Tuesday Seminar on Topology
17:00 -- 18:30 Graduate School of Mathematical Sciences,
The University of Tokyo
Tea: 16:30 -- 17:00 Common Room
Last updated January 12, 2016
October 6 -- Room 056, 17:00 -- 18:30
Sho Saito (Kavli IPMU)
Delooping theorem in K-theory
There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tatefs work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.
October 20 -- Room 056, 17:30 -- 18:30
Bruno Scárdua (Universidade Federal do Rio de Janeiro)
On the existence of stable compact leaves for
transversely holomorphic foliations
One of the most important results in the theory of foliations is
the celebrated Local stability theorem of Reeb :
A compact leaf of a foliation having finite holonomy group is
stable, indeed, it admits a fundamental system of invariant
neighborhoods where each leaf is compact with finite holonomy
group. This result, together with the Global stability theorem of Reeb
(for codimension one real foliations), has many important consequences
and motivates several questions in the theory of foliations. In this talk
we show how to prove:
A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable
leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.
This is a joint work with César Camacho.
October 27 -- Room 056, 15:00 -- 16:30
Jianfeng Lin (UCLA)
The unfolded Seiberg-Witten-Floer spectrum and its applications
Following Furuta's idea of finite dimensional approximation in
the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer
stable homotopy type for rational homology three-spheres in 2003. In
this talk, I will explain how to construct similar invariants for a
general three-manifold and discuss some applications of these new
invariants. This is a joint work with Tirasan Khandhawit and Hirofumi
October 27 -- Room 056, 17:00 -- 18:30
Yuanyuan Bao (The University of Tokyo)
Heegaard Floer homology for graphs
Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and OfDonnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.
November 10 -- Room 056, 17:30 -- 18:30
Kiyonori Gomi (Shinshu University)
Topological T-duality for "Real" circle bundle
Topological T-duality originates from T-duality in superstring theory,
and is first studied by Bouwkneght, Evslin and Mathai. The duality
basically consists of two parts: The first part is that, for any pair
of a principal circle bundle with `H-flux', there is another `T-dual'
pair on the same base space. The second part states that the twisted
K-groups of the total spaces of principal circle bundles in duality
are isomorphic under degree shift. This is the most simple topological
T-duality following Bunke and Schick, and there are a number of
generalizations. The generalization I will talk about is a topological
T-duality for "Real" circle bundles, motivated by T-duality in type II
orbifold string theory. In this duality, a variant of Z_2-equivariant
November 17 -- Room 056, 17:00 -- 18:30
Atsuko Katanaga (Shinshu University)
Topology of some three-dimensional singularities related to algebraic geometry
In this talk, we deal with hypersurface isolated singularities. First, we will recall
some topological results of singularities. Next, we will sketch the classification of
singularities in algebraic geometry. Finally, we will focus on the three-dimensional
case and discuss some results obtained so far.
November 24 -- Room 056, 17:00 -- 18:30
Masatoshi Sato (Tokyo Denki University)
On the cohomology ring of the handlebody mapping class group of genus two
The genus two handlebody mapping class group acts on a tree
constructed by Kramer from the disk complex,
and decomposes into an amalgamated product of two subgroups.
We determine the integral cohomology ring of the genus two handlebody
mapping class group by examining these two subgroups
and the Mayer-Vietoris exact sequence.
Using this result, we estimate the ranks of low dimensional homology
groups of the genus three handlebody mapping class group.
December 1 -- Room 056, 17:00 -- 18:30
Takayuki Okuda (The University of Tokyo)
Monodromies of splitting families for singular fibers
A degeneration of Riemann surfaces is a family of complex curves
over a disk allowed to have a singular fiber.
A singular fiber may split into several simpler singular fibers
under a deformation family of such families,
which is called a splitting family for the singular fiber.
We are interested in the topology of splitting families.
For the topological types of degenerations of Riemann surfaces,
it is known that there is a good relationship with
the surface mapping classes, via topological monodromy.
In this talk,
we introduce the "topological monodromies of splitting families",
and give a description of those of certain splitting families.
December 8 -- Room 056, 17:00 -- 18:30
Yuichi Yamada (The Univ. of Electro-Comm.)
Lens space surgery and Kirby calculus of 4-manifolds
The problem asking "Which knot yields a lens space by Dehn surgery" is
called "lens space surgery". Berge's list ('90) is believed to be the
complete list, but it is still unproved, even after some progress by
Heegaard Floer Homology.
This problem seems to enter a new aspect: study using 4-manifolds, lens
space surgery from lens spaces, checking hyperbolicity by computer.
In the talk, we review the structure of Berge's list and talk on our
study on pairs of distinct knots but yield same lens spaces, and
4-manifolds constructed from such pairs. This is joint work with Motoo
Tange (Tsukuba University).
December 15 -- Room 056, 17:00 -- 18:30
Constantin Teleman (University of California, Berkeley)
The Curved Cartan Complex
The Cartan model computes the equivariant cohomology of a smooth manifold X with
differentiable action of a compact Lie group G, from the invariant polynomial
functions on the Lie algebra with values in differential forms and a deformation
of the de Rham differential. Before extracting invariants, the Cartan differential
does not square to zero and is apparently meaningless. Unrecognised was the fact
that the full complex is a curved algebra, computing the quotient by G of the
algebra of differential forms on X. This generates, for example, a gauged version of
string topology. Another instance of the construction, applied to deformation
quantisation of symplectic manifolds, gives the BRST construction of the symplectic
quotient. Finally, the theory for a X point with an additional quadratic curving
computes the representation category of the compact group G, and this generalises
to the loop group of G and even to real semi-simple groups.
January 12 -- Room 056, 16:30 -- 18:30
Morimichi Kawasaki (The University of Tokyo)
Heavy subsets and non-contractible trajectories
For a compact set Y of an open symplectic manifold (N,ω) and a free
homotopy class α ∈ [S1, N],@
Biran, Polterovich and Salamon
defined the relative symplectic capacity CBPS(N,Y;α) which
measures the existence of non-contractible 1-periodic trajectories of
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,α) (i.e. the
existence of non-contractible 1-periodic trajectories under some setting)
by using heaviness.
Ryo Furukawa (The University of Tokyo)
On codimension two contact embeddings in the standard spheres
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
S5 and give some results, for example, any contact structure on
S3 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.
January 19 -- Room 056, 15:00 -- 16:00
Hikaru Yamamoto (The University of Tokyo)
Ricci-mean curvature flows in gradient shrinking Ricci solitons
A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean
curvature flow and a Ricci flow.
In this talk, we consider a Ricci-mean curvature flow in a gradient
shrinking Ricci soliton, and give a generalization of a well-known result
of Huisken which states that if a mean curvature flow in a Euclidean space
develops a singularity of type I, then its parabolic rescaling near the singular
point converges to a self-shrinker.
February 16 -- Room 056, 17:00 -- 18:30
Luc Menichi (University of Angers)
String Topology, Euler Class and TNCZ free loop fibrations
Let M be a connected, closed oriented manifold.
Chas and Sullivan have defined a loop product and a loop coproduct on
H*(LM; F), the homology of the
free loops on M with coefficients in the field F.
By studying this loop coproduct, I will show that if the free loop
ΩM → LM → M
is homologically trivial, i.e. the induced map
i* : H*(LM; F) →
H*(ΩM; F) is onto,
then the Euler characteristic of M is divisible by the characteristic
of the field F
(or M is a point).