| 22.05.2012
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H.-C. Herbig (Aarhus), On orbifold criteria for singular symplectic toric quotients
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| 08.05.2012
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A. Alldridge (Köln), Twisted equivariant matter
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| 02.05.2012
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A. Alldridge (Köln), Twisted equivariant matter
Vorsicht! Zeit- und Raumänderung: Mittwoch, 10:15 Uhr im Seminarraum B der Chemie
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| 24.04.2012
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M. Zirnbauer (Köln), Twisted equivariant matter
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| 17.04.2012
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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.
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| 6.12.2011
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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.
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| 22.11.2011
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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).
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| 15.11.2011
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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. |
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| 25.10.2011
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M. Schulte (Bochum), Elementary aspects of the
topology of compact complex symmetric spaces: A discussion via examples (2)
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| 18.10.2011
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M. Schulte (Bochum), Elementary aspects of the
topology of compact complex symmetric spaces: A discussion via examples (1)
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| 06.07.2011
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M. Huruguen (Institut Fourier, Grenoble), Toric varieties and spherical embeddings over an arbitrary field
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| 01.06.2011
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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. |
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| 25.05.2011
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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. |
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| 18.05.2011
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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. |
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| 08.12.2010
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S. Kousidis (Köln), Weight distribution in Demazure modules of sl2hat
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| 24.11.2010
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M. Stolz (Bochum) , Random matrices, Bergman kernels, and point
processes on compact complex manifolds
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| 17.11.2010
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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. |
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| 10.11.2010
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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). |
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| 20.10.2010
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B. Camus (Bochum), Semiclassical analysis and quantum ergodicity
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| 9.06.2010
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A. Huckleberry (Bochum), A complex geometric Ansatz for constructing
Sp_n-ensembles of elliptic elements
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| 2.06.2010
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G. Marinescu (Köln), Witten deformations and holomorphic Morse inequalities
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| 19.05.2010
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H. Sebert (Bochum), Semiclassical limits in the Bargmann-Fock-quantization:
Toric varieties as models for asymptotic analysis
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| 21.04.2010
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E. Vishnyakova (Bochum), Parabolic subalgebras of Lie superalgebras
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letzte Änderung am
12. April 2012