24.07.2019

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



11.07.2019 (5:00 pm)

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



18.06.2019

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



28.05.2019

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



21.05.2019

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



30.04.2019

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



13.03.2019

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


22.1.2019

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



15.1.2019

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



18.12.2018

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



27.11.2018

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



20.11.2018

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



6.11.2018

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


30.10.2018

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



23.10.2018

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


19.06.2018

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



29.05.2018

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



10.04.2018

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


30.01.2018

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



23.01.2018

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



19.12.2017

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



5.12.2017

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



21.11.2017

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



07.11.2017

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




19.07.2016

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



05.07.2016

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



21.06.2016

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



14.06.2016

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



07.06.2016

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



31.05.2016

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



24.05.2016

G. Marinescu (Köln), Geometric quantization



10.05.2016

G. Marinescu (Köln), Laplacians on Hermitian manifoldsII



03.05.2016

F. Lapp (Köln), Laplacians and Dirac operatorsII



26.04.2016

F. Lapp (Köln), Laplacians and Dirac operatorsI



01.02.2016

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



25.01.2016

G. Marinescu (Köln), Laplacians on Hermitian manifolds



18.01.2016

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



11.01.2016

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



07.12.2015

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



30.11.2015

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



23.11.2015

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



16.11.2015

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



09.11.2015

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



02.11.2015

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



26.10.2015

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


14.07.2015

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



07.07.2015

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



30.06.2015

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



19.05.2015

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



12.05.2015

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



29.04.2015

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



28.04.2015

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



21.04.2015

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


20.01.2015

H. Geiges (Köln), Integrable Systeme und Kontaktgeometrie



13.01.2015

I. Burban (Köln), LaxPaare und algebrogeometrische Methoden



09.12.2014

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



02.12.2014

C. Lange (Köln), Geodätischer Fluss



25.11.2014

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



18.11.2014

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



11.11.2014

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



04.11.2014

T. Quella (Köln), Impulsabbildung und NoetherSätzeII



28.10.2014

A. Alldridge (Köln), Impulsabbildung und NoetherSätzeI



21.10.2014

G. Marinescu (Köln), Übersicht über symplektische Geometrie



14.10.2014

AG Kunze (Köln), HamiltonJacobi Gleichung



07.10.2014

M. Kunze (Köln), Übersicht über Hamiltonsche Systeme


08.07.2014

George Marinescu, Metric aspects of Okounkov bodies



01.07.2014

Vorbesprechung WS 14/15



17.06.2014

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



03.06.2014

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



27.05.2014

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



20.05.2014

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



13.05.2014

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



06.05.2014

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



29.04.2014

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



22.04.2014

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



15.04.2014

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

10.12.2013

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



03.12.2013

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



26.11.2013

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



22.10.2013

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


25.02.2013

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



14.12.2012

W. Kirwin (Köln), Polarisierung in geometrischer Quantisierung



27.11.2012

S. Klevtsov (Köln), Introduction to random metrics



13.11.2012

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


12.06.2012

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



22.05.2012

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



08.05.2012

A. Alldridge (Köln), Twisted equivariant matter



02.05.2012

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



24.04.2012

M. Zirnbauer (Köln), Twisted equivariant matter



17.04.2012

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


6.12.2011

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



22.11.2011

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



15.11.2011

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


25.10.2011

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



18.10.2011

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


06.07.2011

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



01.06.2011

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


25.05.2011

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


18.05.2011

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

08.12.2010

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



24.11.2010

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



17.11.2010

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


10.11.2010

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


20.10.2010

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


9.06.2010

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



2.06.2010

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



19.05.2010

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



21.04.2010

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


letzte Änderung am
2. September 2014