Seminar semiklassische Analysis und Darstellungstheorie

Wintersemester 2026

29.09.26, 1.10.26 and 6.10.2610:00 -11:30 in Room 313

Mingchen Xia (Hefei) , An introduction to I-good singularities (Minicourse)

The singularity type of a quasi-plurisubharmonic function on a compact Kähler manifold can be measured in two rather different ways: analytically, through non-pluripolar Monge–Ampère masses, and algebraically, through multiplier ideal sheaves. The two do not agree in general. The potentials for which they do agree are the I-good ones. They form the class in which global pluripotential theory behaves the way algebraic geometry predicts, and although non-I-good potentials exist, they are essentially invisible in geometric applications, much as non-measurable functions are invisible in everyday analysis.

In this mini-course I will give a general introduction to the theory of I-good singularities. After discussing the relevant envelope operators and the ways of comparing singularities, I will explain the theorem characterizing I-goodness, in terms of approximation by analytic singularities, of an identity between non-pluripolar mass and volume, and of the agreement of the analytic and the algebraic envelope. If time permits, I will indicate some applications, for instance to partial Bergman kernels.

Only the basic theory of plurisubharmonic functions will be assumed.

Wintersemester 2025

17.02.26

Sayani Bera (Indian Association for the Cultivation of Science, Kolkata) , Hénon maps, attracting basins and stable manifolds

The goal of this talk is to discuss the affirmative answer to Bedford's conjecture in $\mathbb{C}^2$ (formulated by Fornaess-Stensones in 2004 on Fatou-Bieberbach domains), which consequently also answers Bedford's original question (2000) on generalisation of stable manifold theorem in the holomorphic setup. The idea of the proof manifests from dynamical properties of appropriate non-autonomous (or iterative) families of Hénon maps and associated attracting basins, which naturally or canonically are either Fatou-Bieberbach domains or the Short $\mathbb{C}^2$'s. Further, we will also briefly discuss on generalizing the above idea to $\mathbb{C}^n$, $n>2$, via the (non-autonomous) dynamical properties of an appropriate class of weakly-regular maps.This is a joint work with Kaushal Verma

12.02.26

Siarhei Finski (Polytechnique) , Toeplitz Operators: Equidistribution, Algebraic Properties, and Localization

The spectral theory of Toeplitz operators originates in Szegö's classical work on the eigenvalue distribution of Toeplitz matrices. Boutet de Monvel and Guillemin later established an analogous theory in the setting of general complex manifolds. In this minicourse, we present a generalization of these results in the framework of arbitrary Bernstein-Markov measures on complex manifolds. We show that, for such general measures, the vector space of Toeplitz operators is asymptotically closed under composition and therefore forms an algebra in an asymptotic sense, thereby generalizing a result of Bordemann-Meinrenken-Schlichenmaier previously done in the setting of Boutet de Monvel and Guillemin. The corresponding equidistribution results then follow from this algebraic structure together with previously established equidistribution results for the diagonal Bergman kernel. A key ingredient in the proof of the asymptotic algebra property is a detailed analysis of the off-diagonal behavior of the associated Bergman kernel, which was previously conjectured by Zelditch.

02.02.26

Bowoo Kang (KAIST) , Pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flow

In this talk, I will present my recent results on the pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flow. We extend parabolic pluripotential theory introduced by Guedj, Lu, and Zeriahi, and prove the existence and uniqueness of the weak solution to the Cauchy-Dirichlet problem when the given measure is dominated by a Monge-Ampere measure of some bounded plurisubharmonic function.

21.10.25

Duc-Bao Nguyen (National University of Singapore) , Quantitative hyperbolicity of complex manifolds via numerical invariants

We introduce numerical invariants measuring the hyperbolicity of Kähler manifolds, using directed positive currents. Basic properties of these invariants are studied and compared with the notion of hyperbolicity of Kobayashi, Brody and Demailly. We also show that if a complex manifold X satisfies Demailly's condition of negative jet curvature, then it is hyperbolic in our sense. We then compute some examples, discuss some related problems, as well as a list of open questions. This is a joint work with Tien-Cuong Dinh and Duc-Viet Vu.

21.10.25

Xing Xu (University of Cologne) , The $H^2$-Corona problem on delta-regular domains

In 1962, L. Carleson solved the Corona problem on the unit disk. Since then, Corona problems in various function spaces have been extensively studied. Later, T. Wolff simplified Carleson's proof and provided estimates for the corresponding control constants of the solutions. In this talk, we present a proof of the $H^2$-Corona problem on delta-regular domains and give estimates for the associated control constants. This class of domains includes pseudoconvex domains whose defining functions are plurisubharmonic on the boundary and pseudoconvex domains of finite D'Angelo type. This talk is based on joint work with Professor Bo-Yong Chen.

21.10.25

Kuang-Ru Wu (Erdős center) , Mean curvature of direct image bundles

Let $E\to X$ be a vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$. It is known that if the line bundle $O_{P(E^*)}(1)$ over the projectivized bundle $P(E^*)$ is positive, then $E\otimes \det E$ is Nakano positive by the work of Berndtsson. In this talk, we give a subharmonic analogue. Let $p:P(E^*)\to X$ be the projection and $\alpha$ be a Kähler form on $X$. If the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $\Theta$ positive on every fiber and $\Theta^r\wedge p^*\alpha^{n-1}> 0$, then $E\otimes \det E$ carries a Hermitian metric whose mean curvature is positive.
As an application, we show that the following subharmonic analogue of the Griffiths conjecture is true: if the line bundle $O_{P(E^*)}(1)$ admits a metric $h$ with curvature $\Theta$ positive on every fiber and $\Theta^r\wedge p^*\alpha^{n-1}> 0$, then $E$ carries a Hermitian metric with positive mean curvature.

Sommersemester 2025

16.09.25

Nguyen Ngoc Cuong (KAIST) , Weak convergence of Monge–Ampère measures on compact Hermitian manifolds

We give a sufficient condition on a sequence of uniformly bounded ω-plurisubharmonic functions, ω being a Hermitian metric, for which the sequence of associated Monge–Ampère measures converges weakly. This criterion can be used to obtain a bounded ω-plurisubharmonic solution to the Monge–Ampère equation. This is joint work with S. Kołodziej.

24.06.25

Zhangchi Chen (East China Normal University, China) , Recent progress on hypercyclic holomorphic mappings with slow growth

A holomorphic map $h : X \to Y$ is called hypercyclic with respect to a generalised translation $T \in \operatorname{Aut}(X)$ if the orbit $\{h \circ T^n\}_{n \ge 1}$ is dense in the space $\mathcal{O}(X, Y)$ equipped with the compact-open topology. Dinh-Sibony asked what is the slowest growth rate of the Nevanlinna characteristic functions of hypercyclic entire curves $\mathbb{C} \to \mathbb{P}^1$.
We construct holomorphic entire curves $h : \mathbb{C} \to \mathbb{P}^m$ hypercyclic with respect to countably many translations with optimal slow growth rate. This result answers Dinh-Sibony’s question.
An entire curve $h : \mathbb{C} \to Y$ is called frequently hypercyclic with respect to some translation $T_a$ for some $a \in \mathbb{C}^*$ if for any non-empty open subset $U \subset \mathcal{O}(\mathbb{C}, Y)$, the set of positive integers $k \ge 1$ with $h(\,\cdot + k a) \in U$ has positive lower density. Any frequently hypercyclic entire curve into projective spaces $h : \mathbb{C} \to \mathbb{P}^m$ has order at least 1.
We construct order 1 entire curves $h : \mathbb{C} \to \mathbb{P}^m$ frequently hypercyclic with respect to translations in countably many directions. This result on the slow growth rate is optimal.

03.06.25

Frank Vallentin (University of Cologne) , Universally optimal point configurations on the unit sphere

How can we distribute N points on the unit sphere to minimize potential energy?
In general, this is a very difficult question. However, in some cases, the answer is surprisingly independent of the specific choice of potential function. That is, there are point configurations on the unit sphere that minimize potential energy for a broad class of potential functions. Such point configurations are called universally optimal.
In this lecture I want to give an introduction to the theory of universally optimal point configurations, developed by Cohn and Kumar in the paper arXiv:math/0607446 .
This theory draws on the following mathematical concepts: Linear programming bounds, Harmonic analysis on the unit sphere, Hermite interpolation, Spherical designs, Completely monotonic functions.

29.04.25

Koen van den Dungen (Universitat Bonn) , Dirac-Schroedinger operators and a Toeplitz index theorem

Dirac-Schroedinger operators are given by Dirac-type operators on a smooth manifold, together with a potential. I will describe a general notion of Dirac-Schroedinger operators with arbitrary signatures (with or without gradings), which allows us to study index pairings and spectral flow simultaneously. There is a general Callias Theorem, which computes the Fredholm index (or the spectral flow) of Dirac-Schroedinger operators in terms of well-known index pairings on a suitable compact hypersurface. I will then focus on the associated Toeplitz operators, which are obtained by compressing the potential to the kernel of the Dirac operator, and I will provide a general index theorem (and a spectral flow theorem) relating Toeplitz operators on the original manifold to Toeplitz operators on the compact hypersurface.

18.03.25

Hisatoshi Kodani (Kyushu University, Japan) , Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loops

Chern-Simons perturbation theory is one of the ways to extract topological invariants of 3-manifolds from Chern-Simons gauge theory. In the theory, topological invariants are given by integrations of differential forms associated with trivalent graphs. After reviewing mainly Bott-Cattaneo and Cattaneo-Shimizu’s works which are versions of the theory, I will explain our results which state that the latter invariants can be defined using only trivalent graphs without self-loops. This talk is based on joint work with Bingxiao Liu.

Wintersemester 2024

21.01.25

Weixia Zhu (University of Vienna) , Deformation of CR Structures and Stability of CR Embeddings

The interplay between deformation of complex structures and stability of the spectrum for the complex Laplacian on compact complex manifolds was extensively studied by Kodaira and Spencer in the 1950s. In this talk, we will discuss analogous problems for compact CR manifolds, with a focus on the connection between spectral stability and embedding stability for 3-dimensional pseudoconvex CR manifolds of finite type. This talk is based on joint work with Siqi Fu and Howard Jacobowitz.

10.12.24

Nguyen Duc Bao (National University of Singapore) , Pluripotential theory and diameter estimate in Kaehler geometry

In this talk, I will discuss some applications of pluripotential theory in the diameter problem in Kaehler geometry. More precisely, I will explain how to use the modulus of continuity of the solution of complex Monge-Ampere equation to obtain the diameter estimate, the application of complex Sobolev space, and the diameter estimate in the big cohomology class setting.

26.11.24

Song-Yan Xie (Chinese Academy of Sciences) , Hole probabilities of random zeros on compact Riemann surfaces

We establish a convergence speed estimate for hole probabilities of zeros of random holomorphic sections on compact Riemann surfaces. The proof is based on a density formula of Zelditch, the Abel-Jacobi theory, Fekete points theory, and a new perturbation method. This is joint work (arXiv:2406.19114) with Hao Wu (NUS).

19.11.24

Eva Miranda (UPC Barcelona) , When Fluid Computers Meet TQFTs

We present two geometric models as solutions to classical fluid dynamics equations: Beltrami fields—stationary solutions of the Euler equations—linked to Reeb vector fields and contact structures, and stationary solutions of the Navier-Stokes equations with cosymplectic geometry. These structures allow us to implement Turing-complete systems,which we call "Fluid Computers." We’ll discuss how these constructions can be unified using 2-categories to form advanced computational systems. Inspired by Topological Quantum Field Theory (TQFT), we explore the quantization of Turing-complete systems,introducing Topological Kleene Field Theory (TKFT), which combines computation and field theory with “flubits” from 3D Euler and Navier-Stokes flows. This talk is based on joint work with Ángel González-Prieto and Daniel Peralta-Salas.

12.11.24

Yi-Hsin Tsai (National Taiwan University) , Scaling Method of Bergman Kernel

We introduce a scaling method for the Bergman kernel on complex manifolds to derive its leading term, extending the approach to orbifold cases. From this method, we can explore applications such as the leading term of Toeplitz operator and the Kodaira-Baily embedding theorem.

05.11.2024

Vladimir Lazic (Universitat des Saarlandes) , Minimal metrics and the Abundance conjecture

The Abundance conjecture, in its simplest form, predicts that on a minimal projective manifold X some multiple of the canonical bundle of $X$ is globally generated. When the Euler-Poincare characteristic of $X$ does not vanish, I will present a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of $K_X$. I will also give strong evidence that an important class of currents with minimal singularities - supercanonical currents - is central to the completion of the proof of the Abundance conjecture for minimal varieties with non-vanishing Euler-Poincare characteristic.

29.10.2024

Sheng Meng (East China Normal University) , On the multiplicity of the ramification divisor

A critical point x of a differentiable self-map f of $X=R^n$ or manifolds means the determinant of the Jacobian $|J_f(x)|=0$. We collect these points and call it the critical set $C_f$. This set $C_f$ also carries algebraic data, denoted as $R_f$ which is nothing but the zeros (counting multiplicities) of the equation $|J_f(x)|=0$, called the ramification. In the algebraic geometric setting, $R_f$ turns out to be an effective divisor, but not necessarily smooth anymore. In this talk, I shall show you the behaviour of the coefficients of $R_f$ (multiplicity in codimension 1), and also the much more complicated, the singularities of $R_f$ (multiplicity in higher codimension). Some applications will be discussed if time permits.

22.10.2024

Huan Wang (Institute of Mathematics, Henan Academy of Sciences) , Holomorphic Morse inequalities revisited

The history of holomorphic Morse inequalities was initiated from the seminal work of J.-P. Demailly, which was influenced by the Witten’s analytic proof of classical Morse inequalities and Siu's solution of Grauert-Riemenschneider conjecture on Moishezon manifolds. These inequalities provide a flexible way to produce holomorphic sections of high tensor powers of line bundle under mild positivity assumption and have various generalizations in complex geometry. We will give an introduction and our result with D. Coman and G. Marinescu on singular holomorphic Morse inequalities on non-compact complex manifolds.

08.10.2024

Hamid Al-Saqban (Paderborn) , On the Unique Ergodicity for Foliations of Generic Abelian Differentials

In Teichmuller theory, a theorem due to H. Masur and W. Veech states that for generic Abelian differentials, the leaves of the horizontal foliation are uniquely ergodic. Said differently, the orbits of the horizontal straight-line flow on a generic translation surface are uniquely ergodic. Subsequently, G. Forni proved an effective form of this statement, establishing in particular precise power-laws for the deviations of ergodic averages of smooth functions from the power law (as predicted by the ergodic theorem). The main goal of this talk is to introduce and motivate translation surfaces, and to explain the main ideas behind a (microlocal) analytic approach to effective unique ergodicity for straight-line flows on a generic translation surface. This is a joint work in progress with D. Galli (University of Zurich).

04.10.2024

Marco Vergamini (Pisa) , Mixing and central limit theorems for Henon maps

Let f be a complex Henon map and $\mu$ its unique measure of maximal entropy. Recently, Bianchi-Dinh proved that $\mu$ is exponentially mixing of all orders for all Helder observables, and that all such observables satisfy the central limit theorem with respect to $\mu$. De Thelin-Vigny generalized these results for a certain class of bounded plurisubharmonic observables. We prove that these properties hold for all, not necessarily bounded, plurisubharmonic observables. This is a joint work with Hao Wu.

Sommersemester 2024

05.07.2024

Gian Maria Dall'Ara (Pisa) , An uncertainty inequality for the dbar-operator

I will present an inequality for functions of one-complex variable that I introduced a few years ago in my work on off-diagonal estimates for Bergman kernels in dimension greater than one, and that recently provided the key analytical tool to establish sharp subelliptic estimates for the dbar-Neumann problem on a class of dilation-invariant domains (in collaboration with Samuele Mongodi, University of Milano-Bicocca). In both cases the main purpose of the uncertainty inequality is to prove spectral gap estimates of appropriate complex Laplacians, under a weak plurisubharmonicity assumption on a weight (a.k.a. semipositivity of the relevant curvature).

25.06.2024

Shuang Su (Cologne) , Volumes of components of Lelong upper level sets

Let $X$ be a compact Kähler manifold, and let $T$ be a closed positive $(1,1)$-current in a nef cohomology class on $X$. The Lelong number is an important notion for measuring the singularities of currents on $X$. In this talk I will discuss some recent work with Duc Viet Vu.We establish an optimal upper bound for the volume of components of Lelong upper level sets of $T$ in terms of cohomology classes of non-pluripolar self-products of $T$.

11.06.2024

Wangjian Jian (AMSS) , A brief tour to the analytic MMP and some recent progresses

We will give a brief introduction to Song-Tian’s Analytic Minimal Model Program, which aims at using Ricci flow to study the MMP in birational geometry. We will also talk about some recent progresses of finite time singularities of the Kähler-Ricci flow.

16.04.2024

Bo Liu (East China Normal University) , Family APS Index Theorem and Higher Spectral Flow

In 1975, Atiyah-Patodi-Singer established the index theorem for manifold with boundary. The family extension of the APS index theorem was given by Bismut-Cheeger in invertible case and extended to the general case by Melrose-Piazza. The boundary term there is the eta form with perturbation. In this talk, we will generalize the eta form to the equivariant case for a fiberwise compact Lie group action and discuss the properties of it by using Dai-Zhang higher spectral flow. We will also discuss some other problems about the family index theorem with boundary.

18.03.2024

Giuliano Angelo Zugliani (State University of Campinas, Brazil) , Topological properties of closed 1-forms and its applications.

This talk will address some results obtained on the study of closed 1-forms defined on manifolds, and related to a conjecture of V. I. Arnol’d. Our initial interest was to produce a class of globally solvable systems of linear PDEs on compact manifolds.

Wintersemester 2023

30.01.2024

Alexander Rashkovskii (University of Stavanger) , Plurisubharmonic geodesics and plurisubharmonic interpolation

For any negative psh function on a bounded domain of $\mathbb{C}^n$, we construct a psh function determined by the asymptotical behavior of the given function near its singularity points, both inside the domain and on its boundary. We study properties of such ‘residual functions’ and their relations to asymptotic psh rooftops. The considerations are motivated by a problem on when two given psh functions can be connected by a psh geodesic.

23.01.2024

Ngoc Cuong Nguyen (KAIST, Korea) , Regularity of the Siciak-Zaharjuta extremal function on compact Kähler manifolds

We prove that the regularity of the extremal function of a compact subset of a compact Kähler manifold is a local property, and that the continuity and Hoelder continuity are equivalent to classical notions of the local L-regularity and the locally Hoelder continuous property in pluripolential theory. As a consequence we give an effective characterization of the ($C^\alpha$, $C^\alpha$')-regularity of compact sets, the notion introduced by Dinh, Ma and Nguyen. Using this criterion all compact fat subanalytic subsets in $\mathbb{R}^n$ are shown to be regular in this sense.

30.11.2023 (14:00)

Bingyuan Liu (University of Texas Rio Grande Valley) , The Diederich-Fornaess index and the dbar-Neumann problem

A domain is said to be pseudoconvex if -log(-d(z)) is plurisub- harmonic in the domain, where d is a signed distance function of the domain. The study of global regularity of dbar-Neumann problem on bounded pseudoconvex domains is dated back to the 1960s. However, a complete understanding of the regu- larity is still absent. On the other hand, the Diederich-Fornaess index was in- troduced in 1977 originally for seeking bounded plurisubharmonic functions. Through decades, enormous evidence has indicated a relationship between global regularity of the dbar-Neumann problem and the Diederich-Fornaess in- dex. Indeed, it has been a long-lasting open question whether the trivial Diederich-Fornaess index implies global regularity. In this talk, we will intro- duce the backgrounds and motivations. The main theorem of the talk proved recently by Emil Straube and me answers this open question for (0, n-1) forms.

28.11.2023

Shin Kikuta (Kogakuin Universit) , Residue and volume growth of Kahler-Einstein metric over quasi-projective manifolds with boundary of maximal or minimal Kodaira dimension

In this talk, I report some progress of our research about a boundary behavior of the almost-complete Kahler-Einstein metric of negative Ricci curvature on a quasi-projective manifold with semiample log-canonical bundle. In particular, its volume growth near the boundary is investigated in terms of the Kodaira dimension of the boundary when the boundary is of general type or Calabi-Yau. A relation between the residue of the metric along the boundary and the generalized Kahler-Einstein metric is also discussed.

21.11.2023

Yasushi Homma (Waseda University) , Rarita-Schwinger fields on Einstein manifolds

Spin geometry studies the Dirac operator and spinors on spin manifolds. One famous theorem is that there exists no non-trivial harmonic spinor on a positive scalar curvature manifold because of Lichnerowicz's formula. What happens for the spin 3/2 case? As stated in the physics literature, the Rarita-Schwinger operator on spin $3/2$ fields is an analog of the Dirac operator, and Rarita-Schwinger (RS) fields, which are in kernel of the RS operator, are regarded as harmonic spinors with spin $3/2$. In contrast to spin $1/2$ case, positive scalar curvature is not the condition to rule out the existence of non-trivial RS fields. In fact, we can find examples of compact Einstein manifolds with/without RS fields, where the key is to use a variety of Weitzenböck formulas. In this talk, I will talk recent results about RS fields with $3/2$-spin on Einstein manifolds, if time permits, also talk about spinor fields with higher spin on manifolds of constant curvature.

31.10.2023

Max Reinhold Jahnke (Koeln) , Left-invariant involutive structures on compact Lie groups

It is well known that the De Rham cohomology of a compact Lie group is isomorphic to the cohomology of the Chevalley-Eilenberg complex. While the former is a topological invariant of the Lie group, the latter can be computed by using simple linear algebra methods. In this talk, we discuss how to obtain an injective homomorphism between the cohomology spaces associated with left-invariant involutive structures and the cohomology of a generalized Chevalley-Eilenberg complex. We discuss some cases in which the homomorphism is surjective, such as the Dolbeault cohomology and certain elliptic and CR structures. The results provide new insights regarding the general theory of involutive structures as, for example, they reveal algebraic obstructions for solvability for the associated differential complexes.

17.10.2023

Cezar Joita (Institute of Mathematics of the Romanian Academy) , On germs of morphisms between complex spaces

We will talk about images of germs of complex spaces through germs of holomorphic maps with particular emphasis on surjectivity conditions. Based on joint works with Mihnea Coltoiu and Mihai Tibar.

Sommersemester 2023

01.08.2023

Chung-Ming Pan (Toulouse), Kaehler-Einstein metrics on families of Fano varieties.

In this talk, I will introduce a pluripotential approach to study uniform a priori estimates of Kähler-Einstein (KE) metrics on families of Fano varieties. After briefly recalling the basic tools and the variational approach, I will define a notion of convergence of quasi-plurisubharmonic functions in families of normal varieties and extend several classical properties under this context. Finally, I will explain how these elements lead to an analytic proof of the openness of existing singular KE metrics and a uniform estimate of KE potentials. This is joint work with Antonio Trusiani.

01.08.2023

Andrea Galasso (Milan), Star products on CR manifolds.

We present a recent joint work with Chin-Yu Hsiao. Let X be a connected orientable compact CR manifold with non-degenerate Levi form. We study the algebra of Toeplitz operators on X and we establish star product for a certain class of symbols on X.

04.07.2023

Christiaan van de Ven (Wuerzburg), Quantum theories and their classical limit: a C*-algebraic approach.

Quantization in general refers to the transition from a classical to a corresponding quantum theory. The inverse issue, called the classical limit of quantum theories, is considered a much more difficult problem. A rigorous and natural framework that addresses this problem exists under the name strict (or C*-algebraic) deformation quantization. In this talk, I will first introduce this concept by means of the relevant definitions. Subsequently, I will show how this can be applied as a tool to study the classical limit of quantum theories. More precisely, the so-called quantization maps allow one to take the limit of suitable sequence of algebraic states indexed by a semi-classical parameter in which the sequence may converge to a probability measure on the pertinent phase space, as this parameter approaches zero. In addition, since this C*-algebraic approach allows for both quantum and classical theories, it provides a convenient way to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off this parameter. These ideas are illustrated with mean-field quantum spin systems indexed by the number of lattice sites. Finally ,a short summary on how to extend this algebraic framework in the context of quantum spin systems with local interactions (e.g. the Heisenberg model) is provided.

02.05.2023

Tomasso Pacini (Torino), Pluri-potential theory and the geometry of submanifolds.

here are several well-known relationships between convexity and pluri-potential theory. We shall review various cases in which such relationships help understand geometric questions regarding submanifold geometry, both in the context of totally real/Lagrangian submanifolds and in the context of calibrated geometry. We shall also consider the recent extension of pluri-potential theory to more general geometric settings, due to Harvey-Lawson, and ponder its possible geometric applications.

Wintersemester 2022

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 arXiv:2201.03249 joint with Philipp Schmitt and arXiv: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 Kähler manifold $X$ converging to an infinite Galois covering $\phi : \widetilde{X} \rightarrow X$, the sequence of push-forward 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 IBS-CCG.

The talk takes place in Seminarraum 2 at 10:30 am.

13.12.2022

Pau Mir Garcia (Barcelona), Bohr-Sommerfeld quantization of b-symplectic toric manifolds

Bohr-Sommerfeld 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 non-compact 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 Bohr-Sommerfeld 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

b-Structures and other generalizations (such as E-symplectic structures) are ubiquitous and sometimes hidden, unexpectedly, in a number of problems including the space of pseudo-Riemannian geodesics and regularization transformations of the three-body problem. E-symplectic 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 E-symplectic 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 E-symplectic 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 brand-new perspective.

Sommersemester 2022

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 (Perron-Frobenius) 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 Tien-Cuong 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 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.

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 Kähler manifold.

10.5.2022

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.

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.

Wintersemester 2021

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 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.

1.12.2021

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.

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.

Sommersemester 2021 (talks on Zoom)

20.07.2021

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.

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 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.

06.07.2021

Michela Egidi (Ruhr-Universitat Bochum), The observability problem for generalized rectangles

We will introduce the (internal) observability problem for generalized rectangles in $\mathbb{R}^d$, i.e. domains which are exhausted by hyperrectangles in $\mathbb{R}^d$, 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 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.

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 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.

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 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.

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 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.

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 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).

01.12.2020

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.

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 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)

07.07.2020

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 $\mathfrak{so}(2,3)$. 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 $\mathfrak{so}(2,3)$, 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 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.

23.06.2020

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.

16.06.2020

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

27.02.2020

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)

04.02.2020

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.

21.01.2020

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.

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 C-Y. Hsiao.

3.12.2019

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

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 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 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.

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 Cheeger-Mueller 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), 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.

13.03.2019

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

Wintersemester 2018/19

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

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 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.

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 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.

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 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.

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 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.

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.

Sommersemester 2018

19.06.2018

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.

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 finite-genus 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 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

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 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.

19.12.2017

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.

5.12.2017

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.

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

07.11.2017

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.

19.07.2016

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

05.07.2016

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

21.06.2016

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

14.06.2016

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

07.06.2016

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

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 manifolds-II

03.05.2016

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

26.04.2016

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

Wintersemester 2015/16

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

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), Non-commutative 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 theta-functions

23.11.2015

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

16.11.2015

A. Alldridge (Köln), Non-commutative 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

Sommersemester 2015

14.07.2015

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

07.07.2015

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

30.06.2015

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

Sommersemester 2014/15

This semester the seminar focuses on Integrable Systems.

20.01.2015

H. Geiges (Köln), Integrable Systeme und Kontaktgeometrie

13.01.2015

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

09.12.2014

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

02.12.2014

C. Lange (Köln), Geodätischer Fluss

25.11.2014

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

18.11.2014

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

11.11.2014

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

04.11.2014

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

28.10.2014

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

21.10.2014

G. Marinescu (Köln), Übersicht über symplektische Geometrie

14.10.2014

AG Kunze (Köln), Hamilton-Jacobi Gleichung

07.10.2014

M. Kunze (Köln), Übersicht über Hamiltonsche Systeme

Sommersemester 2014

08.07.2014

George Marinescu, Metric aspects of Okounkov bodies

01.07.2014

Vorbesprechung WS 14/15

17.06.2014

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

03.06.2014

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

13.05.2014

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

06.05.2014

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

29.04.2014

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

22.04.2014

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

15.04.2014

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

Wintersemester 2013/14

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 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.

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 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.

22.10.2013

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

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 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.

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), Complex-time evolution in geometric quantization

Sommersemester 2012

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 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

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 non-conformal 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 $\mathfrak{gl}(q|r+s)$

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 $\mathfrak{su}(2)$ 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 $\mathfrak{gl}(q|r+1)$ (for r>q or high enough highest weight). A necessary condition for a representation to be spherical is given for any $\mathfrak{gl}(q|r+s)$ and in fact also for any strongly reductive symmetric pair. The method of prove is similar to the one used by Schlichtkrull ('84).

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 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.

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)

Sommersemester 2011

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 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.

25.05.2011

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.

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 $\mathfrak{sl}_n$ case, give a short introduction to crystals and motivate the construction through concrete examples.

Wintersemester 2010/11

08.12.2010

S. Kousidis (Köln), Weight distribution in Demazure modules of $\widehat{\mathfrak{sl}_2}$

24.11.2010

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

17.11.2010

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.

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 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).

20.10.2010

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

Sommersemester 2010

9.06.2010

A. Huckleberry (Bochum), A complex geometric Ansatz for constructing $Sp_n$-ensembles 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 Bargmann-Fock-quantization: Toric varieties as models for asymptotic analysis

21.04.2010

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