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. Another work in progress by Huang is the con-
struction of a Floer-type theory for foliations using contact topology.
Contact manifolds are smooth odd-dimensional mani-
folds equipped with contact structures, which are maxi-
mally non-integrable hyperplane distributions. Contact
manifolds can be thought of as the odd-dimensional
analogue of symplectic manifolds, and they are in fact
closely related to each other. Roughly speaking, in col-
laboration with K. Honda, my recent work establishes
a possible way to investigate contact manifolds using
Morse theory.
Indeed, the marriage between Morse theory and sym-
plectic topology has been proved to be successful and
fruitful. An (exact) symplectic manifold together with
a compatible, in an appropriate sense, Morse function
is called a Weinstein manifold. Following the work
of Y. Eliashberg, M. Gromov, A. Weinstein and many
others, Weinstein manifolds are studied from many dif-
ferent points of view: complex geometrically using the
theory of Stein manifolds; topologically using handle
decomposition and surgery; algebraically using Fukaya
categories, just to list a few.
The attempt to establish a reasonable contact Morse
theory originated in the work of E. Giroux and prospered
in dimension 3, by the work of Giroux, Honda and many
others, in classifying 3-dimensional contact structures.
In particular, following Giroux, the regular level hypersur-
faces in a contact (Morse) manifold are known as con-
vex hypersurfaces, and they play a fundamental role in
the understanding of 3-dimensional contact topology.
In dimension greater than 3, such constructions are
much harder and much less is known. In collaboration
with Honda, we succeeded in constructing a special
contact Morse function with precisely two critical points
in the middle dimensions, which are topologically can-
celing but not contactly canceling in general. See the
figure. Generalizing the 3-dimensional situation, our
special contact Morse function, conventionally called
a bypass attachment, is conjecturally the fundamen-
tal building block of contact structures in any (odd)
dimension. My earlier work on the structural theory of
coisotropic submanifolds may also serve as a potential
tool to detect the existence of bypass attachment in
given contact manifolds.
A gradient vector field of a Morse function with a canceling
pair of critical points.
Motivated by my work on coisotropic embeddings in
contact manifolds, I am developing a Floer type the-
ory for codimension-1 foliation in any smooth mani-
fold. The main idea is to embed a foliated manifold
into certain higher dimensional contact manifold as a
coisotropic submanifold. It turns out that the leaves of
the foliation are Legendrian submanifolds of the con-
tact manifold. Then the Floer theory can be thought of
as a 1-parametric version of the Legendrian contact
homology in the spirit of Symplectic Field Theory of
Eliashberg, Givental and Hofer. Conceivably, the Floer
theoretic invariants are invariants of the foliation.
Continuing my collaboration with J. Ge on the rela-
tionship between contact topology and Riemannian
geometry, we try to generalize our results on the cur-
vature pinching constant on positively curved contact
manifolds in two directions: one in dimension 3 using
geometric flows compatible with contact structure, and
the other in dimension greater than 3 understanding
the relationship between the pinching constant and the
flexibility of contact structures in the sense of M. Borman,
Eliashberg and E. Murphy.
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,
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