AFSNITSNAVN
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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 and P. Pandit he worked out many details in their ongoing
project on Kähler stability in triangulated categories over local fields.
Now Ψ : Ã J Â J is a morphism in cdga
C . For this to descend to a
morphism in cdga C , the simplest condition is that Ã J should be cofibrant and
Â J fibrant in the model category cdga C . Here Â J is fibrant, as all objects
are, but Ã J may not be cofibrant, i.e. a free cdga. However, Ã J is almost
cofibrant,
it is free spaces
in negative
Virtual fundamental classes
for as moduli
of degrees, and (4.6) says that Ψ does descend
to a morphism in cdga C in degree 0. Together these imply that Ψ descends to
sheaves on CalabiYau a morphism
four-folds
Ψ : Ã J Â J in cdga C .
Let us summarize our progress so far:
We
have from
defined
an explicit
cdga
Ã J , papers
which is
very
similar to the cdga
Dominic
Joyce
University
of Oxford
wrote
with
postdocs
A J we want eventually to construct, but lacks the generators S J n , n =
D. Borisov on virtual cycles for -2-shifted symplectic derived schemes,
1, 2, . . . , which we have not yet chosen.
leading to new Donaldson-Thomas type invariants counting (semi) 0
We have defined an explicit smooth morphism β J : B Ã J , and explicit
stable coherent
sheaves
Calabi-Yau
4-folds, and with Pavel Sa-
submersions
Φ JK on
: A
K J, all part of a
K Ã J in cdga C for all =
fronov on
a Lagrangian
neighbourhood
for Lagrangians in
commutative
diagram
Γ in cdga C theorem
.
k-shifted
symplectic
derived schemes.
We
have constructed
a non-explicit cdga Â J , with morphisms Ψ : Ã J
: Spec Â J X in dSch C , which is a Zariski open
Â J in cdga C and α J
inclusion with image iJ Im α i , satisfying the homotopies (4.4).
University of Oxford
β J
Spec Â J
φ J3
φ J1
φ J2
Spec β 1
φ J1,2
Z =
Spec
B
Spec A 1
Spec
Spec β 1,2
Φ 1,21
β
Spec
Spec β 1,3
2
Spec Φ 1,22
Spec A
Spec A 1,2
2
Spec β 2,3
2
Spec Φ J2,3
φ J1,3
22
Spec Φ J2
Spec
2
Φ J1
2
Spec β 3
Spec Φ 2,32
Spec
Φ 1,31
2 2
β J
Spec
2
2
A
Spec A 3
1,3 222
Φ 1,33
Spec
Spec
2 2 Spec Φ J1,2
222
α 1,3
22
Spec Φ 2,33
Spec Φ J3
222 Spec Φ J1,3
Spec A 2,3
222
Spec Ã J
π α α 1 222 Spec Φ J2,3
2 22 α 1,2
α J
α 2,3
α 3
2 22
X
ψ
Figure 4.2: Graph Spec Γ when J = 1, 2, 3, plus X and Spec Â J
We illustrate the diagram Spec Γ plus vertices X, Spec Â J and edges π,
α K , α J , β J , φ JK , ψ in Figure 4.2 when J = 1, 2, 3. Note that this diagram
is not completely homotopy commutative, as the morphisms α K Spec Φ JK :
Spec Ã J X may depend on = K J, even up to homotopy.
Next, by induction on decreasing n = 1, 2, . . . we will choose the data
S J n , δ J n in Hypothesis 4.1(a),(e). Here is our inductive hypothesis:
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