TQFT Club Meeting 25-Nov-1999

12.30 Lunch party meets at Restaurante "Rota do Colombo" (previously called "O Mimo"). All welcome.

Afternoon Session

Room 3.10 Mathematics Department, IST

14.00 - 15.00 Rui Vilela Mendes (Grupo de Física Matemática, Universidade de Lisboa)

Summary: The algebras of non-relativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanics algebra is also unstable. Its stabilization requires the non-commutativity of the space-time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanics algebra has important consequences on the geometry of space-time, on quantum stochastic calculus and on the construction of quantum fields. I will descuss some of these effects.

References:

1. R. Vilela Mendes, "Geometry, stochastic calculus and quantum fields in a non-commutative space-time", math-ph/9907001.
2. R. Vilela Mendes, "The noncommutative geometry of stable relativistic space-time", Preprint IFM-6-94
3. R. Vilela Mendes, "Deformations, stable theories and fundamental constants", J. Phys. A27 (1994) 8091.

15.00 - 16.00 Roger Picken (Instituto Superior Técnico)

Summary: In this talk we describe some recent results relating, on the one hand, to a method of quantizing a model of gravity in two space and one time dimensions, and on the other, to an algebraic structure - quantum matrix pairs - appearing in this context, which has many similarities with quantum groups.

References:

1. J. E. Nelson and R. F. Picken, Quantum holonomies in (2+1)-dimensional gravity, gr-qc/9911005.
2. J. E. Nelson and R. F. Picken, Quantum matrix pairs, math.QA/9911015.

16.00 - 16.30 Tea & chocolate biscuits!

16.30 - 17.30 Paulo Semião (Universidade do Algarve)

Summary: A Topological quantum field theory (TQFT) in dimension n+1 assigns, among other things, a vector space V_{\Sigma} to each closed oriented n-manifold \Sigma, and a vector Z(M) in V_{\Sigma} to each oriented n+1-manifold M with boundary \Sigma, and this assignment must satisfy certain axioms, known as the Atiyah axioms. In this talk I will try to describe, very briefly, the Atiyah's and Quinn's definitions of a TQFT and give an example of a class of TQFT's based on the Euler characteristic. One of the most important properties of a TQFT is the gluing axiom. I will try to show a new way of gluing spaces and the possibility of incorporating these ideas in a categorical language.

References:

1. M. Atiyah, "Topological Quantum Field Theories", Publ. Math. Inst. Hautes Etudes Sci., 68 (1989) 175.
2. John C. Baez and James Dolan, "Higher dimensional algebra and topological quantum field theory", J. Math. Phys., 36 (1995) 60.
3. F. Quinn, "Geometry and Quantum Field Theory", American Mathematical Society,1995, Vol. I, Mathematical Series.
4. F. Quinn, "Group categories and their field theories", math.QA/9811047.

DATE: Thursday, 25/11/99

VENUE: Mathematics Department, Room 3.10

URL: http://www.math.ist.utl.pt/~rpicken/tqft