burns (Ann Arbor), Holomorphic extension and the MongeAmpère equation
Abstract: A famous theorem of Boutet de Monvel (1978) gives an exact relation between the maximal domain of extension of a holomorphic function and the domain of existence of a solution of the homogeneous complex MongeAmpère equation. We will present the subsequent results in this area, especially those which deal with the geometry of Grauert tubes, with Ricciflow and with the characterization of algebraic manifolds by means of solutions of the MongeAmpère equation. The new results presented are joint work, parts with each of R. Aguilar, V. Guillemin and Z. Zhang. 


25.05.2011

P. Ramacher (Marburg), Integral operators on the Oshima compactification of a Riemannian symmetric space of noncompact type. Kernel asymptotics and regularized traces
Abstract: Let $X=G/K$ be a Riemannian symmetric space of noncompact type, where $G$ denotes a connected real semisimple Lie group, and $K$ a maximal compact subgroup. Let $\widetilde X$ be the Oshima compactification of $X$, and $\pi$ the regular representation of $G$ on $\widetilde X$. We study integral operators on $\widetilde X$ of the form $\pi(f)$, where $f$ is a rapidly falling function on $G$, and characterize them within the framework of totally characteristic pseudodifferential operators, describing the singular nature of their kernels, which originates in the nontransitivity of the underlying $G$action. Since the holomorphic semigroup generated by a strongly elliptic operator associated to the representation $\pi$, as well as its resolvent, can be characterized as integral operators of the mentioned type, we obtain a description of the asymptotic behavior of the corresponding semigroup and resolvent kernels. In addition, a regularized trace for the convolution operators $\pi(f)$ is defined, and in case that $f$ has compact support in a certain set of transversal elements, a fixed point formula for this trace analogous to the AtiyahBott formula for the global character of an induced representation of $G$ is obtained. 


18.05.2011

B. Schumann (Köln), Geometric Realization of Crystals
Abstract: The talk will be about a geometric realization of crystals via the quiver varieties of Lusztig and Nakajima. Here the vertices are irreducible components of Lagrangian subvarieties and the crystal operators will be realized in a geometric way. We will restrict ourselves to the sl_n case, give a short introduction to crystals and motivate the construction through concrete examples. 

08.12.2010

S. Kousidis (Köln), Weight distribution in Demazure modules of sl2hat



24.11.2010

M. Stolz (Bochum) , Random matrices, Bergman kernels, and point
processes on compact complex manifolds



17.11.2010

W. Kirwin (Köln), Halfform quantization of toric varieties in the large complex structure limit
Abstract: I will discuss the large complex structure limit of halfform corrected quantizations of toric varieties. In particular, we will see that holomorphic sections concentrate on BohrSommerfeld orbits of the real torus action, and that these orbits are all associated to points in the interior of the moment polytope. One consequence is that the RiemannRoch number of the quantization can be correctly computed by counting integral points inside the moment polytope. 


10.11.2010

W. Kirwin (Köln), Asymptotic unitarity in "quantization commutes with symplectic
reduction"
Abstract: Let M be a compact Kaehler manifold equipped with a Hamiltonian of a compact Lie group G. Under some appropriate conditions, a classical result of Guillemin and Sternberg states that Ginvariant subspace of the Kaehler quantization of M is isomoprhic to the Kaehler quantization of the symplectic quotient M//G. Indeed, they construct a very natural isomorphism between the two spaces. This result is known as "quantization commutes with symplectic reduction". It turns out, though, that the GuilleminSternberg isomorphism is not, in general, unitary. As unitary equivalence is the natural notion of equivalence for Hilbert spaces, it is important to understand how badly unitarity fails. The next best thing to a unitary isomorphism would be an isomorphism which is unitary to leading order in hbar. Unfortunately, the GuilleminSternberg isomorphism is not even unitary to leading order in hbar. On the other hand, if one includes halfforms (the socalled metaplectic correction), the analogue of the GuilleminSternberg map *is* unitary to leading order in hbar. One can even compute the higher asymptotics of the obstruction to exact unitarity (as hbar goes to zero). In this talk, I will explain the background of the problem as well as the proof of the "asymptotic unitarity" of the GuilleminSternberg map. Then, I would like to discuss the higher asymptotics, leading to current work and potential applications in Kaehler geometry. This is partially joint work with Brian Hall (U. Notre Dame). 


20.10.2010

B. Camus (Bochum), Semiclassical analysis and quantum ergodicity


9.06.2010

A. Huckleberry (Bochum), A complex geometric Ansatz for constructing
Sp_nensembles of elliptic elements



2.06.2010

G. Marinescu (Köln), Witten deformations and holomorphic Morse inequalities



19.05.2010

H. Sebert (Bochum), Semiclassical limits in the BargmannFockquantization:
Toric varieties as models for asymptotic analysis



21.04.2010

E. Vishnyakova (Bochum), Parabolic subalgebras of Lie superalgebras


letzte Änderung am
2. September 2014