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Grant: Moduli spaces of submanifolds and motion groups
Identifier: PN-III-P4-ID-PCE-2020-2798
Duration: 3 years (2021–2023)
Financed by: UEFISCDI
Contract number: PCE 124/2021
Grant director / Principal Investigator: Martin Palmer-Anghel
Team Members:
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Abstract
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.
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Reports
Publications
- Martin Palmer, Ulrike Tillmann, Configuration-mapping spaces and homology stability
Research in the Mathematical Sciences vol. 8 (2021) no. 38
- Martin Palmer, Arthur Soulié, The Burau representations of loop braid groups
Comptes Rendus. Mathématique vol. 360 (2022) pp. 781-797
- Martin Palmer, Ulrike Tillmann, Point-pushing actions for manifolds with boundary
Groups, Geometry, and Dynamics vol. 16 no. 4 (2022) pp. 1179-1224
- Geoffroy Horel, Martin Palmer, Motivic homological stability of configuration spaces
Bulletin of the London Mathematical Society vol. 55 no. 2 (2023) pp. 892-913
- Cristina Anghel, U_q(sl(2))-quantum invariants from an intersection of two Lagrangians in a symmetric power of a surface
Transactions of the American Mathematical Society vol. 376 no. 10 (2023) pp. 7139-7185
- Takuro Abe, Denis Ibadula, Anca Măcinic, On some freeness-type properties for line arrangements
To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
Preprint version available at arXiv:2101.02150
- Jacques Darné, Martin Palmer, Arthur Soulié, When the lower central series stops: a comprehensive study for braid groups and their relatives
To appear in the Memoirs of the American Mathematical Society
Preprint version available at arXiv:2201.03542
- Cristina Anghel, Witten-Reshetikhin-Turaev invariants for 3-manifolds from Lagrangian intersections in configuration spaces
To appear in Quantum Topology
Preprint version available at arXiv:2104.02049
Preprints
- Cristina Anghel, Martin Palmer, Lawrence-Bigelow representations, bases and duality
arXiv:2011.02388
- Christian Blanchet, Martin Palmer, Awais Shaukat, Heisenberg homology on surface configurations
arXiv:2109.00515
- Martin Palmer, Arthur Soulié, Topological representations of motion groups and mapping class groups – a unified functorial construction
arXiv:1910.13423 (v4: 17.11.2021)
- Geoffroy Horel, Manuel Krannich, Alexander Kupers, Two remarks on spaces of maps between operads of little cubes
arXiv:2211.00908
- Martin Palmer, Arthur Soulié, The pro-nilpotent Lawrence-Krammer-Bigelow representation
arXiv:2211.01855
- Geoffroy Horel, Binomial rings and homotopy theory
arXiv:2211.02349
- Martin Palmer, Xiaolei Wu, On the homology of big mapping class groups
arXiv:2211.07470
- Martin Palmer, Ulrike Tillmann, Homology stability for asymptotic monopole moduli spaces
arXiv:2212.11799
- Martin Palmer, Xiaolei Wu, Big mapping class groups with uncountable integral homology
arXiv:2212.11942
- Anca Măcinic, On torsion freeness for the decomposable Orlik-Solomon algebra
arXiv:2212.12275
- Joana Cirici, Geoffroy Horel, Formality of hypercommutative algebras of Kähler and Calabi-Yau manifolds
arXiv:2302.08492
- Martin Palmer, Arthur Soulié, Polynomiality of surface braid and mapping class group representations
arXiv:2302.08827
- Christian Blanchet, Martin Palmer, Awais Shaukat, Action of subgroups of the mapping class group on Heisenberg homologies
arXiv:2306.08614
- Anca Măcinic, Jean Vallès, A geometric perspective on plus-one generated arrangements of lines
arXiv:2309.10501
- Anca Măcinic, Piotr Pokora, On plus-one generated conic-line arrangements with simple singularities
arXiv:2309.15228
Talks at conferences
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials through Lagrangian intersections in configuration spaces", Homological and quantum invariants, CIRM, Marseille, 8–12 February 2021.
- Geoffroy Horel, "Homotopy theory of operads and the Grothendieck-Teichmüller group", Homotopic and Geometric Galois Theory, Mathematische Forschungsinstitut Oberwolfach, 7–13 March 2021.
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", BMC-BAMC: British Mathematical Colloquium, Glasgow, 6–9 April 2021.
- Martin Palmer-Anghel, "Representations of the Torelli group via the Heisenberg group", Workshop for Young Researchers in Mathematics, 20–21 May 2021.
- Geoffroy Horel, "Weight structures and formality" (mini-course of 3 talks), Higher Structures and operadic calculus, CRM, Barcelona, 21–25 June 2021.
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", Remote Rendezvous for Quantum Topologists, 9–13 August 2021.
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", Perspectives on quantum link homology theories, Regensburg, 9–13 August 2021.
- [*] Martin Palmer-Anghel, "Mapping class group representations via Heisenberg homology", Réunion annuelle du GDR de topologie algébrique, Strasbourg, 26–29 October 2021.
- Cristina Palmer-Anghel, "Coloured Jones and coloured Alexander polynomials unified by a graded Lagrangian intersection", Quantum Field Theories and Quantum Topology Beyond Semisimplicity, Banff International Research Station, Alberta, 1–5 November 2021.
- Cristina Palmer-Anghel, "Uq(sl(2))-invariants quantiques via l’intersection de deux lagrangiennes dans une puissance symétrique d’une surface", Nantes-Orsay seminar on symplectic and contact geometry, Paris-Saclay, 18–19 November 2021.
- [*] Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", Fall Southeastern Sectional Meeting, University of South Alabama, 20–21 November 2021.
The symbol [*] indicates that participation was financially supported by the grant.
Invited talks at seminars
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", Topology Seminar, Stanford, 16 February 2021.
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", Séminaire d'algèbre et de géométrie, Caen, 11 May 2021.
- Cristina Palmer-Anghel, "The Witten-Reshetikhin-Turaev invariants for 3-manifolds as state sums of Lagrangian intersections", Knot theory seminar, Warsaw, 28 May 2021.
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", Séminaire AGATA, Montpellier, 3 June 2021.
- Cristina Palmer-Anghel, "Uq(sl(2))-quantum invariants for links and 3-manifolds from Lagrangian intersections in configuration spaces", Knots and Representation Theory Seminar, Moscow, 19 July 2021.
- Cristina Palmer-Anghel, "Renormalized Witten-Reshetikhin-Turaev invariants and M-traces associated to the super quantum group Uq(sl(m|n)) at roots of unity", CUNY Representation Theory Seminar, New York, 1 October 2021.
- Martin Palmer-Anghel, "On homological representations of mapping class groups", Algebra/Topology seminar, Copenhagen, 8 October 2021.
- Cristina Palmer-Anghel, "Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces", Caltech geometry and topology seminar, Pasadena, 22 October 2021.
- Cristina Palmer-Anghel, "Uq(sl(2))-quantum invariants for links and 3-manifolds from Lagrangian intersections in configuration spaces", Quantum topology seminar, Indiana University Bloomington, 25 October 2021.
- Geoffroy Horel, "Tour de Goodwillie-Weiss algébrique et invariants de nœuds type fini", Séminaire de topologie algébrique, UCLouvain, 16 November 2021.
- Cristina Palmer-Anghel, "Coloured Jones and coloured Alexander invariants from two Lagrangians intersected in a symmetric power of a surface", [K-OS] Knot Online Seminar, CNRS/Paris/Regensburg, 2 December 2021.
- Cristina Palmer-Anghel, "Uq(sl2)-quantum invariants via the intersection of two Lagrangians in a symmetric power of a surface", Séminaire de topologie algébrique, UCLouvain, 7 December 2021.
- Cristina Palmer-Anghel, "Coloured Jones and coloured Alexander invariants from two Lagrangians intersected in a symmetric power of a surface", Knots and representation theory seminar, Moscow, 24 January 2022.
- Cristina Palmer-Anghel, "Coloured Jones and coloured Alexander invariants from two Lagrangians intersected in a symmetric power of a surface", Geometry and Topology seminar, Michigan State University, 8 February 2022.
- Martin Palmer-Anghel, "Mapping class group representations via Heisenberg, Schrödinger and Stone-von Neumann", Fudan Topology Seminar, Shanghai, 10 March 2022.
- Martin Palmer-Anghel, "Mapping class group representations via Heisenberg, Schrödinger and Stone-von Neumann", Topology seminar, New York University, Abu Dhabi, 30 March 2022.
Scientific visits of team members
Scientific visits (invited guests at IMAR)
Posters at conferences
Participation in conferences and workshops
(other than those listed further above)
- Anca Măcinic, Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications, Mathematische Forschungsinstitut Oberwolfach, 24–30 January 2021.
- Martin Palmer-Anghel, Homological and quantum invariants, CIRM, Marseille, 8–12 February 2021.
- Cristina Palmer-Anghel, Workshop for Young Researchers in Mathematics, 20–21 May 2021 (co-organiser).
- Martin Palmer-Anghel, Perspectives on quantum link homology theories, Regensburg, 9–13 August 2021.
- Martin Palmer-Anghel, Workshop: "Cohomology of Arithmetic Groups: Duality, Stability, and Computations", Banff International Research Station, Alberta, 10–15 October 2021.
- Geoffroy Horel, Réunion annuelle du GDR de topologie algébrique, Strasbourg, 26–29 October 2021.
- Martin Palmer-Anghel, Workshop: "Homology and homotopy of configuration spaces", Copenhagen, 15–19 November 2021.
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