**email: jmourao@tecnico.ulisboa.pt**

* Keywords:* Mathematical physics, quantum gravity,
theories of connections, rigorous quantum theory, symplectic
geometry, algebraic geometry, functional analysis, difeomorphism
invariant theories

Member of the Center for Mathematical Analysis, Geometry, and Dynamical Systems, CAMGSD.

President of the Mathematics Department of IST

From October 25 to November 24, 2015, I visited the Mathematics Department of Geneva University.

From July 1, 2014 to February 15, 2015 I was guest professor at the Institute for Quantum Gravity of the Friedrich-Alexander Universität Erlangen-Nürnberg, Germany.

From September 1994 to July 1996 I was visiting Assistant Professor at the Physics Department of the University of Algarve.

From September 1993 to July 1994 I was on sabatical leave at the Center for Gravitational Physics and Geometry of the Penn State University.

I am interested in different aspects of the interaction between
Quantum Field Theory and Modern Geometry.

One thing that
fascinates me is the role of Feynman Path Integral in Modern
Mathematics. Some reflections about PI in the proceedings
of the XII
Fall Workshop on Geometry and Physics, Coimbra
University, 2003.

Presently I am, with João
Pimentel Nunes and other collaborators, investigating evolution
in complex time both in quantum physics and in geometry.

Here
are some notes
(see also below) of a minicurse at the International
School on Geometry, Grupoids and Quantization, University
of Hong Kong, 2013.

April 2018: Non-uniqueness of quantization, (no-)reality conditions, complex time evolution and generalized coherent state transforms

Radboud UniversityDecember 2016: Complexified Hamiltonian Symplectomorphisms and Solutions of the Homogeneous Complex Monge-Ampere Equation

Hong Kong Geometry ColloquiumDecember 2015: Quantum Mechanics versus Complex Geometry: Lecture Notes.

Lectures at the Physics Department of the Beijing Normal University.Fall 2015: Imaginary time in Kähler geometry, quantization and tropical amoebas: Notes for the

Seminar (overview of the course) and

Lectures: Index, L1, L2, L3, L4, L5.

at the Mathematics Department of Geneva University.Fall 2014: Quantum Mechanics and Kähler Geometry, Lectures 1-3 (notes), Lectures 4, 5 (notes).

Lectures at the Institute for Quantum Gravity, Friedrich-Alexander Universität Erlangen-Nürnberg.February 3, 2014: Imaginary time flow in geometric quantization and in Kähler geometry, degeneration to real polarizations and tropicalization,

Seminar at the Mathematics Department, University of Illinois at Urbana-Champaign.January 31, 2014: Imaginary time flow in geometric quantization and in Kähler geometry degeneration to real polarizations and tropicalization,

Felix Klein Seminar at the Mathematics Department of the University of Notre Dame.January 24, 2014: Imaginary time flow in geometric quantization and in Kähler geometry, degeneration to real polarizations and tropicalization,

Seminar at the Mathematics Department of the University of Michigan.December 18, 2013: Imaginary time flow in geometric quantization, degeneration to real polarizations and tropicalization,

Seminar at the WaGaRy workshop of the Institute of Mathematics of the Polish Academy of Sciences.November 2, 2013: Decomplexification of integrable systems, quantization and Kähler geometry,

Minicourse given at the International School on Geometry, Grupoids and Quantization, University of Hong Kong..October 17, 2013: Geodesics on the space of Kähler metrics, geometric tropicalization and quantization,

Seminar at the Mini-Workshop on Geometry, Chinese University of Hong Kong.October 16, 2013: Complex time evolution in geometric quantization,

Seminar at the Faculty of Sciences and Technology, University of Macau.September 5, 2013: Tropicalization in symplectic geometry and degeneration to real polarizations,

Seminar at the Conference on Tropical Aspects in Geometry and Topology, Max Planck Institute for Mathematics, Bonn.

(1990-1994)

João Nuno Tavares, Associate Professor at the Departamento de Matematica Pura, Faculdade de Ciencias, Universidade do Porto on:

*Mathematical methods on the quantization of first class constrained systems.*(1990-1993; co-supervisor. The supervisor was Paulo Crawford)

Paulo Moniz, Full Professor at the Departamento de Fisica da Universidade da Beira Interior on:

*Symmetries in general relativity and exact solutions.*(1990-1993; co-supervisor. The supervisor was Alfredo B. Henriques)

Paulo Sá, Associate Professor at Área Departamental de Física, Universidade do Algarve on:

*Inflationary models.*(1995-2001; supervisor. The co-supervisor was Nenad Manojlovic)

José Manuel Velhinho Assistant Professor at the Departamento de Fisica da Universidade da Beira Interior on:

*Quantization of systems with non-trivial phase spaces.*(2003 - 2006)

Pedro Matias on:

*Application of geometric quantization techniques to quantum field theory.*

(2000 - 2002)

Fernando Passos on:

*Representations
of the Virasoro Algebra and Conformal Field Theories. *

I completed my undergraduate studies in June 1986 in the Physics Department of Moscow State University (MSU). Wrote a a M.Sc. thesis on propagation of electromagnetic waves in crystals under the supervision of Vladimir Ivanovich Grigoriev.

In 1985 I was very lucky to meet and attend classes on differential geometry, Lie groups and Lie algebras by Yuri Kubyshin and Igor Volobuev. They, together with Dmitrii Vassilievitch Shirkov, became in September 1986 the supervisors of my PhD thesis which I completed in MSU in June 1988.

Defended habilitation in Physics at the IST in June 1995.

jmourao@tecnico.ulisboa.pt