Research

Areas of research: Algebraic geometry, differential geometry, algebraic topology.

- Gauge theory and manifolds of dimension 4. Proof of the Atiyah-Floer Conjecture and the finite type property of the Donaldson invariants of 4-manifold. Recently, study of Spin (7)-instantons on tori of dimension 8.

- Rational homotopy. Concept of s-formality. Construction of the first symplectic simply-connected non-formal 8-manifold. Property of formality for manifolds with different geometric structures (Sasakian, K-contact, co-symplectic, co-Kähler).

- Symplectic geometry and Kähler geometry. Asymptotically holomorphic embedding of symplectic manifolds in grassmannians, and symplectic Lefschetz pencils adapted to Lagrangian submanifolds. Construction of the first symplectic and complex simply-connected 6-manifold that is not Kähler. Symplectic resolution of orbifolds.

- Sasakian manifolds. Study of formality of Sasakian and K-contact manifolds. Classification of simply-connected 5-manifolds (Smale-Barden manifolds) with positive, negative and null Sasakian structures. Construction of the first Smale-Barden manifold that is K-contact but not Sasakian, solving a question of Boyer-Galicki.

- Moduli spaces in algebraic geometry. Moduli spaces on complex curves, such as moduli spaces of vector bundles, pairs, Higgs bundles, triples, algebraic connections, etc. Calculation of Poincaré polynomials, Hodge-Deligne polynomials, irreducible components, Hodge conjecture for certain moduli, Torelli-type theorems, etc.

- Character varieties. Moduli spaces of representations using geometric techniques to parameterize matrices (stratifications, fibrations, etc.). Hodge-Deligne polynomials for SL(2,C)-character varieties of fundamental groups of surfaces, development of a Topological Quantum Field Theory for character varieties. Geometry of SL(3,C)-character varieties for free groups, torus knots and other knot groups. Character varieties of torus knots for SU(2), SU(3), SL(4,C), etc.

Positions

- 1/1997 - 5/1997. Assistant lecturer (Profesor Asociado). Departament of Anaylisis and Applied Mathematics. Universidad de Malaga
- 4/1998 - 10/1998. Assistant lecturer (Ayudante). Departament of Analysis and Applied Mathematics. Universidad de Malaga
- 10/1998 - 3/1999. Assistant lecturer (Ayudante). Departament of Algebra, Geometry and Topology. Universidad de Malaga
- 3/1999 -  10/2001. Lecturer (Titular de Universidad Interino. Departamento de Matemáticas. Universidad Autonoma de Madrid
- 10/2001 - 5/2004. Lecturer (Titular de Universidad. Departamento de Matemáticas. Universidad Autonoma de Madrid
- 5/2004 - 11/2009 Scientific Researcher (Investigador Cientifico) CSIC. Instituto de Ciencias Matematicas ICMAT
- 2004/2005, 2005/2006 Honorary professor (Profesor Honorario), Universidad Autonoma de Madrid
- 2008/2009 Colaborate professor (Profesor Colaborador), Universidad Complutense de Madrid
- II Term 2006/2007 Member of Institute for Advanced Studies IAS, Princeton (US)
- 11/2009 - 7/2018 Full Professor (Catedratico). Departament of Algebra, Geometry and Topology. Universidad Complutense de Madrid
- 10/2013 - 6/2016 Researcher associated to Instituto de Ciencias Matemáticas ICMAT
- 2014/2015 Visiting Professor. Universite Paris XIII (France)
- 7/2018 - 7/2022 Full Professor (Catedratico). Departament of Algebra, Geometry and Topology. Universidad de Malaga
- Since 7/2022, Full Professor (Catedratico). Departament of Algebra, Geometry and Topology. Universidad Complutense de Madrid

Ongoing research projects

- Geometric structures in riemannian and semi-riemannian geometry, PID2020-118452GB-I00, 01/2022 - 12/2024. Principal researchers: Vicente Muñoz and José Luis Flores (Universidad de Málaga). Number of researchers: 9

Doctoral students

- Marina Logares, "Moduli of parabolic Higgs bundles". Universidad Autonoma de Madrid. May 2006

- Giovanni Bazzoni. "Homotopy theory and geometric structures". Universidad Complutense de Madrid. November 2012. Premio Extraordinario de Doctorado UCM

- Jonathan Sánchez. "Motives of moduli spaces of pairs and applications". Universidad Complutense de Madrid. June 2014.

- Javier Martínez. "Hodge polynomials of character varieties", Universidad Complutense de Madrid. June 2015.

- José Angel González Prieto (cosupervisor; Marina Logares). "Topological quantum eld theories for character varieties". Universidad Complutense de Madrid. Octubre 2018. Premio Extraordinario de Doctorado UCM, Prize Vicent Casellles 2021 (RSME)

- Juan Angel Rojo Carulli. "Orbifolds and geometric structures". Universidad Complutense de Madrid. May 2019

- Lucía Martín Merchán (cosupervisor: Giovanni Bazzoni), "Spin(7) structures, spinors and nilmanifolds". Universidad de Malaga. December 2021

- Alejandro Cañas Muñoz (cosupervisor: Antonio Viruel). "Topology of singularities and algebraic surfaces". Universidad de Malaga. In progress.

Editorial

- Editor of the Newsletter of the European Mathematical Society, 2005-2008. Editor-in-Chief of the Newsletter of the European Mathematical Society, 2008-2012.

- Editor of Tbilisi Mathematical Journal, 2007-2016, Managing editor 2014-2015.

- Member of the Scientific Editorial Committee of Ediciones Electolibris. electolibris.es, 2014-2017.

- Editor of Revista Iberoamericana de Matemática, 2012-2017.

- Managing editor (Geometry section) of Mathematics (ISSN 2227-7390), 01/2020-06/2022.

- Managing editor (Geometry section) of Revista Matemática Complutense (since 2011),

- Editor of Mediterranean Journal of Mathematics (since 2021).

Volumes as editor:

- Modern geometry: A celebration of the work of Simon Donaldson, Proceedings of Symposia in Pure Mathematics, Vol. 99, AMS, 2018. (Editors: V. Muñoz, I. Smith and R. Thomas)

- Proceedings of the VII Workshop on Symplectic and Contact Topology. Geometriae Dedicata, Vol. 132, 2008. (Editors Y. Eliashberg, K. Fukaya, V. Muñoz and F. Presas)

- Proceedings of GESTA 2011: New Trends in Symplectic and Contact Geometry. Geometriae Dedicata, Vol. 165, no. 1, 2013. (Editors: O. Garcia-Prada, V. Ginzburg, W. Goldman, E. Miranda and V. Muñoz)

Transference to society/industry

- R+D+i Contract "Mathematics for Business Solutions". Company: CONENTO (November 2014-June 2015), Art. 83

- R+D+i Contract "Multiple Calibration of Heliostats". Company: BCB Informática y Control, S.L. (10/6/2014 to 11/5/2014). ICMAT (Reference: 000166140023)

- Patent "System to determine the potential solar exposure of the leaves of a tree".. P201200481. Priority country: Spain. Date: May 10, 2012. Holder entity: UCM

Inventors (p.o. of signature): Luis Balaguer, Adrián Escribano, Agustina Ventre, Carlos Granado Yela, Antonio Lopez-Pintor, Juan Antonio Delgado Sáez, Vicente Muñoz, Felix Martínez, Gabriel Dorado

Evaluation and committees

- Member of the Panel Geometry and Topology for the Agencia Nacional de Evaluación y Prospective (ANEP), Ministry of Economy and Competitiveness (MINECO), Spain. Area MTM-Mathematics, 2012-2014.

- Evaluation of applications for La Caixa Foundation scholarship program, 2014.

- Panel Member for the Stimulus of Scientific Employment, Individual Support, Fundaçao para a Ciência e a Tecnologia (FCT), Portugal. Open calls years 2017, 2018 and 2020.

- Moderator of Panel "Algebra, Geometry, Topology and Number Theory".  2021 CRM Postdocs Call. Unidad de Excelencia Maria de Maeztu, Centre de Recerca Matematica, 2022

- Committee Mathematical Olympiad of the Royal Spanish Mathematical Society (RSME), 2006-2013.

- Scientific Committee of the Royal Spanish Mathematical Society (RSME), 2010-2015.

- Executive Committee of the European Mathematical Society (EMS), 2017-2020

- Governing Board of the Royal Spanish Mathematical Society (RSME), 2016-2019

- President of the Organizing Committee of the II Joint Meeting RSME-Unión Matemática Argentina, Ronda, Malaga, December 2022