Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

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

Algebra, Geometry and Physics seminar, joint with Y.I.Manin at MPIM Bonn
Tuesdays 1.45-3.15pm (hybrid)

 

Research topics in recent years  
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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

Rémi Cocou Avohou
Raum 1.325
Combinatorics of graphs, matroids, maps

Guillaume Baverez
Raum 1.325
My 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 1.407
W-algebras, topological recursion and supersymmetric gauge theories


Interns

Thomas Buc-d'Alché (WiSe 21-22)
Integrability in matrix models and applications to algebraic geometry
Inès Combes-Castex
(SoSe 22)
Matrix models as a toy model in QFT, statistical physics on the random lattice


Next activities organized

SCGP Thematic month: Integrability, enumerative geometry and quantization
Stony Brook, 22. Aug-23. Sep. 2022

Group retreat: Matamzee
Hiddensee, 3-8 Oct. 2022.

CIMPA school: Crossroads of representation theory, geometry and higher structures
13-26 Mar. 2023, Puerto Madryn, Argentina. We welcome participant applications, especially from international master and PhD students, especially from South America and from developing countries in the spirit of CIMPA.

KMPB spring school: 3 facets of gravity
HU Berlin, 8-12 May 2023.
Participant applications will be welcome at a later time.

Group retreat: Matamberg
Villa Garibald, 4 Aug.-10 Aug. 2023

CIME summer school: Enumerative geometry, quantization and moduli spaces
Cetraro, 4-10 Sep. 2023. Participant applications will be welcome at a later time.


Outreach

Lange Nacht der Wissenschaft
2. Jul. 2022


Visitors

Bruno Vallette (27 Feb.-3 Mar. 2022)
Stavros Garoufalidis (2-5 Mai 2022)
Reinier Kramer (24-27 Mai 2022)
Campbell Wheeler (24-27 Mai 2022)
Elba Garcia-Failde (31 Mai-3 Jun. 2022)
Sergey Shadrin (31 Mai-3 Jun. 2022)
Felix Leid (31 Mai-3 Jun. 2022)
Séverin Charbonnier (31 Mai-3 Jun. 2022)
Ran Tessler (24-28 Okt. 2022)