Grant: The homology and representation theory of moduli spaces

Identifier: PN-IV-P1-PCE-2023-2001
Duration: 3 years (2025–2027)
Financed by: UEFISCDI
Grant director / Principal Investigator: Martin Palmer-Anghel
Team Members:

• Anca Măcinic
• Cristina Palmer-Anghel
• Martin Palmer-Anghel
• Arthur Soulié

Abstract

The domain of this research project lies at the various rich interfaces between algebraic topology, low-dimensional topology, quantum topology, group theory and representation theory. The unifying theme is the study of moduli spaces, their homology and the properties (group-theoretical and representation-theoretical) of their fundamental groups, which include braid groups and mapping class groups.

Moduli spaces are ubiquitous throughout mathematics and have a wide range of applications, from analytic number theory, knot theory and theoretical physics to more applied topics such as robotics and motion planning algorithms. Moreover, the representation theory of their fundamental groups – notably mapping class groups – is extremely rich.

Within this overarching theme, we have 5 directions of study:
(1) The homology of configuration spaces on closed manifolds. Although this is a very classical topic, much is still unknown here.
(2) Representations of mapping class groups of surfaces, inspired by the long-standing open question of whether these groups are linear. We will investigate the kernels of new families of representations.
(3) The homology of mapping class groups of infinite-type surfaces, with connections to dynamical systems.
(4) Non-commutative extensions of quantum representations of knots and links.
(5) Lower central series of arrangement groups (fundamental groups of complements of complex hyperplane arrangements).