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 tensor category, so
k1
πi 2
πpj
2
as to obtain modular functors in the ρ (S)(ẽ ) =
ẽ p
sin
ρ k,1 (T )(ẽ j ) = e πi/4 e 2k j ẽ j .
j
k,1
k
k
sense of Kevin Walker.
p=1
work with J.E. Andersen, Turaevs
η t,0
(S)(f )(x)
QU A N
O
DU
LI
M
2πk(bb)x 2
=
2k e
2
R
f (y) cos(4πkyx)e 2πk(bb)y dy,
2
Jens Kristians research is focused on the Together with J.E. An-
identification of the Hitchin connection dersen, Kenneth Ras-
with the KZ connection in ongoing joint mussen has constructed
work with J.E. Andersen. In Aug. 2015
CENTRE
G
S
Q
E
M
ETR
= i 2k e
2πk(bb)x 2
f (y) sin(4πikyx)e 2πk(bb)y dy,
R
2
2k
πi/4
2πk(bb)x 2
The
duality
isomorphism
η t,0 (T )(f )(x) = η t,1 (T )(f )(x) = e
e
f (y)e 2πik(xy) e 2πk(bb)y
i
R
η t,1
(S)(f )(x)
Hitchin connections for genus 0
quantum representations
he finished his PhD thesis with the title:
T 12 T 13 T 23 = T 23 T 12
a Hitchin connection
with weaker criterion on
"Hitchin connections for genus 0 quantum the family of complex
representation". Shortly afte he was offe- structures:
red a position at the Max Planck Institute,
Jens Kristian Egsgaard
August 10, 2015
where he now conti-
V = TV + u(V )
Supervisor: Jørgen Ellegaard Andersen
Centre for Quantum Geometry of Moduli Spaces
Faculty of Science and Technology, Aarhus University
u(V ) =
1
G β (V ) + 2 G β (V )dF i(2k + n) β(V )
2(2k + n)
+ 4kV F 2ikdF β(V ) ikδ(β(V ))
+ 2k(k + n)φ(V ) + ikψ(V ))
Alexander Shapiro & Gus Schrader constructed an embedding of the quan-
tum group U q (g) into the quantized algebra of functions on the big symplec-
tic leaf of the group G. As a corollary, they obtain a presentation of U q (g) in
terms of q-commuting generators, which has immediate applications to the
Gelfand-Kirillov conjecture.
1
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