Tuesday Seminar on Topology

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Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s) KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

2016/01/12

16:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Morimichi Kawasaki (The University of Tokyo) 16:30-17:30
Heavy subsets and non-contractible trajectories (JAPANESE)
[ Abstract ]
For a compact set Y of an open symplectic manifold $(N,¥omega)$ and a free
homotopy class $¥alpha¥in [S^1,N]$, Biran, Polterovich and Salamon
defined the relative symplectic capacity $C_{BPS}(N,Y;¥alpha)$ which
measures the existence of non-contractible 1-periodic trajectories of
Hamiltonian isotopies.

On the hand, Entov and Polterovich defined heaviness for closed subsets
of a symplectic manifold by using spectral invarinats of the Hamiltonian
Floer theory on contractible trajectories.
Heavy subsets are known to be non-displaceable.

In this talk, we prove the finiteness of $C(M,X,¥alpha)$ (i.e. the
existence of non-contractible 1-periodic trajectories under some setting)
by using heaviness.
Ryo Furukawa (The University of Tokyo) 17:30-18:30
On codimension two contact embeddings in the standard spheres (JAPANESE)
[ Abstract ]
In this talk we consider codimension two contact
embedding problem by using higher dimensional braids.
First, we focus on embeddings of contact $3$-manifolds to the standard $
S^5$ and give some results, for example, any contact structure on $S^3$
can embed so that it is smoothly isotopic to the standard embedding.
These are joint work with John Etnyre. Second, we consider the relative
Euler number of codimension two contact submanifolds and its Seifert
hypersurfaces which is a generalization of the self-linking number of
transverse knots in contact $3$-manifolds. We give a way to calculate
the relative Euler number of certain contact submanifolds obtained by
braids and as an application we give examples of embeddings of one
contact manifold which are isotopic as smooth embeddings but not
isotopic as contact embeddings in higher dimension.