PRR Project

Summary

Research in Geometry and Topology has always been a driving force for innovation in Mathematics and a source of inspiration for developments in other fields such as Theoretical Physics. Conversely, the geometrical and topological concepts stemming from General Relativity, Yang-Mills theories, the Standard model, Supersymmetry, Quantum Field Theories and string theories have surged a renewed interest in all subdomains of Geometry and Topology.There is an important tradition of research in Geometry at the Mathematics Department of FCUL (DM-FCUL), but in the last decade, two full professors have retired and their groups have not been reinforced. On the other hand (within CMAF-cIO and/or DM-FCUL), new research lines (with clear connections with Mathematical Physics) have been developed recently: (i) complex geometry, geometric quantization, moduli spaces of bundles and representations, invariant theory, (ii) enumerative geometry, Gromov-Witten invariants, stable maps, (iii) integrable systems, ODE/IM correspondence, Opers, (iv) Frobenius manifolds, isomonodromic deformations, Riemann-Hilbert correspondence, quantum cohomology, and (v) matrix models, enumerative geometry, Hurwitz numbers.One research line that should be developed in CMAF-cIO is the field of Algebraic Geometry (AG). This field is clearly underrepresented in Portugal, and FCUL is the perfect place for the development of such a mid/long term project led by a CMAF-cIO member, who is one of the two portuguese members of a leading European group in AG called VBAC - Vector Bundles on Algebraic Curves which has recently founded a new journal dedicated to Moduli Theory, sponsored by Compositio Foundation and published by Cambridge U.P.Members of CMAF-cIO / DM-FCUL, are the local organizers of an important international event in this field, “Moduli spaces and Integrability” in honor of Leon Takhtajan (where two CMAF-cIO members are main speakers), and is a satellite conference of the 9th European Congress of Mathematicians, Seville 2024.In order to potentiate the research in Geometry and Topology, and increase the visibility of these areas at FCUL, as well as to create the conditions for long term sustainability of research groups working in these fields, it is fundamental to hire a candidate not only with a strong background on the geometric aspects in theoretical physics, but also in the multidisciplinary interactions between Geometry/Topology, Algebra, and Mathematical Physics.Some of the topics of research that are encompassed by this profile are the following:Enumerative geometry (GW-DT correspondence, quantum cohomology, Frobenius manifolds)Topology of Moduli spaces (stability, moduli of bundles, character varieties)Quiver varieties (Nakajima varieties, knot-quiver correspondence)Representation theoryMatrix and integrable modelsThe successful candidate will have to perform research work on one or more of the topics mentioned above, and is expected to publish his/her research in the top journals of the field on a regular basis. Within CMAF-cIO, he/she will be part of the active group ALG - Algebra, Logic and Geometry, and will be strongly encouraged to apply for funding from FCT and from other European Sources.It is also expected that the candidate will enroll in teaching of advanced courses in Geometry or Topology, in the Mathematics Department of FCUL, at the Master or PhD level, and develop his own research group with the help of young researchers, funded by CMAF-cIO grants.After an initial adaptation period, it is also expected that the candidate collaborates scientifically with other colleagues, both in the Unit CMAF-cIO and in the Mathematics Department, and participates in organization of scientific events, such as seminars and workshops/conferences. The participation in hiring committees of postdoctoral researchers is also encouraged.