Seminar semiklassische Analysis und Darstellungstheorie

A. Huckleberry, P. Littelmann, G. Marinescu, N. Savale, D.-V. Vu, M. Zirnbauer

Dienstags, 10:15 -11:45, Cohn-Vossen Raum des Mathematischen Instituts (Raum 313)

Sommersemester 2022

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 n-dimensional 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 non-pluripolar volume of the I-model potential associated to u. We give applications to the quantization of partial equilibrium measures. Joint work with Mingchen Xia.

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.

Slawomir Kolodziej (Jagiellonian University), Complex Monge-Ampere and Hessian equations and their applications

I would like to give an overview of existence and stability results for complex Monge-Amp`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.

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.

Wintersemester 2021

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.

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.

Ved Datar (Indian Institute of Science), Some new rigidity results in Kahler geometry

Two basic comparison theorems from classical Riemannian geometry are Bishop-Gromov volume comparison theorem and the Bonnet-Myers' 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 Gromov-Hausdorff 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.

Hoang Chinh Lu (Paris Saclay), Monge-Ampere volumes on compact complex manifolds

We investigate in depth the behaviour of Monge-Ampere volumes of quasi-psh functions on a given compact hermitian manifold. We shall prove that the property for these Monge-Ampere volumes to stay bounded away from zero or infinity is a bimeromorphic invariant. We shall show in particular that a conjecture of Boucksom-Demailly-Paun-Peternell holds true if and only if such Monge-Ampere volumes stay bounded away from infinity.

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.

Sommersemester 2021 (talks on Zoom)

Ksenia Fedosova (Albert-Ludwigs-Universitat Freiburg), Spectral theory of infinite-volume 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.

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 well-studied. 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.

Michela Egidi (Ruhr-Universitat 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.

Yihan Li (Nankai University), An Index Theorem for End-Periodic Toeplitz Operators

In this talk, I will present a recent result on the index theorem for End-Periodic 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 odd-dimensional analogue of the index theorem for end-periodic Dirac operators by Mrowka-Ruberman-Saveliev. In particular, we find a new eta-type invariant in the result and we will show its relation with the eta-type invariant introduced by Dai-Zhang. The approach follows mainly the heat kernel method with a b-calculus-like modification. In the proof, we also introduce a eta-type invariant for end-periodic Dirac operators and a variation formula for it. This is a joint work with professor Guangxiang Su.

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.

Alix Deleporte (Universite Paris-Saclay), 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.

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.

Cyril Letrouit (Laboratoire Jacques-Louis 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 sub-Riemannian geometry.

Wintersemester 2020 (talks on Zoom)

16.03.2021 (18:00)    
Ming Xiao (University of California, San Diego), Normal Stein spaces with Bergman-Einstein 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 Kahler-Einstein 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.

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).

Guokuan Shao (Sun Yat-sen University), G-equivariant Szego kernel asymptotics on CR manifolds

Let $X$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature, which admits a connected compact Lie group $G$ action. In this talk, we will discuss a Boutet de Monvel-Sjostrand 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.

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 non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in a compact Kaehler surface. This is a joint work with Severine Biard (arXiv:2011.06379).

Xiaoshan Li (Wuhan University), Bergman-Einstein 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 ahler-Einstein. This talk is based on a joint work with Professor Xiaojun Huang.

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 CR-singularities 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.

Sommersemester 2020 (talks on Zoom)

Wei Guo Foo (Chinese Academy of Sciences), Equivalence problem of 5-dimensional real hypersurfaces of type $C_{2,1}$

We consider the CR equivalence problem of 5-dimensional real hypersurfaces in C3 that are Levi degenerate of constant rank 1, and are 2 non-degenerate in the sense of Freeman; and we will review some of the progress made in this area. A well-known 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 The-Anh Ta, we study such manifolds in the rigid setting, giving rise to 7-dimensional Lie sub-algebra of so(2,3), with 2 primary invariants.

Huan Wang (Academia Sinica), Semipositive Line Bundles and Growth of Dimension of Cohomology

Firstly, we recall Siu and Demailly's proof of Grauert-Riemenschneider conjecture, and Berndtsson's refined estimates on the cohomology of semipositive line bundles over compact complex manifolds. Secondly, on possibly non-compact 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.

Bo Liu (East China Normal University), Differential K-theory and localization formula for $\eta$ invariants

We obtain a localization formula in differential K-theory for S1-action. 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 K-theory which is interesting in its own right.

Wei-Chuan 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 Cauchy--Riemann 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 complex-valued 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 R-equivariant CR functions on an irregular Sasakian manifold is non-trivial in semi-classical limit.

Wintersemester 2019/20

Maxim Braverman (Northeastern University), Geometric quantization of non-compact and b-symplectic manifolds

We introduce a method of geometric quantization for of non-compact symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We then apply it to a class of compact manifolds with singular symplectic structure, called b-symplectic manifolds. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``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)

Benjamin Kuster (Orsay), Resonances of the frame flow generator on hyperbolic 3-manifolds

The frame flow on hyperbolic 3-manifolds 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 3-manifolds, based on Fourier analysis and a semiclassical calculus for powers of line bundles.

Son Duong (Universitat Wien), Semi-isometric CR immersions of CR manifolds into Kahler manifolds

In this talk, we discuss a notion of semi-isometric 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.

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 C-Y. Hsiao.

Mihajlo Cekic (MPIM Bonn/Orsay), Resonant spaces for volume-preserving 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 3-manifolds. In this talk, I will discuss an extension of this result to volume-preserving 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.

Sommersemester 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 Chin-Yu 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 Riemann-Roch

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 well-defined 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 Riemann-Roch 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.

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 Cheeger-Mueller theorem to spaces with isolated conical singularities by generalising the proof of Bismut and Zhang to the singular setting.

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.

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.

S. Finski (IMJ, Paris 7), Riemann-Roch-Grothendieck theorem for families of surfaces with hyperbolic cusps and its applications to the moduli space of curves

We generalize Riemann-Roch-Grothendieck 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 Weil-Petersson form in terms of the first Chern form of the Hodge line bundle, which generalizes the result of Takhtajan-Zograf. Our result gives also some non-trivial consequences on the growth of the Weil-Petersson form near the compactifying divisor of the moduli space, which permits us to give a new approach to some well-known results of Wolpert on the Weil-Petersson geometry of the moduli space of curves.

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

Wintersemester 2018/19

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: \[ (q-3)\,T_f(r) \leq \sum_{i=1}^qN_f^{[2]}(r,L_i) + o(T_f(r)). \] 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.

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.

N. Savale (Köln), Bochner Laplacian and Bergman kernel expansion of semi-positive 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 semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.

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 electro-magnetic Laplacian $(-ih\nabla-A)^2-hB$ on a bounded, regular, and simply-connected 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 low-lying eigenvalues in the semiclassical limit $h\to 0$. We will show the crucial role of the magnetic Cauchy-Riemann operators (and of their ellipticity), of the Bergman-Hardy spaces, and of the Riemann mapping theorem in the description of the spectrum. This is a joint work with Jean-Marie Barbaroux, Loïc Le Treust, and Edgardo Stockmeyer.

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 Ray-Singer analytic torsion, which can be related to the topological torsion by the Cheeger-Müller 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 W-invariants 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 Bismut-Ma-Zhang's results to the equivariant case.

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

N. Romao (Augsburg), Quillen metrics and geometric quantization of vortex moduli

Abstract: The vortex equations provide an equivariant generalization of Gromov-Witten 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 sigma-models, 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 finite-dimensional, and they can be interpreted as spaces of multi-spinors 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.

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.

Sommersemester 2018

K. Fritsch (Bochum), On equivariant embeddings for non-proper group actions

Abstract: There are several well-known 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 Lie-group 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 non-proper group actions. I will also give a small introduction into the theory of moment maps and their connection to CR manifolds.

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 finite-genus Riemann surfaces.

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 Callias-type 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 bulk-boundary correspondence in the Graf-Porta model of topological insulators is a special case of our result.

Wintersemester 2017/18

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).

G. Borot (MPI Bonn), Matrix models, topological and geometric recursion

Abstract: I will first review the structure of the large N expansion in 1d log-gases/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 Guionnet-Kozlowski, and Andersen-Orantin.

N. Savale (Köln), The Gauss-Bonnet-Chern theorem: a probabilistic perspective

Abstract: We prove a probabilistic refinement of the Gauss-Bonnet-Chern 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 zero-locus of an appropriate random section of the bundle.

L. Ioos (Paris), Lagrangian states in Berezin-Toeplitz quantization

Abstract: A quantization is a process which, given a classical dynamical system, produces the underlying quantum dynamics. In the case of Berezin-Toeplitz 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 so-called semi-classical 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 so-called Bohr-Sommerfeld condition. In this talk, I will construct these "Lagrangian states" in the framework of Berezin-Toeplitz quantization and study their semi-classical 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 Chern-Simons theory.

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 higher-dimensional generalizations and form a field called "stochastic Kähler geometry".

A. Drewitz (Köln), Recent developments in some percolation models with long-range correlations

Abstract: We will introduce the Gaussian free field and the model of random interlacements as prototypical examples for percolation models with long-range 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.

Sommersemester 2017

Mathematical Aspects of the Quantum Hall Effect

Wintersemester 2016/17

Mathematical Aspects of the Quantum Hall Effect

Sommersemester 2016

In this semester we continue to discuss Mathematical Aspects of the Quantum Hall Effect.

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

A. Alldridge (Köln), Non-commutative geometry and the quantum Hall effect-IV

A. Alldridge (Köln), Non-commutative geometry and the quantum Hall effect-III

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

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

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

G. Marinescu (Köln), Geometric quantization

G. Marinescu (Köln), Laplacians on Hermitian manifolds-II

F. Lapp (Köln), Laplacians and Dirac operators-II

F. Lapp (Köln), Laplacians and Dirac operators-I

Wintersemester 2015/16

This semester the seminar focuses on Mathematical Aspects of the Quantum Hall Effect.

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

G. Marinescu (Köln), Laplacians on Hermitian manifolds

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

A. Alldridge (Köln), Non-commutative geometry and quantum Hall effect (III)

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

I. Burban (Köln), Landau problem on a torus, line bundles and theta-functions

A. Alldridge (Köln), Non-commutative geometry and quantum Hall effect (II)

A. Alldridge (Köln), Non-commutative geometry and quantum Hall effect (I)

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

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

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

Sommersemester 2015

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

I. Burban (Köln), Derived categories and Fourier-Mukai transforms II

I. Burban (Köln), Derived categories and Fourier-Mukai transforms I

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

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

X. Ma (Paris/Köln), Introduction to geometric quantization

Vorsicht! Zeit- und Raumänderung: Mittwoch, 16:00 Uhr im Hörsaal des MI

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

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

Sommersemester 2014/15

This semester the seminar focuses on Integrable Systems.

H. Geiges (Köln), Integrable Systeme und Kontaktgeometrie

I. Burban (Köln), Lax-Paare und algebro-geometrische Methoden

L. Galinat (Köln), Adler-Kostant Schema und die klassische Yang-Baxter Gleichung

C. Lange (Köln), Geodätischer Fluss

A. Huckleberry (Bochum), Liouville-Arnold Integrabilität und Winkel-Wirkung Koordinaten-II

A. Huckleberry (Bochum), Liouville-Arnold Integrabilität und Winkel-Wirkung Koordinaten-I

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

T. Quella (Köln), Impulsabbildung und Noether-Sätze-II

A. Alldridge (Köln), Impulsabbildung und Noether-Sätze-I

G. Marinescu (Köln), Übersicht über symplektische Geometrie

AG Kunze (Köln), Hamilton-Jacobi Gleichung

M. Kunze (Köln), Übersicht über Hamiltonsche Systeme

Sommersemester 2014

George Marinescu, Metric aspects of Okounkov bodies

Vorbesprechung WS 14/15

W. Kirwin (Köln), Complex-time flows in toric geometry

I. Burban (Köln), Survey of the theory of the classical Yang-Baxter equation

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

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 N-flation 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.

P. Littelmann (Köln), Introduction to Newton-Okounkov bodies II

X. Ma (Paris und Köln), Atiyah-Singer Index Theorem VI

P. Littelmann (Köln), Introduction to Newton-Okounkov bodies I

X. Ma (Paris und Köln), Atiyah-Singer Index Theorem III

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

Wintersemester 2013/14

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 Cohen-Macaulay modules of covariants for the action of SL(n) on the set of m n-vectors. It also allows to describe the equivariant simple D-modules, 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.

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.

Dr. A. Jarosz (Holon Institute of Technology, Israel), Quantum dots and Jack polynomials

Abstract: The talk will discuss a random-matrix approach to quantum transport in chaotic quantum dots with one non-ideal 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.

D. Ostermayr (Bonn), Bott-Periodizität in KR-Theorie via äquivarianten Gamma-Räumen

Abstract: Atiyah's Reelle K-Theorie, oder KR-Theorie, ist eine Verallgemeinerung topologischer K-Theorie auf Räume mit Involution, die reelle und komplexe topologische K-Theorie als Spezialfälle enthält. Nach Einführung der grundlegenden Definitionen, werde ich ein Modell für das konnektive Cover kr als C_2-Spektrum vorstellen, welches auf Segal und Suslin zurückgeht. Anschließend werde ich, hierbei Suslin folgend, zeigen, wie aus Segal's Arbeit zu Gamma-Räumen eine Version der (1, 1)-Periodizität für KR-Theorie folgt.

Sommersemester 2013

This semester the seminar concentrates on spin models, with the following schedule.

Wintersemester 2012/13

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 high-energy 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 high-energy 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, Gromov-Witten 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.

W. Kirwin (Köln), Polarisierung in geometrischer Quantisierung

S. Klevtsov (Köln), Introduction to random metrics

W. Kirwin (Köln), Complex-time evolution in geometric quantization

Sommersemester 2012

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

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

A. Alldridge (Köln), Twisted equivariant matter

A. Alldridge (Köln), Twisted equivariant matter

Vorsicht! Zeit- und Raumänderung: Mittwoch, 10:15 Uhr im Seminarraum B der Chemie

M. Zirnbauer (Köln), Twisted equivariant matter

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 G-action 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.

Wintersemester 2011/12

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 non-conformal field theory is coupled to gravity, and generalize the standard Liouville model in two dimensions.

S. Schmittner (Köln), Spherical representations of the Lie super algebra gl(q|r+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 non-compact 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 non-zero K-invariant vector if and only if the highest weight vector is M-invariant, 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(q|r+1) (for r>q or high enough highest weight). A necessary condition for a representation to be spherical is given for any gl(q|r+s) and in fact also for any strongly reductive symmetric pair. The method of prove is similar to the one used by Schlichtkrull ('84).

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 non-abelian 1-cohomology; the study of such sheaves on projective superspaces, in particular, generalization of the Barth - Van de Ven - Tyurin Theorem for super-case; 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.

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

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

Sommersemester 2011

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

D. Burns (Ann Arbor), Holomorphic extension and the Monge-Ampère 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 Monge-Ampère equation. We will present the subsequent results in this area, especially those which deal with the geometry of Grauert tubes, with Ricci-flow and with the characterization of algebraic manifolds by means of solutions of the Monge-Ampère equation. The new results presented are joint work, parts with each of R. Aguilar, V. Guillemin and Z. Zhang.

P. Ramacher (Marburg), Integral operators on the Oshima compactification of a Riemannian symmetric space of non-compact type. Kernel asymptotics and regularized traces

Abstract: Let $X=G/K$ be a Riemannian symmetric space of non-compact 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 non-transitivity 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 Atiyah--Bott formula for the global character of an induced representation of $G$ is obtained.

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.

Wintersemester 2010/11

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

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

W. Kirwin (Köln), Half-form quantization of toric varieties in the large complex structure limit

Abstract: I will discuss the large complex structure limit of half-form corrected quantizations of toric varieties. In particular, we will see that holomorphic sections concentrate on Bohr-Sommerfeld 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 Riemann-Roch number of the quantization can be correctly computed by counting integral points inside the moment polytope.

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 G-invariant 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 Guillemin-Sternberg 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 h-bar. Unfortunately, the Guillemin-Sternberg isomorphism is not even unitary to leading order in h-bar. On the other hand, if one includes half-forms (the so-called metaplectic correction), the analogue of the Guillemin-Sternberg map *is* unitary to leading order in h-bar. One can even compute the higher asymptotics of the obstruction to exact unitarity (as h-bar 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 Guillemin-Sternberg 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).

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

Sommersemester 2010

A. Huckleberry (Bochum), A complex geometric Ansatz for constructing Sp_n-ensembles of elliptic elements

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

H. Sebert (Bochum), Semiclassical limits in the Bargmann-Fock-quantization: Toric varieties as models for asymptotic analysis

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

George Marinescu

letzte Änderung am

2. September 2014