Geometry/topology group, Department of Mathematical Sciences, NTNU

Seminar Autumn 2009

Pedro de M. Rios (University of São Paulo): Topics on Spherical Quantization
19. november 2009, KJL22, Kjelhuset, 14.15 - 16.00
Abstract: The word quantization has various meanings, depending on context. Here it means the construction of a non-commutative product of functions on the 2-sphere, which is SO(3)- equivariant. We will briefly review three basic approaches to this problem, originated in the quantization of R2n, to concentrate on the approach via symbol correspondence, outlining some results.
Eldar Straume: Spin-j-systems and Symbol Correspondences
13. november 2009, 656, Sentralbygg 2, 10.15 - 12.00
Abstract:  By a spin-j quantum mechanical system we mean a Hilbert space H of dimension n + 1, where n = 2j and j is integral or half-integral, and H is endowed with an irreducible unitary representation of SU(2). The space of linear operators on H identifies with the tensor product HH, or also the algebra of square matrices of dimension n + 1, with an induced action of SU(2). The latter is, in fact, a representation of SO(3). On the other hand, the 2-sphere is the classical phase space for spin-j-systems, and here SO(3) acts by rotations, which induces an SO(3)-action on the polynomial functions of various degree. In fact, the space of polynomials of degree at most n can be identified, by a linear and SO(3)-equivariant map, with the matrix space HH. This map is referred to as a symbol correspondence. We shall define and explain the basic ideas behind all this, with focus on the family of Stratonovich-Weyl correspondences. For a given correspondence Ff, multiplication of operators F,G corresponds to the (non-commutative) star-product f*g of the associated functions on the sphere. The identity operator is represented by the function 1, and Hermitean operators are represented by the real functions. The star-product depends also on n, and a precise calculation is very difficult as n increases. In the limit, as n tends to infinity, f*g becomes the usual product fg. The study is a joint project with Pedro de M. Rios.
Haaken Annfelt Moe: K-ads II
30. oktober 2009, 656, Sentralbygg 2, 10.15 - 12.00
Haaken Annfelt Moe: K-ads
23. oktober 2009, 656, Sentralbygg 2, 10.15 - 12.00
Abstract:  K-ads and ad theories, basic definitions and some examples. We start with supporting definitions.
Marius Thaule: On the Classifying Space of Some Cobordism Categories
2. oktober 2009, 656, Sentralbygg 2, 10.15 - 12.00
Abstract: Baas, Cohen and Ramirez determined the homotopy type of the category of open and closed strings. In this talk I will relate their findings to the category of open strings and closed strings with "windows" and prove that these have homotopy equivalent classifying spaces. Furthermore, I will prove that the category of closed strings and the category of open strings without "windows" have the same homotopy type, i.e. their classifying spaces are homotopy equivalent.

Part of this work is joint with Elizabeth Hanbury.
Helge Maakestad: On Canonical Filtrations, Jet Bundles and Irreducible SL(E)-Modules
18. september 2009, 734, Sentralbygg 2, 10.15 - 12.00
Abstract: Assume E is a complex vector space of finite dimension and let SL(E) be the special linear group on E. Let V be an irreducible finite dimensional SL(E)-module with highest weight vector vV. Let g = Lie(SL(E)) be the Lie algebra of SL(E) and let P ⊆ SL(E) be the subgroup of elements stabilizing v. It follows that P is a parabolic subgroup. Let Uk(g)vV for k ≥ 1 be the canonical filtration of V. It follows that Uk(g)v is a P-module. There is an equivalence of categories between the category of rational P-modules and the category of locally free sheaves on SL(E)/P with an SL(E)-linearization. I will use this equivalence to give a geometric construction of Uk(g)v. I will apply this construction to study syzygies of discriminants of linear systems on projective space and flag varieties.
Martin Raussen (Aalborg Universitet): Directed Algebraic Topology
4. september 2009, 734, Sentralbygg 2, 14.15 - 15.00
Abstract:  Concurrency theory in Computer Science studies the effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods to deal with e.g. "the state space explosion problem". In recent years, models with a combinatorial/topological flavour have been introduced and investigated as tools in the analysis of concurrent processes. It is a common feature of these models that an execution corresponds to a directed path (d-path), and that homotopies preserving the directions have equivalent computations as a result.

In this talk, I will discuss a particular example of a directed space arising as a model of a Higher Dimensional Automaton. For such a space, I will describe a new method yielding the homotopy type of the space of traces (executions) as a prodsimplicial complex - with products of simplices as building blocks. This opens up for calculations of homology groups and other invariants from algebraic topology.

The prodsimplicial model arises from a covering of the trace space by contractible subspaces. Nonempty intersections of these subspaces form a poset with a classifying space that corresponds to the barycentric subdivision of the prodsimplicial model. To decide whether a particular simplex product is contained in the model or not depends on a list of inequalities.
Tore A. Kro (Høgskolen i Østfold): Principal Two-Bundles and Manifoldbundles
4. september 2009, 734, Sentralbygg 2, 10.15 - 12.00
Jack Morava (The Johns Hopkins University): The Madsen-Tillmann Spectra for 4-Dimensional Spin Cobordisms
12. august 2009, 734, Sentralbygg 2, 10.15 - 12.00