Seminar semiklassische Analysis und Darstellungstheorie

A. Alldridge, I. Burban, A. Huckleberry, P. Littelmann, G. Marinescu, M. Zirnbauer

Dienstags, 10:15 -11:45, Seminarraum 3 des Mathematischen Instituts (Raum 314)

Wintersemester 2016/17

Mathematical Aspects of the Quantum Hall Effect

Sommersemester 2016

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

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

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

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

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

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

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

G. Marinescu (Köln), Geometric quantization

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

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

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

Wintersemester 2015/16

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

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

G. Marinescu (Köln), Laplacians on Hermitian manifolds

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

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

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

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

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

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

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

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

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

Sommersemester 2015

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

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

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

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

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

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

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

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

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

Sommersemester 2014/15

This semester the seminar focuses on Integrable Systems.

H. Geiges (Köln), Integrable Systeme und Kontaktgeometrie

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

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

C. Lange (Köln), Geodätischer Fluss

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

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

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

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

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

G. Marinescu (Köln), Übersicht über symplektische Geometrie

AG Kunze (Köln), Hamilton-Jacobi Gleichung

M. Kunze (Köln), Übersicht über Hamiltonsche Systeme

Sommersemester 2014

George Marinescu, Metric aspects of Okounkov bodies

Vorbesprechung WS 14/15

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

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

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

Th. Bachlechner (Heidelberg), Inflation in UV theories

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

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

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

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

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

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

Wintersemester 2013/14

J. Weyman (Essen), Local cohomology supported in determinantal varieties

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

A. Kahle (Bochum), An elementary theorem for Field Theories

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

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

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

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

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

Sommersemester 2013

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

Wintersemester 2012/13

N. Orantin (Lisabon), From random matrix theory to enumerative geometry, a journey into integrable systems

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

W. Kirwin (Köln), Polarisierung in geometrischer Quantisierung

S. Klevtsov (Köln), Introduction to random metrics

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

Sommersemester 2012

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

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

A. Alldridge (Köln), Twisted equivariant matter

A. Alldridge (Köln), Twisted equivariant matter

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

M. Zirnbauer (Köln), Twisted equivariant matter

G. Schwarz (Brandeis), Automorphisms of quotients

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

Wintersemester 2011/12

S. Klevtsov (Brussels), 2D gravity and random Kahler metrics

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

S. Schmittner (Köln), Spherical representations of the Lie super algebra gl(q|r+s)

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

E. Vishnyakova (MPI Bonn), Locally free sheaves on complex supermanifolds

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

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

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

Sommersemester 2011

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

D. Burns (Ann Arbor), Holomorphic extension and the Monge-Ampère equation

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

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

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

B. Schumann (Köln), Geometric Realization of Crystals

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

Wintersemester 2010/11

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

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

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

Abstract: I will discuss the large complex structure limit of half-form corrected quantizations of toric varieties. In particular, we will see that holomorphic sections concentrate on Bohr-Sommerfeld orbits of the real torus action, and that these orbits are all associated to points in the interior of the moment polytope. One consequence is that the Riemann-Roch number of the quantization can be correctly computed by counting integral points inside the moment polytope.

W. Kirwin (Köln), Asymptotic unitarity in "quantization commutes with symplectic reduction"

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

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

Sommersemester 2010

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

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

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

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

George Marinescu

letzte Änderung am

2. September 2014