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17/09/2015, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática
Jedrzej Swiezewski, University of Warsaw

Geometrical diffeomorphism invariant observables for General Relativity and their applications

During the talk I will present a recent construction of observables for General Relativity invariant under spatial diffeomorphisms. The construction involves introducing a local structure representing the "observer" (based on arXiv:1403.8062). I will also present how those observables can be used to reduce the phase space of canonical General Relativity (based on arXiv:1506.09164).

If time permits, I will argue that the construction is particularly useful in spherically symmetric situations. This realization lead to a proposal of a scheme of reducing Loop Quantum Gravity to its spherically symmetric sector, which completes the standard, midisuperspace approach (based on arXiv:1410.5609).

Ver também

swiezewski.pdf

15/06/2015, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
Kishore Marathe, City University of New York

Physical Mathematics: old and new

In this talk  we will discuss some topics related  to the interaction of physical and mathematical theories that have led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. I coined the term Physical Mathematics to describe this new and fast growing area of research and used it in the title of my paper in Springer's book Mathematics Unlimited: 2001 and beyond.

We will discuss some recent developments in this area. General reference for this talk is my book Topics in Physical Mathematics, Springer (2010).

22/04/2015, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
, University of Erlangen-Nürnberg

Black hole entropy in loop quantum gravity

After briefly introducing the main ingredients of the loop quantum gravity approach, I show how it can applied to the calculation of black hole entropy. I review some well known results and open issues resulting from the interplay with Chern-Simons theory techniques. I then introduce a new analysis of the horizon degrees of freedom in terms of purely LQG methods, which turns out to be dual to a CFT description. I show how this unifying framework allows us to recover the semiclassical Bekenstein-Hawking entropy formula.  

Ver também

Pranzetti_Black_Hole_in_Loop_Quantum_Gravity_Slides_20150422.pdf

25/03/2015, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
, Grupo de Fisica Matematica, Univ. Lisboa

Introduction to Loop Quantum Gravity (part 3)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

In the third lecture we will finish the construction of the spin-knot space and introduce the loop transform. Then we move on to the analysis of geometric observables (distance, area and volume) and the structure of the scalar constraint. Finally, matter coupling will be introduced. If time permits, we will also give a short review of two applications of the formalism: calculation of the black hole entropy, and the Big Bounce model of Loop Quantum Cosmology.

18/03/2015, 16:00 — 17:00 — Sala P3.10, Pavilhão de Matemática
, Grupo de Fisica Matematica, Univ. Lisboa

Introduction to Loop Quantum Gravity (part 2)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The second lecture is devoted to the canonical quantization procedure within the LQG framework. We will begin by a short introduction to the notion of background independence, and differences between perturbative and nonperturbative quantization. We will then rewrite general relativity in the canonical space+time formulation and introduce Ashtekar variables, as preparation for the canonical quantization. Then the main step is the quantization itself, and the construction of the appropriate Hilbert space of the theory based on the notions of spin networks and spin-knots.

Please note the change of date from Wednesday 11th March to Wednesday 18th March.

04/03/2015, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática
, Grupo de Fisica Matematica, Univ. Lisboa

Introduction to Loop Quantum Gravity (part 1)

This series of lectures is intended to give an elementary introduction to the topic of the canonical quantization of the gravitational field, in the context of the Loop Quantum Gravity approach.

The first lecture will be devoted to the formulation of the problem of quantization of the gravitational field. We will give an overview of perturbative quantization, discuss the issue of nonrenormalizability, and provide a general classification of most prominent approaches to constructing a theory of quantum gravity. One such approach is Loop Quantum Gravity, which will be studied in more detail in subsequent lectures.

26/02/2015, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
Saikat Chatterjee, Institut des Hautes Études Scientifiques, France

Twisted actions of categorical groups

We develop a theory of twisted actions of categorical groups using the notion of semidirect product of categories. I will present many examples of semi-direct product of categories. If time permits I will also work-out an example of twisted action involving the Poincaré 2-group. Specializing to the case of representations, where the the category on which categorical group acts has some kind of a vector space structure, we will establish a categorical analogue of Schur's lemma.

This is a joint work with A. Lahiri and A. Sengupta.

17/06/2014, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
, The Australian National University, Canberra

Holography for parallel conformal data

The Fefferman-Graham ambient metric construction, with some technical asterisks, positively resolves the Dirichlet problem for compactification of asymptotically hyperbolic Einstein metrics, the compactification that occurs in the AdS/CFT correspondence. We show that data on the conformal boundary parallel with respect to Cartan's normal conformal connection — which is nearly the same thing as a holonomy reduction of the conformal structure — can be extended (again with an asterisk) to data parallel with respect to a natural connection on a corresponding bundle over the bulk, which in particular enables holographic study of such data. As an application, we use this extension result to construct metrics of exceptional holonomy.

Note unusual day/time and room

27/05/2014, 14:30 — 15:30 — Sala P3.10, Pavilhão de Matemática
Jonathon Funk, University of the West Indies, Barbados

Grothendieck topologies for C*-algebras

We investigate a (contravariant) functor from C*-algebras to toposes and geometric morphisms that generalizes the Gelfand spectrum in the commutative case. The functor produces a locale, presented by means of a Grothendieck topology on an inf-semilattice of 'Gelfand' opens \([U;a]\).

27/05/2014, 13:30 — 14:30 — Sala P3.10, Pavilhão de Matemática
Andreas Döring, Friedrich-Alexander-Universität Erlangen-Nürnberg

The Spectral Presheaf as the Spectrum of a Noncommutative Operator Algebra

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from the category of unital C*-algebras to a category of presheaves that contains the spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures can lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.

26/05/2014, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática
, Thammasat University, Bangkok, Thailand

Higher Categories of Operator Algebras

A satisfactory marriage between “higher” categories and operator algebras has never been achieved: although (monoidal) C*-categories have been systematically used since the development of the theory of superselection sectors, higher category theory has more recently evolved along lines closer to classical higher homotopy.

We present axioms for strict involutive \(n\)-categories (a vertical categorification of dagger categories) and a definition for strict higher C*-categories and Fell bundles (possibly equipped with involutions of arbitrary depth), that were developed in collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.

In order to treat some very natural classes of examples arising from the study of hypermatrices and hyper-C*-algebras, that would be otherwise excluded by the standard Eckmann-Hilton argument, we suggest a non-commutative version of exchange law and we also explore alternatives to the usual globular and cubical settings.

Possible applications of these non-commutative higher C*-categories are envisaged in the algebraic formulation of Rovelli's relational quantum theory, in the study of morphisms in Connes' non-commutative geometry, and in our proposed “modular” approach to quantum gravity (arXiv: 1007.4094).

Note: unusual time/day and room

14/05/2014, 16:00 — 17:00 — Sala P4.35, Pavilhão de Matemática
Sara Tavares, University of Nottingham, United Kingdom

Two-dimensional state sum models and spin structures

Topological field theories are very special in two dimensions: they have been classified and provide a rich class of examples. In this talk I will discuss a new state sum construction for these models that considers not just the topology of surfaces but also their spin structure. Emphasis is given to partition functions: I will detail general properties of these manifold invariants and discuss some non-trivial examples.

02/04/2014, 16:00 — 17:00 — Sala P4.35, Pavilhão de Matemática
Aleksandar Mikovic, Universidade Lusófona and Grupo de Física Matemática

2-BF Theories

We will describe 2-BF topological field theories, which are categorical generalization of the BF theories. We will also explain how to construct invariants of manifolds by using 2-BF theory path integrals.

References

  1. João Faria Martins, Aleksandar Mikovic,. Lie crossed modules and gauge-invariant actions for 2-BF theories. Adv. Theor. Math. Phys. Volume 15, Number 4, 1059-1084 (2011).
  2. Aleksandar Mikovic, Marko Vojinovic, Poincaré 2-group and quantum gravity,. Class. Quant. Grav. 29, 165003 (2012).

01/04/2014, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
, University of Zagreb

Coherent states for quantum groups

Quantum groups at roots of unity appear as hidden symmetries in some conformal field theories. For this reason I. Todorov has (in 1990s) used coherent state operators for quantum groups to covariantly build the field operators in Hamiltonian formalism. I tried to mathematically found his coherent states by an analogy with the Perelomov coherent states for Lie groups. For this, I use noncommutative localization theory to define and construct the noncommutative homogeneous spaces, and principal and associated bundles over them. Then, in geometric terms, I axiomatize the covariant family of coherent states which enjoy a resolution of unity formula, crucial for physical applications. Even the simplest case of quantum \(\operatorname{SL2}\) is rather involved and the corresponding resolution of unity formula involves the Ramanujan's \(q\)-beta integral. The correct covariant family differs from ad hoc proposed formulas in several published papers by earlier authors.

Ver também

lispr1.pdf
Note: unusual time/day and room

12/03/2014, 16:00 — 17:00 — Sala P4.35, Pavilhão de Matemática
, Grupo de Fisica Matemática, Universidade de Lisboa

Introduction to renormalization in QFT (part III)

In the previous talk we discussed the renormalization procedure on the example ϕ 4 scalar field theory. In this lecture we will conclude the analysis of that example, construct the final renormalized state sum, and discuss the renormalization group equations. At the end we will give some final general remarks about renormalization in QFT.

Ver também

https://math.tecnico.ulisboa.pt/seminars/download.php?fid=9

26/02/2014, 16:00 — 17:00 — Sala P4.35, Pavilhão de Matemática
John Huerta, IST, Lisbon

What can higher categories do for physics? Part II

In this follow up to last year's talk, we briefly review the cobordism hypothesis that formed the subject of our first part, and then outline its use for the existence and construction of field theories, in particular Chern-Simons theory, as discussed in a 2009 paper of Freed, Hopkins, Lurie and Teleman.

22/01/2014, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática
, Grupo de Fisica Matemática, Universidade de Lisboa

Introduction to renormalization in QFT (part II)

In the previous talk we gave an overview of the renormalization procedure in Quantum Field Theory. In this lecture we will demonstrate that abstract procedure on a simple explicit example, the so-called ϕ 4 theory of a single real scalar field. We will illustrate the construction of a renormalized state sum using two different regularization schemes, construct the renormalization group equations, and discuss some of their properties.

Ver também

2014-Lisbon-TQFTclub-Renormalization-Lecture.pdf

18/12/2013, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática
Marko Vojinovic, Grupo de Fisica Matemática, Universidade de Lisboa

Introduction to renormalization in QFT

We will give an overview of the renormalization procedure in Quantum Field Theory. The emphasis will be on the general idea of constructing a finite QFT from the one plagued by divergencies, in the standard perturbative approach, and discussing the uniqueness of the resulting QFT. The lecture does not assume much background knowledge in QFT, and should be accessible to a wide audience.

Ver também

https://math.ist.utl.pt/seminars/download.php?fid=9

11/12/2013, 17:00 — 18:00 — Sala P4.35, Pavilhão de Matemática
Carlos Guedes, AEI, Golm-Potsdam

The non-commutative Fourier transform for Lie groups

The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. In quantum mechanics on the Euclidean space, the standard Fourier transform gives a unitary map between the position representation -- functions on the configuration space -- and the momentum representation -- functions on the corresponding cotangent space. That is no longer the case for systems whose configuration space is a more general Lie group. In this talk I will introduce a notion of Fourier transform that extends this duality to arbitrary Lie groups.

arXiv:1301.7750

04/12/2013, 16:30 — 17:30 — Sala P4.35, Pavilhão de Matemática
, Universidade Lusófona and GFM, Universidade de Lisboa

Quantum mechanics in phase space: The Schrödinger and the Moyal representations

I will present some recent results on the dimensional extension of pseudo-differential operators. Using this formalism it is possible to generalize the standard Weyl quantization and obtain, in a systematic way, several phase space (operator) representations of quantum mechanics. I will present the Schrodinger and the Moyal phase space representations and discuss some of their properties, namely in what concerns the relation with deformation quantization.

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Organizadores correntes: Roger Picken, Marko Stošić.

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

CAMGSD FCT