CENTRE FOR QUANTUM
GEOMETRY OF MODULISPACES (QGM)
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QGM was established in 2009 as a Center of Ex- Mission statement
cellence funded by the Danish National Research - To contribute to defining quantum field theory as a
Foundation. The research objective is to address fun-
damental mathematical problems at the interface
between geometry and theoretical physics.
mathematical entity
- To contribute to unify quantum theory with
quantum gravity.
Based at Aarhus University and directed by Professor
Jørgen Ellegaard Andersen, QGM hosts a strong team
of high-profile, internationally acclaimed researchers,
and with the continuous generation of groundbreaking
results, the Centre together with its international col-
laborators are recognized throughout the mathematics
community worldwide as one of the leading research
Original main research focus
- The Geometric Langlands Program and Higgs bundle moduli spaces
institutions within this highly specialized field. - Toeplitz operators and geometric quantization of moduli spaces
Background and research mission - Quantum representation theory and pertubative invariants.
- Combinatorial models for moduli spaces and quantum moduli paces
The role of mathematics in our understanding of
nature has been recognized for millenia, and its im-
portance is especially poignant in modern theoretical
New core areas
- Moduli spaces and macromolecules
physics as the cost of experiments escalates, and the - Topological Quantum Field Theory and Topological String Theory
mathematical complexity of physical theories increa- - Quantum computing.
ses. QGM set out to become the world-leading center
in quantum geometry of moduli spaces at the crucial
interface between mathematics and theoretical phy-
New directions
- Wall crossing
sics, with the aim of contributing to the mathematical - Gauge theory
underpinnings of contemporary and future physical - Contact geometry
theories. - Symplectic geometry.
Organization
QGM has on top of the permanent professors at Aarhus
University eleven internationally renowned scholars
based at other research institutions, who regularly visit
QGM and with whom resources and research efforts
are shared. The core staff thus comprise:
Conceptual goals
- To significantly advance the understanding of moduli spaces and their
Prof. Jørgen Ellegaard Andersen, Aarhus University
Prof. Henning Haahr Andersen, Aarhus University
quantization
- To train the next generation of scholars and researchers to build the
Prof. Nicolai Reshetikhin, University of California, Ber- Centre into a world-renowed catalyst for collborative
keley cutting-edge research
Prof. Edward Frenkel, University of California, Berkeley - To develop the quantum geometry of moduli spaces, in order to provide
Prof. Denis Auroux, University of California, Berkeley complete mathematical models for a number of quantum field theories.
Prof. Maxim Kontsevich, Institut des Hautes Études
Scientifique
Prof. Nigel Hitchin, University of Oxford
Prof. Alexander Ritter, University of Oxford
Prof. Dominic Joyce, University of Oxford The academic profile of the Centre is additionally shaped by a steady
Prof. Hirosi Ooguri, Caltech population of postdocs (some shared with Caltech) and PhD students,
Prof. Sergei Gukov, Caltech (some based at University of Oxford and UC Berkeley, respectively).
Prof. Anton Kapustin, Caltech
Prof. Robert Penner, Caltech/IHÉS
AFSNITSNAVN PUBLIKATIONSNAVN QGM HIGHLIGHTS 2015 1
Photo by Christine Dilling Sandbjerg Estate, Nov 2015 CONTENT Message from the Director ........................................................................................................... 3 About QGM.............................................................................................
AFSNITSNAVN 3 PUBLIKATIONSNAVN Message from the Director Photo: AU Communication By Jørgen Ellegaard Andersen 2015 was a very productive year for QGM in many Very talented mathematicians from all over the world ways. QGM researchers and close international col- was present at Aarhus University durin
CENTRE FOR QUANTUM GEOMETRY OF MODULISPACES (QGM) 4 PUBLIKATIONSNAVN AFSNITSNAVN QGM was established in 2009 as a Center of Ex- Mission statement cellence funded by the Danish National Research - To contribute to defining quantum field theory as a Foundation. The research objective is to address fun
AFSNITSNAVN PUBLIKATIONSNAVN 5 HELLOS ... QGM continues to be an attractive workplace for both national and international researchers. Aside from the permanent profes- sors, the Centre hosts a population of 7-8 postdocs and around 15 PhD students at any one time. Qiongling Li Alessandro Malusà Postd
6 = i k πpj 2 e p cos k k p=0 πi SCIENTIFIC BREAKTHROUGH PUBLIKATIONSNAVN AFSNITSNAVN 2 ρ k,1 (S)(ẽ j ) = k k1 sin p=1 πpj ẽ p k 2 ρ k,0 (T )(e j ) = e πi/4 e 2k j e j , πi 2 ρ k,1 (T )(ẽ j ) = e πi/4 e 2k j ẽ j . 2 2 2πk(bb)y Isomorphisms between the combinatorial of Complex Chern-Simons t
AFSNITSNAVN 7 PUBLIKATIONSNAVN With Soibelman, Maxim Kontsevich established an algebraic wall-crossing struc- ture (applicable e.g. to cluster algebras). Kontsevich further found a resurgence structure in the path integral for the free particle on the sphere. Together with F. Haiden, L. Katzarkov an
8 PUBLIKATIONSNAVN AFSNITSNAVN Locality of Gravitational Systems from Entangle- ment of Conformal Field Theories Topological recursion for Gaussian means and cohomological field theories One of Hirosi Ooguris current J.E. Andersen & Leonid Chekhov proved combinato- projects is to derive robust pre-
PHD RESEARCH PROJECTS PUBLIKATIONSNAVN 9 Construction of Modular Functors from Modular Tensor Categories Willam Petersen adapted, in joint 1 6 k 1 6 1 6 1 6 1 6 πi 2 πpj 2 e p ρ k,0 (T )(e j ) = e πi/4 e 2k j e j , ρ k,0 (S)(e j ) = i cos construction of modular functors k k p=0 from modular ten
10 AWARDS 2015 PUBLIKATIONSNAVN AFSNITSNAVN Maxim Kontsevich, Docteur Honoris Causa de l'Universite de Vienne, Austria Maxim Kontsevich, Member of the Selection Committee for the 2015 Breakthrough Prize in Fundamental Physics and the New Horizons in Physics Prize Edward Frenkel, The Euler Book Prize
INTERDISCIPLINARY COLLABORATION WITH iNANO AFSNITSNAVN PUBLIKATIONSNAVN Center leader Jørgen Ellegaard Andersen joined the A molecular biological system comprises the building Modelling and the Medicine Group at iNANO, Aarhus blocks of all living systems. A description of the struc- University in Ma
12 CONFERENCE ON NEW DEVELOPMENTS IN TQFT PUBLIKATIONSNAVN AFSNITSNAVN The historical background for the conference New developments in TQFT is the seminal work of Vaughan Jones and Edward Witten. First Jones defined his knot polynomial based on his tower construction in von Neumann algebras in the
AFSNITSNAVN PUBLIKATIONSNAVN 13 In total 59 participants joined the conference New developments in TQFT SPEAKERS Dror Bar-Natan (University of Toronto) Christian Blanchet (Institut de Mathématiques de Jussieu, U. Paris Diderot) Gaëtan Borot (MPI, Bonn) Steven Bradlow (Univ. of Illinois at Urbana-Cha
14 RESEARCH REPORTS PUBLIKATIONSNAVN By Leonid Chekhov Visiting prof. at QGM In collaboration with Jørgen Andersen, Bob Penner, and Paul Norbury (Melbourne University), we have applied the technique of topological recursion developed in 20042006 by Bertrand Eynard, Nicolas Orantin and myself to calc
AFSNITSNAVN By Yang Huang Postdoc at QGM PUBLIKATIONSNAVN 15 In 2015 Yang Huang constructed, in a joint work in progress with Prof. K. Honda, a contact Morse function in the critical level, which may play a fundamental role in the intrinsic understanding of contact mani- folds in any dimension. Anot
16 PUBLIKATIONSNAVN AFSNITSNAVN PHD PROFILE By Simone Marzioni Simone Marzioni is presently working on his PhD thesis. He expects to finish 31 July 2016. I started my PhD program at the QGM in 2013 as a student of Jørgen Ellegaard Andersen, after a master degree at the University of Bologna. My prev
AFSNITSNAVN 17 PUBLIKATIONSNAVN QGM EVENTS 2015 Conferences, Masterclasses & workshops CENTRE Masterclass by Ivan Loseu (Northeastern University) Quantized quiver March 2015 varieties Title : Quantized quiver varieties Q G M for QUANTUM GEOMETRY of MODULI SPACES D E P A R T M E N T of M A T H E M A
18 PUBLIKATIONSNAVN AFSNITSNAVN PUBLICATIONS 2015 Journal articles Abad, C.A. & Schätz, F., Higher holonomies: comparing two constructions, Differential Geometry and its Applications, vol. 40, 1442 Alexeev, N.V., Andersen, J.E., Penner, R.C. & Zograf P., Enumeration of chord diagrams on many interva
AFSNITSNAVN PUBLIKATIONSNAVN 19 Mandel, T., Tropical theta functions and log CalabiYau surfaces, Paper accepted for publication in Selecta Mathematica Martens, J., Thaddeus, M., Compactifications of reductive groups as moduli stacks of bundles, Compositio Mathematica, vol. 152, no. 1, 62- 98 Mayer,
Center for Quantum Geometry of Moduli Spaces (QGM) Aarhus University Ny Munkegade 118, bldg. 1530 DK-8000 Aarhus C Denmark Phone: +45 8715 5141 Fax: +45 8613 1769 http://qgm.au.dk Center of Excellence funded by the Danish National Research Foundation