Research

Mathematics in Heidelberg

The Heidelberg Experimental Geometry Lab (HEGL) is part of the Research Station Geometry+Dynamics, which is part of the Institute for Mathematics, which is part of the Faculty of Mathematics and Computer Science at Heidelberg University.

Research interests

My research interests lie in the area at the intersection of algebraic and differential geometry. I study Galois and hidden symmetries of Yang-Mills connections, with applications to the topology and geometry of real and complex algebraic varieties.

• Fuchsian groups (30F35)
• Vector bundles on curves and their moduli (14H60)
• Special connections and metrics on vector bundles (53C07)
• Topology of real algebraic varieties (14P25)

Funding

• Excellence grant: Mobility in International Research Collaboration, 2024-2026. Project ID: ExU 11.2.1.45 Mobilität Schaffhauser. Role: PI.
• AEI-DFG joint Spanish-German research project, 2025-2028. Very-stability, Hecke transforms and representation varieties in the presence of symmetry (V-SHARP). Project ID: SCHA 2147_1-1 AOBJ 706923. Role: PI.

Publications

1. With Vincent Voß and Katrin Weiß. Algebra, Logik, und Beweisvisualisierung.   Mitteilungen der Deutschen Mathematiker-Vereinigung, vol. 33, no. 3, 2025, pp. 205-209.  
2. With Juan Sebastián Numpaque-Roa. Flat torsors in complex geometry. In del Ángel R., P., Neumann, F., Schmitt, A. (eds).  Moduli, Motives and Bundles, New Trends in Algebraic Geometry , pp 106-142, LMS Lecture Note Series (Cambridge University Press).  
3. With Juan Martín Pérez. Orbifolds and the modular curve.
   In Ohshika, K., Papadopoulos, A. (eds).   In the Tradition of Thurston III , pp 365–421, Springer (First online: 19 March 2024).  
4. With Daniele Alessandrini and Gye-Seon Lee. Hitchin components for orbifolds.
   J. Eur. Math. Soc. 25 (2023), no. 4, pp. 1285–1347.  
5. With Erwan Brugallé. Maximality of moduli spaces of vector bundles on curves.  
   Épijournal de Géométrie Algébrique (EPIGA), Volume 6 (2023).  
6. Symmetric differentials and the dimension of Hitchin components for orbi-curves. 
   Current Trends in Analysis, its Applications and Computation, Birkhäuser Trends in Mathematics (2022), pp. 129-140.  
7. With Victoria Hoskins. Rational points of quiver moduli spaces.
   Ann. Inst. Fourier Volume 70 (2020) no. 3, pp. 1259–1305.  
8. With Indranil Biswas. Parabolic vector bundles on Klein surfaces.
   Illinois J. Math. 64 (2020), no. 1, pp.105–118.  
9. With Victoria Hoskins. Group actions on quiver varieties and applications.
   Internat. J. Math. 30 (2019), no. 2, p. 1950007, 46.  
10. Finite group actions on moduli spaces of vector bundles.
    Sémin. Théor. Spectr. Géom. (Grenoble) 34 (2016-2017), 33–63.  
11. On the Narasimhan-Seshadri correspondence for Real and Quaternionic vector bundles.
    J. Differential Geom. (2017) 105 (1), 119–162.  
12. Lectures on Klein surfaces and their fundamental group.  In Geometry and Quantization of Moduli Spaces.   Advanced Courses in Mathematics - CRM Barcelona. Springer (2016), 67–108.  
13. With Indranil Biswas. Vector bundles over a real elliptic curve.
    Pacific J. Math. (2016) 283 (1), 43–62.  
14. Differential geometry of holomorphic vector bundles on a curve. In Geometric and Topological Methods for Quantum Field Theory. Proceedings of the 2009 Villa de Leyva Summer School. Cambridge University Press (2013), 39–80.  
15. With Chiu-Chu Melissa Liu. The Yang-Mills equations over Klein surfaces.
    J. Topol. (2013) 6 (3), 569–643.  
16. Real points of coarse moduli schemes of vector bundles on a real algebraic curve.
    J. Symplectic Geom. 10 (2012), no. 4, 503–534.  
17. Moduli spaces of vector bundles over a Klein surface.
    Geom. Dedicata 151 (2011), no. 1, 187–206.  
18. Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups.
     Math. Ann. 342 (2008), no. 2, 405–447.  
19. Anti-symplectic involutions on quasi-Hamiltonian quotients.
    Trav. Math. 17 (2007) 57–64.  
20. A note on quasi-Hamiltonian geometry and representation spaces of surface groups.
    Sem. Math. Sci. 36 (2007), 35–48.  
21. Quasi-Hamiltonian quotients as disjoint unions of symplectic manifolds.  In Non-Commutative Geometry and Physics 2005.   Proceedings of the 2005 International Sendai-Beijing Joint Workshop, 31-54. World Scientific Press (2007).  
22. Un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie.
    C. R. Math. Acad. Sci. Paris 345 (2007), no. 1, 25–30.  
23. Representations of the fundamental group of an L-punctured sphere generated by products of Lagrangian involutions.
    Canad. J. Math. 59 (2007), no. 4, 845–879.  
24. With Elisha Falbel and Jean-Pierre Marco. Classifying triples of Lagrangians in a Hermitian vector space.
    Topology Appl. 144 (2004), no. 1-3, 1–27.  

Preprints

• With Emanuel Roth. Harder-Narasimhan filtrations of decorated vector bundles. arXiv:2511.17371.
• With Tommaso Scognamiglio. Real Bialynicki-Birula flows in moduli spaces of Higgs bundles. arXiv.2507.18613.

Theses

• Habilitation Thesis: Topology of representation varieties of Fuchsian groups (2019).  
• Doctoral Thesis: Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surfaces groups (2005).  

Slides from conference talks and other material

• Algebra, Logic, and Proof Visualization.   (Illustrating Mathematics: Reunion/Expansion, ICERM, 11-15.08.2025).  
• Mathematical libraries of the future   (Visit of the Academic Advisory Council, Heidelberg University, 26.05.2025).  
• Matemáticas, Lógica e Informática   (Universidad de Los Andes, 02-06.12.2024).  
• La correspondencia de Curry-Howard  (Coloquio del Departamento de Matemáticas, Universidad de Los Andes, 11.03.2024).  
• An informal introduction to formal mathematics   (Waseda University Geometry Seminar, 29.11.2023).  
• Welcome to HEGL! Visiting Day of the 10th Heidelberg Laureate Forum (27.09.2023).  
• Formally real fields. Machine-checked mathematics (Lorentz Center, Leiden).  
• Hodge numbers of moduli stacks of principal bundles. Rencontres du GDR GAGC 2022 (CIRM, Luminy).  
• Twisted character varieties. Conference in honour of Elisha Falbel, 2022 (CIRM, Luminy).  
• Coco's Fest 2016: Categorías abelianas con conjugación (Bogotá, 2016).  
• 50 years of the Narasimhan-Seshadri theorem: Real Seifert manifolds and the Narasimhan-Seshadri correspondence (NS@50, Chennai 2015).  
• The Yang-Mills equations over Klein surfaces (Seoul ICM 2014, contributed talk).  
• Monodromy of Knizhnik-Zamolodchikov equations (2007).  

Supervision and mentoring

If you want to know what it is like to write a thesis under my supervision (the good and the bad), talk to my students!

Undergraduate students

1. Sergio Pedraza-Rodríguez, Uniandes (2011). Hurwitz surfaces.
2. Alejandro Rivera, ENS Lyon (2013). Equivariant Morse theory in symplectic geometry.
3. Simón Soto-Ochoa, Uniandes (2016). The modular group of a compact orientable surface.
4. Santiago Cortés-Gómez, Uniandes (2016). Galois theory of separable algebras over a field.
5. Alirio Calderón-Díaz, Universidad Distrital de Bogotá (2017). Gromov's affine non-squeezing theorem.
6. David Jaramillo-Duque, Uniandes (2017). Clifford algebras and spin groups.
7. Nicolás Betancourt-Cardona, Uniandes (2019). Extensions of field of meromorphic functions.
8. Manuel Aragón, Uniandes (2023). Complex projective structures and opers.
9. Liva Diler, Universität Heidelberg (2024). Equivariant cohomology of real algebraic varieties.
10. Chenyi Yang, Universität Heidelberg (2024). Cell complexes in Homotopy Type Theory.
11. Janina Płaneta, Universität Heidelberg (2025). Topology of loop spaces.
12. Felix Lentze, Universität Heidelberg (2025). Formalizing Dedekind cuts in Lean.
13. Sebastian Grafe, Universität Heidelberg (2025). Gaussian elimination in Lean.
14. Daniel Geisler, Universität Heidelberg (2025). Introduction to Dessins d'enfants.

Master students

1. Andrés Jaramillo-Puentes, Uniandes (2012). Uniformisation of real algebraic curves.
2. Ramón Urquijo-Novella, Uniandes (2012). GIT quotients and symplectic quotients, the Kempf-Ness theorem.
3. Camilo Vargas-Contreras, Uniandes (2013). Higgs bundles and hyper-Kähler reduction in infinite dimension.
4. Nicolás Walteros-Vergara, Uniandes (2018). Unitary connexions and holomorphic vector bundles.
5. Juan Martín Pérez-Bernal, Uniandes (2019). The moduli stack of elliptic curves.
6. Juan Sebastián Numpaque-Roa, Uniandes (2021). The Hitchin fibration.
7. Emanuel Roth, Universität Heidelberg (2024). The Harder-Narasimhan reduction of a principal bundle.
8. Till Janke, Universität Heidelberg. Real spectrum compactification of Teichmüller space.

PhD students

1. Marwan Benyoussef, Character varieties for Seifert groups and Higgs bundles over orbifolds, 24.06.2025, Freie Universität Berlin (co-supervision with Alexander Schmitt).

Postdoctoral supervision

1. Leonardo Roa-Leguizamón, Uniandes (2022-2023).
2. Zeinab Toghani, Uniandes (2022).
3. Tommaso Scognamiglio, Universität Heidelberg (2024-2025).
4. Kevin Klinge, Universität Heidelberg (2025-2026).