## Seminar semiklassische Analysis und Darstellungstheorie

### A. Alldridge, A. Huckleberry, S. Klevtsov, P. Littelmann, G. Marinescu, N. Savale, M. Zirnbauer

#### 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 \emph{integral integral affine manifold.} I will discuss the relation of integral integral affine structures to quantization as well as some associated results, in particular the fact that for a regular Lagrangian fibration $M \to B$, the Riemann-Roch number of $M$ is equal to the number of "integral points" in $B$. Along the way we encounter the fact that the volume of $B$ is equal to the number of integral points, a simple claim from "integral integral affine geometry" whose proof turns out to be surprisingly tricky. This is joint work with Yael Karshon and Takahiko Yoshida. 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 gl(q|r+s) Abstract: Spherical representations are interesting because their matrix coefficients are eigenfunctions of invariant differential operators on symmetric spaces. A classical theorem proven by Helgason in his '84 book classifies all spherical representations of a non-compact semisimple Lie group with Iwasawa decomposition G=KAN. It states that a given finite dimensional irreducible representation (which is necessarily a highest weight representation) contains a non-zero K-invariant vector if and only if the highest weight vector is M-invariant, where M is the centralizer of A in K. This immediately translates into a condition on the highest weight. For the simplest example, the spherical representations of su(2) containing a u(1) invariant vector are exactly those with even highest weight. (For physicists: The multiplets with integer total spin are exactly those containing a state with vanishing magnetic quantum number.) In my Diploma thesis we could fully generalize this results to the case of gl(q|r+1) (for r>q or high enough highest weight). A necessary condition for a representation to be spherical is given for any gl(q|r+s) and in fact also for any strongly reductive symmetric pair. The method of prove is similar to the one used by Schlichtkrull ('84). 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 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 sl2hat 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

George Marinescu

letzte Änderung am

2. September 2014