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30/10/2003, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática
Pedro Lopes, Instituto Superior Técnico

Nós e os Quandles I

Apresentação dos resultados da tese de doutoramento.

References:

  1. P. Lopes, Quandles at finite temperatures I, J. Knot Theory Ramifications, 12(2):159-186 (2003), math.QA/0105099
  2. F. M. Dionisio and P. Lopes, Quandles at finite temperatures II, J. Knot Theory Ramifications to appear, math.GT/0205053
  3. J. Bojarczuk and P. Lopes, Quandles at finite temperatures III, submitted to J. Knot Theory Ramifications

15/10/2003, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Igor Kanatchikov, Institute of Theoretical Physics, Free University of Berlin

Precanonical quantization of the Yang-Mills fields and the mass gap problem

We overview the ideas of the precanonical quantization approach and apply it to the Yang-Mills fields. We show how the approach reduces the mass gap problem to a quantum mechanical spectral problem similar to that for the magnetic Schroedinger operator with a Clifford-valued magnetic field. Reference: hep-th/0301001.

23/01/2003, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática
Lina Oliveira, Instituto Superior Técnico

Ideals of nest algebras

Complex Banach spaces are naturally endowed with an algebraic structure, other than that of a vector space. The holomorphic structure of the open unit ball in a complex Banach space A leads to the existence of a closed subspace As of A, known as the symmetric part of A, and of a partial triple product (a,b,c)abc mapping A× As ×A to A. The existence of a Jordan triple identity satisfyied by this algebraic structure relates any complex Banach space to the Banach Jordan triple systems important in infinite-dimensional holomorphy.
A nest algebra, which is a primary example of a non-self-adjoint algebra of operators, is also an interesting case of a complex Banach space whose symmetric part is a proper subspace.
The ideals of nest algebras related to its associative multiplication have been extensively investigated, and whilst it is clear that ideals in the associative sense provide examples of ideals in the partial triple sense, the converse assertion remains in general an open problem. It is the aim of this talk to show that, in a large class of nest algebras, the weak*-closed ideals in the partial triple sense are also weak*-closed ideals in the associative algebra sense.
A brief overview of how the partial triple produtct arises from the holomorphic structure of the open unit ball of the nest algebra will also be given.

References

  1. Jonathan Arazy, An application of infinite dimensional holomorphy to the geometry of Banach space, Geometrical aspects of functional analysis, Lecture Notes in Mathematics, Vol. 1267, Springer-Verlag, Berlin/Heidelberg/New York, 1987.
  2. Lina Oliveira, Weak*-closed Jordan ideals of nest algebras, Math. Nach., to appear.

23/01/2003, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
, Univ. Beira Interior and CENTRA

Representations of holonomy algebras and shadow states

It has been argued that the Ashtekar-Lewandowski representation of the Ashtekar-Isham holonomy algebra is fundamental, in the sense that any other representation can be obtained by a suitable limit procedure. We propose to clarify that statement, providing, in particular, a canonical way of mapping GNS states to a family of vectors of the Ashtekar-Lewandowski Hilbert space. The so-called family of shadow states thus obtained converges, as states of the algebra, to the original GNS state.

References

  1. M. Varadarajan, Phys. Rev D. 64 , 104003 (2001); gr-qc/0104051
  2. J.M. Velhinho, Commun. Math. Phys. 227, 541 (2002); math-ph/0107002
  3. A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 18, L117 (2001); gr-qc/0107043
  4. T. Thiemann, gr-qc/0206037
  5. H. Sahlmann, gr-qc/0207112
  6. A. Ashtekar, J. Lewandowski and H. Sahlmann, gr-qc/0211012

19/12/2002, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática
João Baptista, Cambridge University

Vortex dynamics on the sphere

I will first give a brief overview of the Bogomolny equations for vortices, their moduli space of solutions, and the method of geodesic approximation. In the case of N vortices on a sphere of radius R2 >N this moduli space is C PN , but the geodesic method cannot be directly applied, because the solutions of the Bogomolny equations are not known explicitly. I will then try to show how to circumvent this problem in the limit where R2 is close to N.

References

  1. J. M. Baptista and N. S. Manton, The dynamics of vortices on S2 near the Bradlow limit, hep-th/0208001.
  2. T. M. Samols, Vortex Scattering, Commun. Math. Phys. 145, 149 (1992).

19/12/2002, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
, Instituto Superior Técnico

Differential geometry on the path space and applications

When trying to construct a Riemannian geometry on the path space of a Riemannian manifold several approaches could be thought about. The local chart approach, considering the path space as an infinite dimensional manifold and the basic tangent space the Cameron-Martin Hilbert space, leads to the study of the so-called Wiener-Riemann manifolds. Several difficulties appear in this study, namely the difficulty of finding an atlas such that the change of charts is compatible with the probabilistic structure (preserves the class of Wiener measures together with the Cameron-Martin type tangent spaces) and the non-availability of an effective computational procedure in the local coordinate system. Indeed, in infinite dimensions, the summation operators of differential geometry become very often divergent series. But the path space is more than a space endowed with a probability: time and the corresponding Itô filtration provide a much richer structure. In particular, the parallel transport over Brownian paths can be naturally defined by a limiting procedure from ODEs to SDEs. The stochastic parallel transport defines a canonical moving frame on the path space: the point of view we have adopted is the one of replacing systematically the machinery of local charts by the method of moving frames (as in Cartan theory). In this way it is possible to transfer geometrical quantities of the path space to the classical Wiener space and use Itô calculus to renormalize the apriori divergent expressions. An effective computational procedure is then achieved, where Stochastic Analysis and Geometry interact, not only on a technical level, but in a deeper way: Stochastic Analysis makes it possible to define geometrical quantities, Geometry implies new results in Stochastic Analysis. An application to assymptotics of the vertical derivatives of the heat kernel associated to the horizontal Laplacian on the frame bundle is discussed.
References
  1. A. B. Cruzeiro and P. Malliavin -"Renormalized differential geometry on path space: structural equation, curvature", J.Funct. Anal. 139 (1996), p. 119 -181.
  2. A. B. Cruzeiro, P. Malliavin and S. Taniguchi - "Ground state estimations in gauge theory", Bull Sci.Math. 125, 6-7 (2001), p. 623-640.

14/11/2002, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática
, Universidade do Algarve

The coadjoint orbits of SL(2,C) - II

In my talks (17/10 and 14/11) I will sketch the construction of the principal series of unitary representations of SL(2,C) and the coadjoint orbits they correspond to. I will also try to explain the geometric quantization of these orbits and show that the infinitesimal representations thus obtained are indeed the ones corresponding to the principal series. If there is any time left I might say something about the Kirillov character formula for SL(2,C).

References

  1. I.M. Gelfand, R.A. Minlos, Z.Ya. Shapiro, Representations of the rotation and Lorentz groups and their applications, Pergamon Press, 1963.
  2. A. Kirillov, Elements of representation theory, Springer.
  3. G.W. Mackey, Induced representations of locally compact groups I, Ann. Math. 55(1):101-139, 1952.

14/11/2002, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Jorge Drumond Silva, Instituto Superior Técnico

How the Weyl quantization brings about an improvement in Egorov's Theorem

Using ideas from the Weyl quantization, we show how the classical theorem by Y. Egorov, on conjugation of pseudo-differential operators by Fourier integral operators, can have its accuracy increased.

References

  1. G. Folland, Harmonic Analysis in Phase Space, Princeton University Press.
  2. J. Duistermaat, Fourier Integral Operators, Birkhauser.
  3. L. Hormander, The Analysis of Linear Partial Differential Operators, Vol. III, IV, Springer-Verlag.

17/10/2002, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática
, Universidade do Algarve

The coadjoint orbits of SL(2,C)

17/10/2002, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
, Instituto Superior Técnico

Introduction to the Chas-Sullivan bracket

11/04/2002, 14:45 — 15:45 — Sala P3.10, Pavilhão de Matemática
, Instituto Superior Técnico

Twisted vector bundles and twisted K-theory

14/03/2002, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Pedro Castelo Ferreira, University of Sussex e Instituto Superior Técnico

From Compact Maxwell Chern-Simons to Azbel Hofstadter

14/03/2002, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
, Instituto Superior Técnico

The Verlinde algebra in conformal field theory, algebraic geometry and K-theory - II

24/01/2002, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
, Instituto Superior Técnico

The Verlinde algebra in conformal field theory, algebraic geometry and K-theory.

24/01/2002, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
Johannes Aastrup, Copenhagen University

Deformation quantization of endomorphism bundles

20/12/2001, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Nuno Romão, Cambridge University

Quantum vortices on a sphere

20/12/2001, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
, Faculdade de Ciências, Universidade de Lisboa

A stochastic reinterpretation of the method of E. Cartan of exterior differential systems for the heat equation

14/11/2001, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
Aleksandar Mikovic, Universidade Lusófona

Spin-foam state-sum models

14/11/2001, 14:15 — 15:15 — Sala P3.10, Pavilhão de Matemática
, University of Porto

The lifting and classification problems for subcoverings

26/07/2001, 15:45 — 16:45 — Sala P3.10, Pavilhão de Matemática
, Dept. of Mathematics and Statistics, Univ. of South Alabama

Mini-course: Knots, Knotted Surfaces, and Quandles - Lecture 5

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Organizadores correntes: Roger Picken, Marko Stošić.

Projecto FCT PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry.

CAMGSD FCT