Research

Preprints

2024

1. arXiv

   Cyclic Higgs bundles and the Toledo invariant

   Oscar García-Prada, and Miguel González

   2024

   arXiv preprint

   Abs arXiv

   Let G be a complex semisimple Lie group and g its Lie algebra. In this paper, we study a special class of cyclic Higgs bundles constructed from a Z-grading g = \oplus_j=1-m^m-1 g_j by using the natural representation G_0 \to GL(g_1 ⊕g_1-m), where G_0 \le G is the connected subgroup corresponding to g_0. The resulting Higgs pairs include G^R-Higgs bundles for G^R \le G a real form of Hermitian type (in the case m=2) and fixed points of the C^*-action on G-Higgs bundles (in the case where the Higgs field vanishes along g_1-m). In both of these situations a topological invariant with interesting properties, known as the Toledo invariant, has been defined and studied in the literature. This paper generalises its definition and properties to the case of arbitrary (G_0,g_1 ⊕g_1-m)-Higgs pairs, which give rise to families of cyclic Higgs bundles. The results are applied to the example with m=3 that arises from the theory of quaternion-Kähler symmetric spaces.

Journal Articles

2024

1. IJM

   Hitchin map on even very stable upward flows

   Miguel González, and Tamás Hausel

   International Journal of Mathematics, 2024

   Abs Website

   We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GLn case, we classify the type (1,…, 1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, 4n-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.

Talks

2024

1. Seminar

   La Clasificación de Schommer-Pries de 2-TQFTs Extendidas con Teoría de Categorías (The Schommer-Pries Classification of Extended 2-TQFTs using Category Theory)

   Miguel González

   May 2024

   Reading seminar on infinity categories, Universidad Complutense de Madrid (UCM)

   PDF
2. MFO

   Introduction to Hodge theory and examples

   Miguel González

   Apr 2024

   Arbeitsgemeinschaft: Geometry and Representation Theory around the P=W Conjecture, Matematisches Forschungsinstitut Oberwolfach (MFO)

   Website
3. Seminar

   Cyclic Higgs bundles and the Toledo invariant

   Miguel González

   Feb 2024

   Hausel group working seminar, Institute of Science and Technology Austria (ISTA)

   Slides
4. Seminar

   Introducción a la teoría de Hodge en geometría algebraica compleja (Introduction to Hodge theory in complex algebraic geometry)

   Miguel González

   Jan 2024

   Seminario de Doctorandos (PhD student seminar), Universidad Complutense de Madrid (UCM)

   Slides

2023

1. Seminar

   The Hitchin Fibration and Spectral Curves

   Miguel González

   Feb 2023

   Hitchin-Ngô Lab Working Group on the Hitchin Fibration, Langlands duality and mirror symmetry, Instituto de Ciencias Matemáticas (ICMAT)

   PDF

Posters

2022

1. ISTA

   Very Stable U(p,q)-Higgs bundles

   Miguel González

   Aug 2022

   Institute of Science and Technology Austria (ISTA) Summer Internship 2022. Supervised by Tamás Hausel.

   PDF

Theses

2023

1. MSc Thesis

   Cyclic Higgs bundles and Vinberg theory

   Miguel González

   Jul 2023

   Mathematics master’s thesis. Supervised by Oscar García-Prada.

   PDF

2022

1. BSc Thesis

   Invariants for the action of Gal(Qalg/Q) on dessins d’enfants

   Miguel González

   Jun 2022

   Mathematics bachelor thesis. Supervised by Ernesto Girondo.

   PDF
2. BSc Thesis

   Reinforcement learning via dueling bandits

   Miguel González

   Jun 2022

   Computer Science bachelor thesis. Supervised by Pablo Castells.

   PDF