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:

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.

representations of mapping class groups and motion groups,

topological models for invariants of knots, links and 3-manifolds,

algebraic models of homotopy theory,

as well as new theorems about:

the behaviour of the lower central series of braid groups and their relatives (welded braid groups, surface braid groups, etc.),

the structure of different freeness-type properties in moduli spaces of line arrangements and of curves in the complex projective plane,

the homology of non-local configuration spaces and moduli spaces of magnetic monopoles,

the homology of mapping class groups of infinite-type surfaces.

Expanding a little more on three of these points:

We give a complete answer — for the family of all (partitioned) surface braid groups, loop braid groups, virtual braid groups and mixed versions of these — to the fundamental but subtle group-theoretical question of whether their lower central series stops. As well as being an important question from the point of view of group theory, the behaviour of these lower central series also controls the possible (higher nilpotent) homological representations that one may construct for these groups.

We construct explicit topological models for quantum invariants of links and of 3-manifolds. This is expected to open up important avenues of research towards the construction of geometric categorifications of these invariants, as well as the study of their asymptotic behaviour, both of which are key goals in this domain of research.

We prove that there are uncountably many (infinite-type) surfaces S whose mapping class groups Map(S) have trivial homology — an example is illustrated on the left below — and, contrastingly, there are uncountably many surfaces S such that the homology of Map(S) is as "large" as possible: its (integral) homology is uncountable in every positive degree — an example is illustrated on the right below.

Cristina Anghel, ADO invariants directly from partial traces of homological representations
To appear in the New York Journal of Mathematics
Preprint version available at arXiv:2007.15616

Cristina Anghel, A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space
To appear in Annales de l'Institut Fourier
Preprint version available at arXiv:2205.07842

Christian Blanchet, Martin Palmer, Awais Shaukat, Action of subgroups of the mapping class group on Heisenberg homologies
To appear in Contemporary Mathematics
Preprint version available at arXiv:2306.08614

Martin Palmer, Arthur Soulié, Topological representations of motion groups and mapping class groups – a unified functorial construction To appear in Annales Henri Lebesgue
Preprint version available at arXiv:1910.13423

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

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

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

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

Anca Măcinic, Piotr Pokora, Addition-deletion results for plus-one generated curves arXiv:2310.19610

Cristina Anghel, Universal coloured Alexander invariant from configurations on ovals in the disc arXiv:2401.17245

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.

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.

Cristina Palmer-Anghel, "U_{q}(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.

Martin Palmer-Anghel, "Mapping class group representations via Heisenberg, Schrödinger and Stone-von Neumann", Workshop on Cobordisms, Strings, and Thom Spectra, Casa Matemática Oaxaca, 10–14 October 2022.

Geoffroy Horel, "Binomial rings and homotopy theory", Homotopical Methods in Geometry, Universitat de Barcelona, 9 November 2022.

Cristina Palmer-Anghel, "A globalisation of the Jones and Alexander polynomials from configurations on arcs and ovals in the punctured disc", Journées de topologie quantique, IMJ-PRG, Paris, 21 November 2022.

Anca Măcinic, "Ziegler restrictions and combinatorics of line arrangements", Workshop for Young Researchers in Mathematics, Universitatea Alexandru Ioan Cuza, Iași, 18–19 May 2023.

Martin Palmer-Anghel, "The homology of big mapping class groups", Homotopy: fruit of the fertile furrow, Isaac Newton Institute, Cambridge, 12–16 June 2023.

Geoffroy Horel, "Binomial rings and homotopy theory", Homotopy theory in Trondheim, NTNU, Trondheim, 26–30 June 2023.

Cristina Palmer-Anghel, "Globalising Jones and Alexander polynomials via Lagrangians in configuration spaces", The Tenth Congress of Romanian Mathematicians, Universitatea din Pitești, 30 June – 5 July 2023.

Martin Palmer-Anghel, "Homological mapping class group representations and lower central series", The Tenth Congress of Romanian Mathematicians, Universitatea din Pitești, 30 June – 5 July 2023.

Cristina Palmer-Anghel, "Globalising Jones and Alexander Polynomials (and their quantum generalisations) via configurations in the punctured disc", Mapping class groups: pronilpotent and cohomological approaches, SwissMAP Research Station, Les Diablerets, 18–22 September 2023.

Geoffroy Horel, "Hypercommutative algebra structures on Kähler and Calabi-Yau manifolds", GDR Théorie de l'Homotopie et Applications, Université de Lille, 24–27 October 2023.

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, "U_{q}(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 U_{q}(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, "U_{q}(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, "U_{q}(sl_{2})-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", 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.

Cristina Palmer-Anghel, "U_{q}(sl(2))-invariants quantiques via l'intersection de deux Lagrangiennes dans une puissance symétrique d'une surface", Séminaire de Topologie et Géométrie, Université de Genève, 7 April 2022.

Geoffroy Horel, "Galois symmetries of knot spaces", Algebraic topology seminar, University of Warwick, 26 April 2022.

Cristina Palmer-Anghel, "Coloured Jones and coloured Alexander invariants from two Lagrangians intersected in a symmetric power of a surface", Topology seminar, Universität Zürich, 30 May 2022.

Martin Palmer-Anghel, "Homological stability for asymptotic monopole moduli spaces", Topology seminar, EPFL, Lausanne, 31 May 2022.

Martin Palmer-Anghel, "Homological representations of motion groups", Geometry and Topology seminar, University of Glasgow, 6 June 2022.

Cristina Palmer-Anghel, "A globalisation of the Jones and Alexander polynomials from configurations on arcs and ovals in the punctured disc", Topology seminar, Texas State University, 28 October 2022.

Geoffroy Horel, "Binomial rings and homotopy theory", Topology seminar, EPFL, Lausanne, 28 October 2022.

Cristina Palmer-Anghel, "Globalising Jones and Alexander polynomials via configurations on arcs and ovals in the disc", Geometry and Topology Seminar, Imperial College London, 26 May 2023.

Geoffroy Horel, "Configuration categories of surfaces and applications to knot theory", Oberseminar Topologie, Universität Münster, 12 July 2023.

Cristina Palmer-Anghel, "Globalising the Jones and Alexander Polynomials (and their quantum generalisations) via configurations in the punctured disc", Melbourne Topology Seminar, University of Melbourne, 2 October 2023.

Martin Palmer-Anghel, "The homology of big mapping class groups", Séminaire de topologie, Grenoble, 24 November 2023.