Research group Prof. Dr. Gaetan Borot
In our group, mathematical physics is often practiced as a part of mathematics which can take inspiration from (but is not limited to) theoretical physics or aims at providing a rigorous understanding of physical models, in particular those coming from quantum field theory or string theory in which a mathematical framework (or even: definition) is lacking. This covers a rich array of techniques and problems, from combinatorics and algebra, geometry and topology, representation theory, probability and functional analysis, with an emphasis on their interrelations.
Recent publications
Mathematical physics seminar
SoSe23: Tuesdays 11am-1pm, BMS Seminar room, Haus 1, RUD 25, Adlershof
Research topics in recent years
Counting surfaces is a general problem having many incarnations: studying intersection theory on moduli space of complex curves, Gromov-Witten theory (the art of counting holomorphic maps from a Riemann surface to a target Kähler manifold, which in physics language relates to topological string theory), studying random hyperbolic surfaces (via the geometry of the Teichmüller space), counting discretisations (e.g. triangulations) of surfaces perhaps carrying a statistical physics model, counting branched covers of the 2-sphere (Hurwitz theory), etc. Topological recursion is an ubiqutious algebraic structure allowing to attack such questions by cut-and-paste methods, and it is actively developed (both for the theory and the applications) in our group.
The relation to theoretical physics (via non-existing path integrals, Feynman diagram expansions, physical dualities, etc.) enrich these topics with many mathematically formulated interrelations, and relations to other topics: periods of algebraic varieties (mirror symmetry, quantum singularity theory), Frobenius manifolds, 2d conformal field theory, 3d Chern-Simons theory and quantum invariants of 3-manifolds, 4d and 5d gauge theories, quantization of moduli spaces, etc. In some sense, topological recursion gives a way to study (and compute!) the quantisation of various geometric objects or field theories.
Large random matrices are particularly interesting, as they are virtually related to all the above topics, either by direct relations or because they can be studied with the same tools (for the understanding of which they provide an instructive playground). For instance: their (random) eigenvalue typically behave as strongly repulsive particles, which are governed by a wealth of universality classes (going way beyond the Gaussian laws that govern independent random variables), which have to do with the geometry of curve singularities. Asymptotic of various eigenvalue statistics is often governed by the topological recursion. Matrix integrals provide well-defined examples of 0-dimensional QFTs, whose Feynman diagrams are discretised surfaces, and as such they generate interesting models of random surfaces. In some special cases matrix integrals exactly solve enumerative problems of interest. Their algebra of observables are often governed by various algebraic structures, in particular relating to 2d conformal field theory integrability or higher structures. It fits in a broader context of operator algebras, where the application of combinatorial tools and topological recursion is an active field of research.
We are interested both in each of these topics per se, as well as in their relations (for instance, apply techniques from a domain to another). An important motivation is to understand their (sometimes surprising) unity.
Postdocs
Guillaume Baverez
Raum 1.325My background is in probability theory, and my research is primarily focused on two-dimensional conformal field theory (CFT). These are quantum field theories with a rich algebra of symmetry, meaning that they are amenable to algebraic tools, such as the representation theory of the Virasoro algebra. While the algebraic approach has been traditionally dominating the field, in recent years modern probabilistic methods have been used in the study of CFT with great success, especially for the Liouville model. My current goals are to bridge the gap between these approaches, and get a probabilistic understanding on CFTs with extended symmetry (W, Kac-Moody).
Pedro Tamaroff
Raum 1.325
During my Licenciatura en Ciencias Matemáticas, I developed a liking for representation theory, rewriting theory, Hopf algebras, deformation theory and Hochschild Cohomology under the direction of Mariano Suarez-Álvarez. Later, I obtained my PhD from Trinity College Dublin in 2021 under the direction of Vladimir Dotsenko, where I worked with algebraic operads, associative algebras, Gröbner bases and higher structures. During May 2021-March 2022, I worked as a postdoctoral researcher in the Non-linear Algebra Group at MPIMiS (Leipzig) led by Bernd Sturmfels. In my free time, I enjoy dancing and singing blues music, folding origami models, baking and reading.
PhD students
Giacomo Umer
Raum 1.328
W-algebras, topological recursion and supersymmetric gauge theories
Davide Scazzuso
Raum 1.328
All genera topological recursion on Seiberg-Witten curves, gauge theories, topological strings
Niklas Martensen
Geometric quantisation from spectral curves and non-perturbative wave functions
Master dissertations
Florian Götz - Master Math. HU - Jan. 2023
Spectral networks and the geometry of surfaces
Niklas Martensen - Master Math. HU - Juli 2023
Non-perturbative topological recursion and geometric quantisation
Johannes Hübers - Master Math. HU - ongoing
Non-perturbative Schwinger-Dyson equations in supersymmetric gauge theories
Kanishka Katipearachchi - Master Math. HU - ongoing
Earthquakes in hyperbolic geometry
Leonard Vetter - Bachelor Info-Math-Phys HU - Aug. 2023
Topological recursion and open-closed TQFTs
Jonas Lautenschläger - Bachelor Physik HU - Dez. 2023
Lp-regularity of the Poisson equation (supervised by Barbara Zwicknagl)
Lea Polonyi - Bachelor Physik HU - Jan. 2024
Cluster algebras (supervised by Pedro Tamaroff)
Julian Komaromy - Bachelor Info-Math-Phys HU - ongoing
Hurwitz numbers
Interns
Thomas Buc-d'Alché - Master Math., ENS Lyon (WiSe 2021-2022)
Integrability in matrix models and applications to algebraic geometry
Inès Combes-Castex - Bachelor Physik, Université de Toulouse (SoSe 2022)
Matrix models as a toy model in QFT, statistical physics on the random lattice
Maïté Carli - Master Math., Universität Bonn (Aug.-Sept. 2023)
Fractional fields in Liouville CFT (supervised by Guillaume Baverez)
Anna Hollands - Master Math., Universite Paris Sorbonne (Jan.-Feb. 2024)
Topological recursion, intersection theory and Hurwitz numbers
Next activities organized
MFO workshop: Riemann surfaces - random, flat and hyperbolic geometry
with Martin Möller and Samuel Grushevsky
Oberwolfach, 9-13 Sep. 2024.
BIRS workshop: Non-commutative geometry meets topological recursion
Hangzhou, 23-27 Sep. 2024.
Outreach (past and coming)
Lange Nacht der Wissenschaft
2nd July 2022: In which dimension do we live in ? (with Alessandro Valentino)
17th June 2023: Random matrices are everywhere.
Tag der Mathematik
6th May 2023, Hauptvortrag: Die Welt der partiellen differentiellen Gleichungen
A history of the domino problem
Around compositions of Michael Winter
13th Nov.-1st Dec. 2023: exhibition by Michael Winter and Mareike Yin-Yee Lee
22nd November 2023: public lecture by Jarkko Kari, concert by the Kali Ensemble
Visitors
Bruno Vallette, Universite Paris 13 (27 Feb.-3 Mar. 2022)
Stavros Garoufalidis, SUSTech and MPIM Bonn (2-5 Mai 2022)
Reinier Kramer, University of Alberta (24-27 Mai 2022)
Campbell Wheeler, MPIM Bonn (24-27 Mai 2022)
Elba Garcia-Failde, Universite Paris 6 (31 Mai-3 Jun. 2022)
Sergey Shadrin, University of Amsterdam (31 Mai-3 Jun. 2022)
Felix Leid, Universität des Saarlandes (31 Mai-3 Jun. 2022)
Séverin Charbonnier, Université de Genève (31 Mai-3 Jun. 2022)
Yvain Bruned, University of Edinburgh (28 Jun. 2022)
Alessandro Giacchetto, IPhT Saclay France (5-8 Jul. 2022)
Vincent Delecroix, Université de Bordeaux (29 Sep.-8 Okt. 2022)
Ran Tessler, Weizmann Institute (24-28 Okt. 2022)
Nezhla Aghaei, SDU Odense (24-25 Okt. 2022)
Arne van Antwerpen, Vrije Universiteit Brussels (8 Nov. 2022)
Jorgen Ellegaard Andersen, SDU Odense (27 Nov.-3 Dez. 2022)
Nicolas Orantin, Université de Genève (27 Nov.-3 Dez. 2022)
Bruno Vallette, Universite Paris 13 (24-27 Jan. 2023)
Alexey Bufetov, Leipzig Universität (6-7 Feb. 2023)
Martin Markl, Czech Academy of Sciences (7 Nov. 2023)
Murad Alim, Edinburgh University (21 Nov. 2023)
Yannik Schüler, Sheffield University (27-30 Nov. 2023)
Thomas Buc-d'Alché, ENS Lyon (3-9 Dez. 2023)
Pietro Longhi, Uppsala University (15-19 Jan. 2024)
Hugo Parlier, Université du Luxembourg (21-23 Jan. 2024)