###
15/10/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Igor Kanatchikov, *Institute of Theoretical Physics, Free University of Berlin*

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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 — Room P3.10, Mathematics Building

Lina Oliveira, *Instituto Superior Técnico*

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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
${A}_{s}$ of
$A$, known as the symmetric part of
$A$, and of a partial triple product
$(a,b,c)\to abc$ mapping
$A\times {A}_{s}\times A$ 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

- 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.
- Lina Oliveira,
*Weak*-closed Jordan ideals
of nest algebras*, Math. Nach., to appear.

###
23/01/2003, 11:00 — 12:00 — Room P3.10, Mathematics Building

José Velhinho, *Univ. Beira Interior and CENTRA*

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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

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

###
19/12/2002, 13:30 — 14:30 — Room P3.10, Mathematics Building

João Baptista, *Cambridge University*

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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
${R}^{2}>N$ this moduli space is
$C{P}^{N}$, 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
${R}^{2}$ is close to
$N$.

### References

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

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19/12/2002, 11:00 — 12:00 — Room P3.10, Mathematics Building

Ana Bela Cruzeiro, *Instituto Superior Técnico*

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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

- A. B. Cruzeiro and P. Malliavin -"Renormalized differential
geometry on path space: structural equation, curvature", J.Funct.
Anal. 139 (1996), p. 119 -181.
- 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 — Room P3.10, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

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The coadjoint orbits of
$\mathrm{SL}(2,C)$ - II

In my talks (17/10 and 14/11) I will sketch the construction of the
principal series of unitary representations of
$\mathrm{SL}(2,C)$ 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
$\mathrm{SL}(2,C)$.
### References

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

###
14/11/2002, 11:00 — 12:00 — Room P3.10, Mathematics Building

Jorge Drumond Silva, *Instituto Superior Técnico*

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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

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

###
17/10/2002, 13:30 — 14:30 — Room P3.10, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

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The coadjoint orbits of SL(2,C)

###
17/10/2002, 11:00 — 12:00 — Room P3.10, Mathematics Building

Roger Picken, *Instituto Superior Técnico*

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Introduction to the Chas-Sullivan bracket

###
11/04/2002, 14:45 — 15:45 — Room P3.10, Mathematics Building

Roger Picken, *Instituto Superior Técnico*

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Twisted vector bundles and twisted K-theory

###
14/03/2002, 16:00 — 17:00 — Room P3.10, Mathematics Building

Pedro Castelo Ferreira, *University of Sussex e
Instituto Superior Técnico*

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From Compact Maxwell Chern-Simons to Azbel Hofstadter

###
14/03/2002, 14:15 — 15:15 — Room P3.10, Mathematics Building

Carlos Florentino, *Instituto Superior Técnico*

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

###
24/01/2002, 16:00 — 17:00 — Room P3.10, Mathematics Building

Carlos Florentino, *Instituto Superior Técnico*

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

###
24/01/2002, 14:15 — 15:15 — Room P3.10, Mathematics Building

Johannes Aastrup, *Copenhagen University*

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Deformation quantization of endomorphism bundles

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20/12/2001, 16:00 — 17:00 — Room P3.10, Mathematics Building

Nuno Romão, *Cambridge University*

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Quantum vortices on a sphere

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20/12/2001, 14:15 — 15:15 — Room P3.10, Mathematics Building

Jean Claude Zambrini, *Faculdade de Ciências, Universidade de Lisboa*

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

###
14/11/2001, 16:00 — 17:00 — Room P3.10, Mathematics Building

Aleksandar Mikovic, *Universidade Lusófona*

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Spin-foam
state-sum models

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14/11/2001, 14:15 — 15:15 — Room P3.10, Mathematics Building

Benjamin Steinberg, *University of Porto*

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The lifting and
classification problems for subcoverings

###
26/07/2001, 15:45 — 16:45 — Room P3.10, Mathematics Building

J. Scott Carter, *Dept. of Mathematics and Statistics, Univ. of South Alabama*

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

###
26/07/2001, 14:00 — 15:00 — Room P3.10, Mathematics Building

J. Scott Carter, *Dept. of Mathematics and Statistics, Univ. of South Alabama*

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Mini-course: Knots, Knotted Surfaces, and Quandles - Lecture 4