Grant: Moduli spaces of submanifolds and motion groups
Duration: 3 years (2021–2023)
Financed by: UEFISCDI
Contract number: PCE 124/2021
Grant director / Principal Investigator: Martin Palmer-Anghel
Our domain of research is at the interaction between algebraic topology, quantum topology and algebraic geometry.
The principal objects of study in this project are moduli spaces of submanifolds. These play a central role in algebraic topology, having profound connections with knot theory (predicted by physics) and homotopy theory. In addition, the representation theory of their fundamental groups – motion groups – is very rich.
In dimension 0, these include configuration spaces and braid groups, whose study is a very well-known domain. In higher dimensions, they are more subtle, and their study has developed rapidly recently, leaving nevertheless many open problems.
The aim of our project is to create new interactions between the topology of moduli spaces, TQFTs and motivic and étale cohomology theories, having 4 main directions:
(1) The stable limit of the homology of moduli spaces of disconnected submanifolds. This is an open question for submanifolds of higher dimension.
(2) New kinds of moduli spaces of submanifolds, inspired by physics and number theory. More precisely, the complement of the submanifold is equipped with a 'field' taking values in a bundle.
(3) The definition of new topological representations of motion groups and extensions towards TQFTs. The aim is to create connections with quantum knot invariants.
(4) Algebraic-geometrical analogues of these moduli spaces, and the study of their étale and motivic cohomology.
- Takuro Abe, Denis Ibadula, Anca Măcinic, On some freeness-type properties for line arrangements, arXiv:2101.02150
- Christian Blanchet, Martin Palmer, Awais Shaukat, Heisenberg homology on surface configurations, arXiv:2109.00515
- Martin Palmer, Arthur Soulié, The Burau representations of loop braid groups, arXiv:2109.11468
- Cristina Palmer-Anghel: Homological and quantum invariants — CIRM, 8–12 February 2021.
- Cristina Palmer-Anghel: Topology Seminar — Stanford, 16 February 2021.
- Cristina Palmer-Anghel: British Mathematical Colloqium — Glasgow, 6–9 April 2021.
- Cristina Palmer-Anghel: Séminaire d'algèbre et de géométrie — Caen, 11 May 2021.
- Martin Palmer-Anghel: Workshop for Young Researchers in Mathematics — 20 May 2021.
- Cristina Palmer-Anghel: Knot theory seminar — Warsaw, 28 May 2021.
- Cristina Palmer-Anghel: Séminaire AGATA — Montpellier, 3 June 2021.
- Cristina Palmer-Anghel: Knots and Representation Theory Seminar — Moscow, 19 July 2021.
- Cristina Palmer-Anghel: Remote Rendezvous for Quantum Topologists — 11 August 2021.
- Cristina Palmer-Anghel: Perspectives on quantum link homology theories — Regensburg, 14 August 2021.
- Cristina Palmer-Anghel: CUNY Representation Theory Seminar — New York, 1 October 2021.
- Martin Palmer-Anghel: Algebra/Topology seminar — Copenhagen, 8 October 2021.
- Cristina Palmer-Anghel: Caltech geometry and topology seminar — Pasadena, CA, 22 October 2021.
- Cristina Palmer-Anghel: Quantum topology seminar — Indiana University Bloomington, 25 October 2021.
- Martin Palmer-Anghel: Réunion annuelle du GDR de topologie algébrique — Strasbourg, 26 October 2021.
- Cristina Palmer-Anghel: Quantum Field Theories and Quantum Topology Beyond Semisimplicity — Banff International Research Station, 1–5 November 2021.
- Cristina Palmer-Anghel: Fall Southeastern Sectional Meeting — University of South Alabama, 20–21 November 2021.
Participation (supported by the grant) in conferences and workshops