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Tuesday Seminar on Topology
Seminar information archive ～05/02｜Next seminar｜Future seminars 05/03～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | HABIRO Kazuo, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
Seminar information archive
2016/02/16
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Luc Menichi (University of Angers)
String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)
Luc Menichi (University of Angers)
String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)
[ Abstract ]
Let $M$ be a connected, closed oriented manifold.
Chas and Sullivan have defined a loop product and a loop coproduct on
$H_*(LM;¥mathbb{F})$, the homology of the
free loops on $M$ with coefficients in the field $¥mathbb{F}$.
By studying this loop coproduct, I will show that if the free loop
fibration
$¥Omega M¥buildrel{i}¥over¥hookrightarrow
LM¥buildrel{ev}¥over¥twoheadrightarrow M$
is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow
H^*(¥Omega M;¥mathbb{F})$ is onto,
then the Euler characteristic of $M$ is divisible by the characteristic
of the field $¥mathbb{F}$
(or $M$ is a point).
Let $M$ be a connected, closed oriented manifold.
Chas and Sullivan have defined a loop product and a loop coproduct on
$H_*(LM;¥mathbb{F})$, the homology of the
free loops on $M$ with coefficients in the field $¥mathbb{F}$.
By studying this loop coproduct, I will show that if the free loop
fibration
$¥Omega M¥buildrel{i}¥over¥hookrightarrow
LM¥buildrel{ev}¥over¥twoheadrightarrow M$
is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow
H^*(¥Omega M;¥mathbb{F})$ is onto,
then the Euler characteristic of $M$ is divisible by the characteristic
of the field $¥mathbb{F}$
(or $M$ is a point).
2016/01/19
15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Hikaru Yamamoto (The University of Tokyo)
Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)
Hikaru Yamamoto (The University of Tokyo)
Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)
[ Abstract ]
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.
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.
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)
On codimension two contact embeddings in the standard spheres (JAPANESE)
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) 17:30-18:30For 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.
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.
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.
2015/12/15
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Constantin Teleman (University of California, Berkeley)
The Curved Cartan Complex (ENGLISH)
Constantin Teleman (University of California, Berkeley)
The Curved Cartan Complex (ENGLISH)
[ Abstract ]
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.
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.
2015/12/08
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Yuichi Yamada (The Univ. of Electro-Comm.)
Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)
Yuichi Yamada (The Univ. of Electro-Comm.)
Lens space surgery and Kirby calculus of 4-manifolds (JAPANESE)
[ Abstract ]
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-maniolds constructed from such pairs. This is joint work with Motoo
Tange (Tsukuba University).
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-maniolds constructed from such pairs. This is joint work with Motoo
Tange (Tsukuba University).
2015/12/01
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Takayuki Okuda (The University of Tokyo)
Monodromies of splitting families for singular fibers (JAPANESE)
Takayuki Okuda (The University of Tokyo)
Monodromies of splitting families for singular fibers (JAPANESE)
[ Abstract ]
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.
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.
2015/11/24
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Masatoshi Sato (Tokyo Denki University)
On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)
Masatoshi Sato (Tokyo Denki University)
On the cohomology ring of the handlebody mapping class group of genus two (JAPANESE)
[ Abstract ]
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.
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.
2015/11/17
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Atsuko Katanaga (Shinshu University)
Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)
Atsuko Katanaga (Shinshu University)
Topology of some three-dimensional singularities related to algebraic geometry (ENGLISH)
[ Abstract ]
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.
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.
2015/11/10
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Kiyonori Gomi (Shinshu University)
Topological T-duality for "Real" circle bundle (JAPANESE)
Kiyonori Gomi (Shinshu University)
Topological T-duality for "Real" circle bundle (JAPANESE)
[ Abstract ]
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
K-theory appears.
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
K-theory appears.
2015/10/27
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Yuanyuan Bao (The University of Tokyo)
Heegaard Floer homology for graphs (JAPANESE)
Yuanyuan Bao (The University of Tokyo)
Heegaard Floer homology for graphs (JAPANESE)
[ Abstract ]
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 O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.
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 O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.
2015/10/27
15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Jianfeng Lin (UCLA)
The unfolded Seiberg-Witten-Floer spectrum and its applications
(ENGLISH)
Jianfeng Lin (UCLA)
The unfolded Seiberg-Witten-Floer spectrum and its applications
(ENGLISH)
[ Abstract ]
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
Sasahira.
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
Sasahira.
2015/10/20
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Bruno Scardua (Universidade Federal do Rio de Janeiro)
On the existence of stable compact leaves for
transversely holomorphic foliations (ENGLISH)
Bruno Scardua (Universidade Federal do Rio de Janeiro)
On the existence of stable compact leaves for
transversely holomorphic foliations (ENGLISH)
[ Abstract ]
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 Cesar Camacho.
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 Cesar Camacho.
2015/10/06
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Sho Saito (Kavli IPMU)
Delooping theorem in K-theory (JAPANESE)
Sho Saito (Kavli IPMU)
Delooping theorem in K-theory (JAPANESE)
[ Abstract ]
There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s 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.
There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s 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.
2015/07/21
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Keiji Tagami (Tokyo Institute of Technology)
Ribbon concordance and 0-surgeries along knots (JAPANESE)
Keiji Tagami (Tokyo Institute of Technology)
Ribbon concordance and 0-surgeries along knots (JAPANESE)
[ Abstract ]
Akbulut and Kirby conjectured that two knots with
the same 0-surgery are concordant. Recently, Yasui
gave a counterexample of this conjecture.
In this talk, we introduce a technique to construct
non-ribbon concordant knots with the same 0-surgery.
Moreover, we give a potential counterexample of the
slice-ribbon conjecture. This is a joint work with
Tetsuya Abe (Osaka City University, OCAMI).
Akbulut and Kirby conjectured that two knots with
the same 0-surgery are concordant. Recently, Yasui
gave a counterexample of this conjecture.
In this talk, we introduce a technique to construct
non-ribbon concordant knots with the same 0-surgery.
Moreover, we give a potential counterexample of the
slice-ribbon conjecture. This is a joint work with
Tetsuya Abe (Osaka City University, OCAMI).
2015/07/14
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Carlos Moraga Ferrandiz (The University of Tokyo, JSPS)
How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)
Carlos Moraga Ferrandiz (The University of Tokyo, JSPS)
How homoclinic orbits explain some algebraic relations holding in Novikov rings. (ENGLISH)
[ Abstract ]
Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.
Here, X is a descending pseudo-gradient, which is said to be adapted to α.
The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.
We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.
The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.
We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.
Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space F_u of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in F_u with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in F_u.
Here, X is a descending pseudo-gradient, which is said to be adapted to α.
The morphism π1(M) → R induced by u (given by the integral of any α in F_u over a loop of M) determines a set of u-negative loops.
We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of F_u which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.
The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg^2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.
We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.
2015/07/07
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Kitayama (Tokyo Institute of Technology)
Representation varieties detect essential surfaces (JAPANESE)
Takahiro Kitayama (Tokyo Institute of Technology)
Representation varieties detect essential surfaces (JAPANESE)
[ Abstract ]
Extending Culler-Shalen theory, Hara and I presented a way to construct
certain kinds of branched surfaces (possibly without any branch) in a 3-
manifold from an ideal point of a curve in the SL_n-character variety.
There exists an essential surface in some 3-manifold known to be not
detected in the classical SL_2-theory. We show that every essential
surface in a 3-manifold is given by the ideal point of a line in the SL_
n-character variety for some n. The talk is partially based on joint
works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.
Extending Culler-Shalen theory, Hara and I presented a way to construct
certain kinds of branched surfaces (possibly without any branch) in a 3-
manifold from an ideal point of a curve in the SL_n-character variety.
There exists an essential surface in some 3-manifold known to be not
detected in the classical SL_2-theory. We show that every essential
surface in a 3-manifold is given by the ideal point of a line in the SL_
n-character variety for some n. The talk is partially based on joint
works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.
2015/06/30
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Makoto Sakuma (Hiroshima University)
The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)
Makoto Sakuma (Hiroshima University)
The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)
[ Abstract ]
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.
2015/06/23
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Matsushita (The University of Tokyo)
Box complexes and model structures on the category of graphs (JAPANESE)
Takahiro Matsushita (The University of Tokyo)
Box complexes and model structures on the category of graphs (JAPANESE)
[ Abstract ]
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.
Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.
In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.
Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.
In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.
2015/06/16
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Masaharu Ishikawa (Tohoku University)
Stable maps and branched shadows of 3-manifolds (JAPANESE)
Masaharu Ishikawa (Tohoku University)
Stable maps and branched shadows of 3-manifolds (JAPANESE)
[ Abstract ]
We study what kind of stable map to the real plane a 3-manifold has. It
is known by O. Saeki that there exists a stable map without certain
singular fibers if and only if the 3-manifold is a graph manifold. According to
F. Costantino and D. Thurston, we identify the Stein factorization of a
stable map with a shadow of the 3-manifold under some modification,
where the above singular fibers correspond to the vertices of the shadow. We
define the notion of stable map complexity by counting the number of
such singular fibers and prove that this equals the branched shadow
complexity. With this equality, we give an estimation of the Gromov norm of the
3-manifold by the stable map complexity. This is a joint work with Yuya Koda.
We study what kind of stable map to the real plane a 3-manifold has. It
is known by O. Saeki that there exists a stable map without certain
singular fibers if and only if the 3-manifold is a graph manifold. According to
F. Costantino and D. Thurston, we identify the Stein factorization of a
stable map with a shadow of the 3-manifold under some modification,
where the above singular fibers correspond to the vertices of the shadow. We
define the notion of stable map complexity by counting the number of
such singular fibers and prove that this equals the branched shadow
complexity. With this equality, we give an estimation of the Gromov norm of the
3-manifold by the stable map complexity. This is a joint work with Yuya Koda.
2015/06/09
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Manabu Akaho (Tokyo Metropolitan University)
Symplectic displacement energy for exact Lagrangian immersions (JAPANESE)
Manabu Akaho (Tokyo Metropolitan University)
Symplectic displacement energy for exact Lagrangian immersions (JAPANESE)
[ Abstract ]
We give an inequality of the displacement energy for exact Lagrangian
immersions and the symplectic area of punctured holomorphic discs. Our
approach is based on Floer homology for Lagrangian immersions and
Chekanov's homotopy technique of continuations. Moreover, we discuss our
inequality and the Hofer--Zehnder capacity.
We give an inequality of the displacement energy for exact Lagrangian
immersions and the symplectic area of punctured holomorphic discs. Our
approach is based on Floer homology for Lagrangian immersions and
Chekanov's homotopy technique of continuations. Moreover, we discuss our
inequality and the Hofer--Zehnder capacity.
2015/05/26
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Ken'ichi Kuga (Chiba University)
Introduction to formalization of topology using a proof assistant. (JAPANESE)
Ken'ichi Kuga (Chiba University)
Introduction to formalization of topology using a proof assistant. (JAPANESE)
[ Abstract ]
Although the program of formalization goes back to David
Hilbert, it is only recently that we can actually formalize
substantial theorems in modern mathematics. It is made possible by the
development of certain type theory and a computer software called a
proof assistant. We begin this talk by showing our formalization of
some basic geometric topology using a proof assistant COQ. Then we
introduce homotopy type theory (HoTT) of Voevodsky et al., which
interprets type theory from abstract homotopy theoretic perspective.
HoTT proposes "univalent" foundation of mathematics which is
particularly suited for computer formalization.
Although the program of formalization goes back to David
Hilbert, it is only recently that we can actually formalize
substantial theorems in modern mathematics. It is made possible by the
development of certain type theory and a computer software called a
proof assistant. We begin this talk by showing our formalization of
some basic geometric topology using a proof assistant COQ. Then we
introduce homotopy type theory (HoTT) of Voevodsky et al., which
interprets type theory from abstract homotopy theoretic perspective.
HoTT proposes "univalent" foundation of mathematics which is
particularly suited for computer formalization.
2015/05/19
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Akishi Kato (The University of Tokyo)
Quiver mutation loops and partition q-series (JAPANESE)
Akishi Kato (The University of Tokyo)
Quiver mutation loops and partition q-series (JAPANESE)
[ Abstract ]
Quivers and their mutations are ubiquitous in mathematics and
mathematical physics; they play a key role in cluster algebras,
wall-crossing phenomena, gluing of ideal tetrahedra, etc.
Recently, we introduced a partition q-series for a quiver mutation loop
(a loop in a quiver exchange graph) using the idea of state sum of statistical
mechanics. The partition q-series enjoy some nice properties such
as pentagon move invariance. We also discuss their relation with combinatorial
Donaldson-Thomas invariants, as well as fermionic character formulas of
certain conformal field theories.
This is a joint work with Yuji Terashima.
Quivers and their mutations are ubiquitous in mathematics and
mathematical physics; they play a key role in cluster algebras,
wall-crossing phenomena, gluing of ideal tetrahedra, etc.
Recently, we introduced a partition q-series for a quiver mutation loop
(a loop in a quiver exchange graph) using the idea of state sum of statistical
mechanics. The partition q-series enjoy some nice properties such
as pentagon move invariance. We also discuss their relation with combinatorial
Donaldson-Thomas invariants, as well as fermionic character formulas of
certain conformal field theories.
This is a joint work with Yuji Terashima.
2015/05/12
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Masayuki Asaoka (Kyoto University)
Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)
Masayuki Asaoka (Kyoto University)
Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)
[ Abstract ]
For any hyperbolic dynamical system, the number of periodic
points grows at most exponentially and the growth rate
reflects statistic property of the system. For dynamics far
from hyperbolicity, the situation is different. In 1999,
Kaloshin proved genericity of super-exponential growth in the
region where dense set of dynamical systems exhibits homoclinic
tangency (so called the Newhouse region).
How does the number of periodic points grow for generic
partially hyperbolic dynamical systems? Such systems are known
to be far from homoclinic tangency. Is the growth at most
exponential like hyperbolic system, or super-exponential by
a mechanism different from homoclinic tangency?
The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved
super-exponential growth of the number of periodic points for
generic one-dimensional iterated function systems under some
reasonable conditions. Such systems are models of dynamics
of partially hyperbolic systems in neutral direction. So, we
expect genericity of super-exponential growth in a region of
partially hyperbolic systems.
In this talk, we start with a brief history of the problem on
growth rate of the number of periodic point and discuss two
mechanisms which lead to genericity of super-exponential growth,
Kaloshin's one and ours.
For any hyperbolic dynamical system, the number of periodic
points grows at most exponentially and the growth rate
reflects statistic property of the system. For dynamics far
from hyperbolicity, the situation is different. In 1999,
Kaloshin proved genericity of super-exponential growth in the
region where dense set of dynamical systems exhibits homoclinic
tangency (so called the Newhouse region).
How does the number of periodic points grow for generic
partially hyperbolic dynamical systems? Such systems are known
to be far from homoclinic tangency. Is the growth at most
exponential like hyperbolic system, or super-exponential by
a mechanism different from homoclinic tangency?
The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved
super-exponential growth of the number of periodic points for
generic one-dimensional iterated function systems under some
reasonable conditions. Such systems are models of dynamics
of partially hyperbolic systems in neutral direction. So, we
expect genericity of super-exponential growth in a region of
partially hyperbolic systems.
In this talk, we start with a brief history of the problem on
growth rate of the number of periodic point and discuss two
mechanisms which lead to genericity of super-exponential growth,
Kaloshin's one and ours.
2015/05/07
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Patrick Dehornoy (Univ. de Caen)
The group of parenthesized braids (ENGLISH)
Patrick Dehornoy (Univ. de Caen)
The group of parenthesized braids (ENGLISH)
[ Abstract ]
We describe a group B obtained by gluing in a natural way two well-known
groups, namely Artin's braid group B_infty and Thompson's group F. The
elements of B correspond to braid diagrams in which the distances
between the strands are non uniform and some rescaling operators may
change these distances. The group B shares many properties with B_infty:
as the latter, it can be realized as a subgroup of a mapping class
group, namely that of a sphere with a Cantor set removed, and as a group
of automorphisms of a free group. Technically, the key point is the
existence of a self-distributive operation on B.
We describe a group B obtained by gluing in a natural way two well-known
groups, namely Artin's braid group B_infty and Thompson's group F. The
elements of B correspond to braid diagrams in which the distances
between the strands are non uniform and some rescaling operators may
change these distances. The group B shares many properties with B_infty:
as the latter, it can be realized as a subgroup of a mapping class
group, namely that of a sphere with a Cantor set removed, and as a group
of automorphisms of a free group. Technically, the key point is the
existence of a self-distributive operation on B.
2015/04/28
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Hidetoshi Masai (The University of Tokyo, JSPS)
Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)
Hidetoshi Masai (The University of Tokyo, JSPS)
Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)
[ Abstract ]
In this talk I will talk about the program called HIKMOT which
rigorously proves hyperbolicity of a given triangulated 3-manifold. To
prove hyperbolicity of a given triangulated 3-manifold, it suffices to
get a solution of Thurston's gluing equation. We use the notion called
interval arithmetic to overcome two types errors; round-off errors,
and truncated errors. I will also talk about its application to
exceptional surgeries along alternating knots. This talk is based on
joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and
A. Takayasu.
In this talk I will talk about the program called HIKMOT which
rigorously proves hyperbolicity of a given triangulated 3-manifold. To
prove hyperbolicity of a given triangulated 3-manifold, it suffices to
get a solution of Thurston's gluing equation. We use the notion called
interval arithmetic to overcome two types errors; round-off errors,
and truncated errors. I will also talk about its application to
exceptional surgeries along alternating knots. This talk is based on
joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and
A. Takayasu.
