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Mario Garcia-Fernandez

Institution: Universidad Autónoma de Madrid

Position: Profesor Ayudante Doctor (Assistant Professor)

Office: 316        Phone:+34 912999 776

E-mail: mario.garcia()icmat.es

Personal Webpage

 

Biographical Review

I am an Assistant Professor (Profesor Ayudante Doctor) at the Universidad Autónoma de Madrid, from September 2017. Previously, I was a Marie Sklodowska-Curie Individual Fellow at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid (2015-2017), and had postdoctoral appointments at the Nigel Hitchin Laboratory of ICMAT (2014-2015), the École Polytechnique Federal de Lausanne (2013-2014), and the Centre for Quantum Geometry of Moduli Spaces (Aarhus, 2010-2012). I have also held visiting positions at the Max Planck Institute for Mathematics (Bonn), at the Humboldt University (Berlin), and at the Hausdorff Research Institute for Mathematics (Bonn). I obtained my PhD in 2009 at the Universidad Autónoma de Madrid.

Research Interests

My research is in the areas of differential geometry and algebraic geometry, and has strong links with geometric analysis and mathematical physics. The main projects I have carried out include special metrics and connections in Kähler geometry (constant scalar curvature Kähler, Hermite-Yang-Mills, vortices, Hitchin equations) and their relation to stability conditions in algebraic geometry (GIT), moduli spaces (bundles and varieties, Higgs bundles, wall-crossing in DT-Theory), special holonomy and mirror symmetry (Calabi-Yau, G2 geometry, Hull-Strominger system, spin geometry, generalized geometry, T-duality).

My current main research interests are complex non-Kähler geometry, special holonomy with torsion, and vertex algebras. In my work I transfer techniques from Kähler geometry to address problems in complex non-Kähler geometry, using the theory of Courant algebroids and other objects reminiscent from higher gauge theory. My research builds towards providing mathematical underpinnings for the interesting phenomenon of (0,2)-mirror symmetry, using special holonomy with torsion and the theory of vertex algebras. In particular, in the last 8 years I have extensively studied the Hull-Strominger system of partial differential equations, which provides a common ground for these three topics.

 

Publications

• Infinitesimal moduli for the Strominger system and Killing spinors in generalized geometry, (with R. Rubio and C. Tipler) Mathematische Annalen 369 (2017) 539–-595.

• Stability data, irregular connections and tropical curves, (with S. Filippini and J. Stoppa) Selecta Mathematica New Ser. 23 (2) (2017) 1355–-1418.

• Gravitating vortices, cosmic strings and the Kähler-Yang-Mills equations, (with L. Álvarez-Cónsul and O. García-Prada) Comm. Math. Phys. 351 (1) (2017) 361--385.

• Balanced metrics on twisted Higgs bundles, (with J. Ross) Mathematische Annalen 367 (3) (2017) 1429-–1471.

• Lectures on the Strominger system, Travaux Mathématiques, Special Issue: School GEOQUANT at the ICMAT, Vol. XXIV (2016) 7--61, arXiv:1609.02615.

• Note on Solutions of the Strominger System from Unitary Representations of cocompact lattices of SL(2;C), (with B. Andreas) Comm. Math. Phys. 332 (2014) 1381--1383.

• Torsion-free generalized connections and Heterotic Supergravity, Comm. Math. Phys. 332 (2014) 89--115.

• Deformation of complex structures and the Coupled Kähler–Yang–Mills equations, (with C. Tipler) J. London Math. Soc. 89 (2014) 779-–796.

• Balanced metrics on vector bundles and polarised manifolds, (with J. Ross) Proc. London Math. Soc. 106 (2013) 1143–-1156.

• Coupled equations for Kähler metrics and Yang–Mills connections, (with L. Álvarez-Cónsul and O. García-Prada) Geometry and Topology 17 (2013) 2731–-2812.

• Heterotic Non-Kähler Geometries via polystable bundles on Calabi-Yau threefolds, (with B. Andreas) J. Geom. Phys. 62 (2012) 183-–188.

• Solutions of the Strominger System via stable bundles on Calabi-Yau threefolds, (with B. Andreas) Comm. Math. Phys. (1) 315 (2012) 153-–168.

 

Preprints

• T-dual solutions of the Hull-Strominger system on non-Kähler threefolds, arXiv:1810.04740 (2018).

• Holomorphic string algebroids, (with R. Rubio and C. Tipler) arXiv:1807.10329 (2018).

• On the Kähler-Yang-Mills-Higgs equations, (with L. Álvarez-Cónsul and O. García-Prada), arXiv:1807.10691 (2018).

• Canonical metrics on holomorphic Courant algebroids, (with R. Rubio, C. Tipler and C. Shahbazi) arXiv:1803.01873 (2018).

• Ricci flow, Killing spinors, and T-duality in generalized geometry, arXiv:1611.08926 (2016).

• Moduli of G2 structures and the Strominger system in dimension 7, (with A. Clarke and C. Tipler) arXiv:1607.01219 (2016).

• Gravitating vortices and the Einstein-Bogomol'nyi equations, (with L. Álvarez-Cónsul, O. García-Prada, and V. Pingali) arXiv:1606.07699 (2016).

• Quantization of Hitchin's equations for Higgs bundles I, (with J. Keller and J. Ross) arXiv:1601.04960 (2016).

• Self-dual generalized metrics for pure N=1 six-dimensional Supergravity, (with C. S. Shahbazi) arXiv:1505.03088 (2015).

 

Some recent and forthcoming activities

• SISSA, Trieste, Italy, June 2020.

• Bridging the gap between Kähler and non-Kähler complex geometry, BIRS, Banff, October 2019.

• Université de Bretagne Occidentale, June-July 2019.

• Università di Firenze, May 2019.

• MIST 2019, Hong Kong, January 2019.

• School and Workshop on Special Metrics and Gauge Theory, ICMAT, December 2018.

• School and Workshop on New Trends in Higgs Bundle Theory, ICMAT, November 2018.

• Group actions in algebraic and symplectic geometry, ICMAT, October 2018.

• Trimester on Moduli Spaces, ICMAT, September-December 2018.