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QuantGeomLangTFT SIGNED

The Quantum Geometric Langlands Topological Field Theory

Total Cost €

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EC-Contrib. €

0

Partnership

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 QuantGeomLangTFT project word cloud

Explore the words cloud of the QuantGeomLangTFT project. It provides you a very rough idea of what is the project "QuantGeomLangTFT" about.

surfaces    geometry    once    algebraic    variety    operators    quantization    spaces    differential    unified    character    first    manifolds    modern    flat    affine    double    shende    thereby    class    quantizations    punctured    special    construct    module    received    moduli    geometric    langlands    rational    rasmussen    mapping    construction    springer    hecke    representations    verlinde    celebrated    conjectural    interesting    groups    quantized    computing    arbitrary    witten    deep    torus    algebra    dubbed    dimensional    genus    closed    bundles    qgl    leverage    shine    quantum    fundamental    recover    unrelated    existence    elliptic    solving    actions    algebras    form    obtain    parameterizing    shedding    constructing    polynomial    surface    analogs    root    deal    varieties    unity    light    previously    realize    yields    connections    oblomkov    invariants    respectively    uncovered    degenerate    spherical    equipped    connection    knots    knot    group    extension    daha    techniques    cherednik    representation    theory    topological   

Project "QuantGeomLangTFT" data sheet

The following table provides information about the project.

Coordinator
THE UNIVERSITY OF EDINBURGH 

Organization address
address: OLD COLLEGE, SOUTH BRIDGE
city: EDINBURGH
postcode: EH8 9YL
website: www.ed.ac.uk

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Project website http://www.maths.ed.ac.uk/
 Total cost 1˙100˙947 €
 EC max contribution 1˙100˙947 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-STG
 Funding Scheme ERC-STG
 Starting year 2015
 Duration (year-month-day) from 2015-06-01   to  2020-05-31

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    THE UNIVERSITY OF EDINBURGH UK (EDINBURGH) coordinator 1˙100˙947.00

Map

 Project objective

We will use modern techniques in derived algebraic geometry, topological field theory and quantum groups to construct quantizations of character varieties, moduli spaces parameterizing G-bundles with flat connection on a surface. We will leverage our construction to shine new light on the geometric representation theory of quantum groups and double affine Hecke algebras (DAHA's), and to produce new invariants of knots and 3-manifolds.

Our previous research has uncovered strong evidence for the existence of a novel construction of quantum differential operators -- and their extension to higher genus surfaces -- in terms of a four-dimensional topological field theory, which we have dubbed the Quantum Geometric Langlands (QGL) theory. By construction, the QGL theory of a surface yields a quantization of its character variety; quantum differential operators form just the first interesting example. We thus propose the following long-term projects:

1. Build higher genus analogs of DAHA's, equipped with mapping class group actions -- thereby solving a long open problem -- by computing QGL theory of arbitrary surfaces; recover quantum differential operators and the (non-degenerate, spherical) DAHA of G, respectively, from the once-punctured and closed two-torus. 2. Obtain a unified construction of both the quantized A-polynomial and the Oblomkov-Rasmussen-Shende invariants, two celebrated -- and previously unrelated -- conjectural knot invariants which have received a great deal of attention. 3. By studying special features of our construction when the quantization parameter is a root of unity, realize the Verlinde algebra as a module over the DAHA, shedding new light on fundamental results of Cherednik and Witten. 4. Develop genus one, and higher, quantum Springer theory -- a geometric approach to constructing representations of quantum algebras -- with deep connections to rational and elliptic Springer theory, and geometric Langlands program.

 Publications

year authors and title journal last update
List of publications.
2017 Andrea Appel, Sachin Gautam
An explicit isomorphism between quantum and classical sl(n)
published pages: , ISSN: , DOI:
2020-01-24
2018 Samuelson, Peter; Cooper, Benjamin
The Hall Algebras of Surfaces I
published pages: , ISSN: 1474-7480, DOI:
Journal of the Institute of Mathematics of Jussieu to appear 2020-01-24
2016 Martina Balagovic, David Jordan
The Harish-Chandra isomorphism for quantum GL_2
published pages: , ISSN: 1661-6952, DOI:
Journal of Noncommutative Geometry to appear 2020-01-24
2018 Andrea Appel, Valerio Toledano-Laredo
Coxeter categories and quantum groups
published pages: , ISSN: , DOI:
2020-01-24
2016 Sabin Cautis, Aaron D. Lauda, Anthony Licata, Peter Samuelson, Joshua Sussan
The Elliptic Hall algebra and the deformed Khovanov Heisenberg category
published pages: , ISSN: 1022-1824, DOI:
Selecta Mathematica to appear 2020-01-24
2017 David Jordan and Monica Vazirani
The rectangular representation of the double affine Hecke algebra via elliptic Schur-Weyl duality
published pages: , ISSN: , DOI:
2020-01-24
2018 David Ben-Zvi, Adrien Brochier, David Jordan
Integrating quantum groups over surfaces
published pages: 873-916, ISSN: 1753-8416, DOI: 10.1112/topo.12072
Journal of Topology 11/4 2020-01-24
2017 David Jordan and Noah White
The center of the reflection equation algebra via quantum minors
published pages: , ISSN: , DOI:
2020-01-24
2018 Yuri Berest, Peter Samuelson
Affine cubic surfaces and character varieties of knots
published pages: 644-690, ISSN: 0021-8693, DOI: 10.1016/j.jalgebra.2017.11.015
Journal of Algebra 500 2020-01-24
2017 Andrea Appel, Valerio Toledano-Laredo
Uniqueness of quasi-Coxeter structures on Kac-Moody algebras
published pages: , ISSN: , DOI:
2020-01-24
2018 David Ben-Zvi, Adrien Brochier, David Jordan
Quantum character varieties and braided module categories
published pages: , ISSN: 1022-1824, DOI: 10.1007/s00029-018-0426-y
Selecta Mathematica 2020-01-24

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