Text only print
Full screen print
Tuesday Seminar on Topology
Seminar information archive ～09/26｜Next seminar｜Future seminars 09/27～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
Seminar information archive
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.
2015/04/21
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Yoshikata Kida (The University of Tokyo)
Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)
Yoshikata Kida (The University of Tokyo)
Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)
[ Abstract ]
This talk is about measure-preserving actions of countable groups on probability
measure spaces and their orbit structure. Two such actions are called orbit equivalent
if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus
on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation
ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial
and geometric group theory. Whether Baumslag-Solitar groups with different p and q can
have orbit-equivalent actions is still a big open problem. I will discuss invariants under
orbit equivalence, motivating background and some results toward this problem.
This talk is about measure-preserving actions of countable groups on probability
measure spaces and their orbit structure. Two such actions are called orbit equivalent
if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus
on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation
ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial
and geometric group theory. Whether Baumslag-Solitar groups with different p and q can
have orbit-equivalent actions is still a big open problem. I will discuss invariants under
orbit equivalence, motivating background and some results toward this problem.
2015/04/14
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Nobuhiro Nakamura (Gakushuin University)
Pin(2)-monopole invariants for 4-manifolds (JAPANESE)
Nobuhiro Nakamura (Gakushuin University)
Pin(2)-monopole invariants for 4-manifolds (JAPANESE)
[ Abstract ]
The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations
which can be considered as a real version of the SW equations. A Pin(2)-mono
pole version of the Seiberg-Witten invariants is defined, and a special feature of
this is that the Pin(2)-monopole invariant can be nontrivial even when all of
the Donaldson and Seiberg-Witten invariants vanish. As an application, we
construct a new series of exotic 4-manifolds.
The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations
which can be considered as a real version of the SW equations. A Pin(2)-mono
pole version of the Seiberg-Witten invariants is defined, and a special feature of
this is that the Pin(2)-monopole invariant can be nontrivial even when all of
the Donaldson and Seiberg-Witten invariants vanish. As an application, we
construct a new series of exotic 4-manifolds.
2015/04/07
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Kazushi Ueda (The University of Tokyo)
Potential functions for Grassmannians (JAPANESE)
Kazushi Ueda (The University of Tokyo)
Potential functions for Grassmannians (JAPANESE)
[ Abstract ]
Potential functions are Floer-theoretic invariants
obtained by counting Maslov index 2 disks
with Lagrangian boundary conditions.
In the talk, we will discuss our joint work
with Yanki Lekili and Yuichi Nohara
on Lagrangian torus fibrations on the Grassmannian
of 2-planes in an n-space,
the potential functions of their Lagrangian torus fibers,
and their relation with mirror symmetry for Grassmannians.
Potential functions are Floer-theoretic invariants
obtained by counting Maslov index 2 disks
with Lagrangian boundary conditions.
In the talk, we will discuss our joint work
with Yanki Lekili and Yuichi Nohara
on Lagrangian torus fibrations on the Grassmannian
of 2-planes in an n-space,
the potential functions of their Lagrangian torus fibers,
and their relation with mirror symmetry for Grassmannians.
2015/03/24
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Mina Aganagic (University of California, Berkeley)
Knots and Mirror Symmetry (ENGLISH)
Mina Aganagic (University of California, Berkeley)
Knots and Mirror Symmetry (ENGLISH)
[ Abstract ]
I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.
I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.
2015/03/10
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Andrei Pajitnov (Univ. de Nantes)
Arnold conjecture, Floer homology,
and augmentation ideals of finite groups.
(ENGLISH)
Andrei Pajitnov (Univ. de Nantes)
Arnold conjecture, Floer homology,
and augmentation ideals of finite groups.
(ENGLISH)
[ Abstract ]
Let H be a generic time-dependent 1-periodic
Hamiltonian on a closed weakly monotone
symplectic manifold M. We construct a refined version
of the Floer chain complex associated to (M,H),
and use it to obtain new lower bounds for the number P(H)
of the 1-periodic orbits of the corresponding hamiltonian
vector field. We prove in particular that
if the fundamental group of M is finite
and solvable or simple, then P(H)
is not less than the minimal number
of generators of the fundamental group.
This is joint work with Kaoru Ono.
Let H be a generic time-dependent 1-periodic
Hamiltonian on a closed weakly monotone
symplectic manifold M. We construct a refined version
of the Floer chain complex associated to (M,H),
and use it to obtain new lower bounds for the number P(H)
of the 1-periodic orbits of the corresponding hamiltonian
vector field. We prove in particular that
if the fundamental group of M is finite
and solvable or simple, then P(H)
is not less than the minimal number
of generators of the fundamental group.
This is joint work with Kaoru Ono.
2015/01/20
16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Toru Yoshiyasu (The University of Tokyo)
On Lagrangian caps and their applications (JAPANESE)
Toru Yoshiyasu (The University of Tokyo)
On Lagrangian caps and their applications (JAPANESE)
[ Abstract ]
In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for
exact Lagrangian embeddings with a concave Legendrian boundary. In this
talk, I will explain a modification of their $h$-principle and show
applications to Lagrangian submanifolds in the complex projective spaces.
In 2013, Y. Eliashberg and E. Murphy established the $h$-principle for
exact Lagrangian embeddings with a concave Legendrian boundary. In this
talk, I will explain a modification of their $h$-principle and show
applications to Lagrangian submanifolds in the complex projective spaces.
2015/01/13
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Ken'ichi Yoshida (The University of Tokyo)
Stable presentation length of 3-manifold groups (JAPANESE)
Ken'ichi Yoshida (The University of Tokyo)
Stable presentation length of 3-manifold groups (JAPANESE)
[ Abstract ]
We will introduce the stable presentation length
of a finitely presented group, which is defined
by stabilizing the presentation length for the
finite index subgroups. The stable presentation
length of the fundamental group of a 3-manifold
is an analogue of the simplicial volume and the
stable complexity introduced by Francaviglia,
Frigerio and Martelli. We will explain some
similarities of stable presentation length with
simplicial volume and stable complexity.
We will introduce the stable presentation length
of a finitely presented group, which is defined
by stabilizing the presentation length for the
finite index subgroups. The stable presentation
length of the fundamental group of a 3-manifold
is an analogue of the simplicial volume and the
stable complexity introduced by Francaviglia,
Frigerio and Martelli. We will explain some
similarities of stable presentation length with
simplicial volume and stable complexity.
2014/12/16
17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)
Norio Iwase (Kyushu University)
Differential forms in diffeological spaces (JAPANESE)
Norio Iwase (Kyushu University)
Differential forms in diffeological spaces (JAPANESE)
[ Abstract ]
The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.
Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.
The idea of a space with smooth structure is first introduced by K. T. Chen in his study of a loop space to employ the idea of iterated path integrals.
Following the pattern established by Chen, J. M. Souriau introduced his version of a space with smooth structure which is now called diffeology and become one of the most exciting topics in Algebraic Topology. Following Souriau, P. I.-Zenmour presented de Rham theory associated to a diffeology of a space. However, if one tries to show a version of de Rham theorem for a general diffeological space, he must encounter a difficulty to show the existence of a partition of unity and thus the exactness of the Mayer-Vietoris sequence. To resolve such difficulties, we introduce a new definition of differential forms.
2014/12/09
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Koji Fujiwara (Kyoto University)
Stable commutator length on mapping class groups (JAPANESE)
Koji Fujiwara (Kyoto University)
Stable commutator length on mapping class groups (JAPANESE)
[ Abstract ]
Let MCG(S) be the mapping class group of a closed orientable surface S.
We give a precise condition (in terms of the Nielsen-Thurston
decomposition) when an element
in MCG(S) has positive stable commutator length.
Stable commutator length tends to be positive if there is "negative
curvature".
The proofs use our earlier construction in the paper "Constructing group
actions on quasi-trees and applications to mapping class groups" of
group actions on quasi-trees.
This is a joint work with Bestvina and Bromberg.
Let MCG(S) be the mapping class group of a closed orientable surface S.
We give a precise condition (in terms of the Nielsen-Thurston
decomposition) when an element
in MCG(S) has positive stable commutator length.
Stable commutator length tends to be positive if there is "negative
curvature".
The proofs use our earlier construction in the paper "Constructing group
actions on quasi-trees and applications to mapping class groups" of
group actions on quasi-trees.
This is a joint work with Bestvina and Bromberg.
2014/12/02
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yosuke Kubota (The University of Tokyo)
The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)
Yosuke Kubota (The University of Tokyo)
The Atiyah-Segal completion theorem in noncommutative topology (JAPANESE)
[ Abstract ]
We introduce a new perspevtive on the Atiyah-Segal completion
theorem applying the "noncommutative" topology, which deals with
topological properties of C*-algebras. The homological algebra of the
Kasparov category as a triangulated category, which is developed by R.
Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal
type completion theorems for equivariant K-homology and twisted K-theory.
This is a joint work with Yuki Arano.
We introduce a new perspevtive on the Atiyah-Segal completion
theorem applying the "noncommutative" topology, which deals with
topological properties of C*-algebras. The homological algebra of the
Kasparov category as a triangulated category, which is developed by R.
Meyer and R. Nest, plays a central role. It contains the Atiyah-Segal
type completion theorems for equivariant K-homology and twisted K-theory.
This is a joint work with Yuki Arano.
2014/11/25
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Masahico Saito (University of South Florida)
Quandle knot invariants and applications (JAPANESE)
Masahico Saito (University of South Florida)
Quandle knot invariants and applications (JAPANESE)
[ Abstract ]
A quandles is an algebraic structure closely related to knots. Homology theories of
quandles have been defined, and their cocycles are used to construct invariants for
classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given
for quandle cocycle invariants and their applications to geometric properties of knots.
The current status of computations, recent developments and open problems will also
be discussed.
A quandles is an algebraic structure closely related to knots. Homology theories of
quandles have been defined, and their cocycles are used to construct invariants for
classical knots, spatial graphs and knotted surfaces. In this talk, an overview is given
for quandle cocycle invariants and their applications to geometric properties of knots.
The current status of computations, recent developments and open problems will also
be discussed.
2014/11/18
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Charles Siegel (Kavli IPMU)
A Modular Operad of Embedded Curves (ENGLISH)
Charles Siegel (Kavli IPMU)
A Modular Operad of Embedded Curves (ENGLISH)
[ Abstract ]
Modular operads were introduced by Getzler and Kapranov to formalize the structure of gluing maps between moduli of stable marked curves. We present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. This is joint work with Satoshi Kondo and Jesse Wolfson.
Modular operads were introduced by Getzler and Kapranov to formalize the structure of gluing maps between moduli of stable marked curves. We present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. This is joint work with Satoshi Kondo and Jesse Wolfson.
2014/11/11
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Kenneth Baker (University of Miami)
Unifying unexpected exceptional Dehn surgeries (ENGLISH)
Kenneth Baker (University of Miami)
Unifying unexpected exceptional Dehn surgeries (ENGLISH)
[ Abstract ]
This past summer Dunfield-Hoffman-Licata produced examples of asymmetric, hyperbolic, 1-cusped 3-manifolds with pairs of lens space Dehn fillings through a search of the extended SnapPea census.
Examinations of these examples with Hoffman and Licata lead us to coincidences with other work in progress that gives a simple holistic topological approach towards producing and extending many of these families. In this talk we'll explicitly describe our construction and discuss related applications of the technique.
This past summer Dunfield-Hoffman-Licata produced examples of asymmetric, hyperbolic, 1-cusped 3-manifolds with pairs of lens space Dehn fillings through a search of the extended SnapPea census.
Examinations of these examples with Hoffman and Licata lead us to coincidences with other work in progress that gives a simple holistic topological approach towards producing and extending many of these families. In this talk we'll explicitly describe our construction and discuss related applications of the technique.
2014/11/04
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Brian Bowditch (University of Warwick)
The coarse geometry of Teichmuller space. (ENGLISH)
Brian Bowditch (University of Warwick)
The coarse geometry of Teichmuller space. (ENGLISH)
[ Abstract ]
We describe some results regarding the coarse geometry of the
Teichmuller space
of a compact surface. In particular, we describe when the Teichmuller
space admits quasi-isometric embeddings of euclidean spaces and
half-spaces.
We also give some partial results regarding the quasi-isometric rigidity
of Teichmuller space. These results are based on the fact that Teichmuller
space admits a ternary operation, natural up to bounded distance
which endows it with the structure of a coarse median space.
We describe some results regarding the coarse geometry of the
Teichmuller space
of a compact surface. In particular, we describe when the Teichmuller
space admits quasi-isometric embeddings of euclidean spaces and
half-spaces.
We also give some partial results regarding the quasi-isometric rigidity
of Teichmuller space. These results are based on the fact that Teichmuller
space admits a ternary operation, natural up to bounded distance
which endows it with the structure of a coarse median space.
2014/10/21
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Toshiyuki Akita (Hokkaido University)
Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)
Toshiyuki Akita (Hokkaido University)
Vanishing theorems for p-local homology of Coxeter groups and their alternating subgroups (JAPANESE)
[ Abstract ]
Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.
Given a prime number $p$, we estimate vanishing ranges of $p$-local homology groups of Coxeter groups (of possibly infinite order) and alternating subgroups of finite reflection groups. Our results generalize those by Nakaoka for symmetric groups and Kleshchev-Nakano and Burichenko for alternating groups. The key ingredient is the equivariant homology of Coxeter complexes.
2014/10/07
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Kei Irie (RIMS, Kyoto University)
Transversality problems in string topology and de Rham chains (JAPANESE)
Kei Irie (RIMS, Kyoto University)
Transversality problems in string topology and de Rham chains (JAPANESE)
[ Abstract ]
The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.
It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.
One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.
In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.
The starting point of string topology is the work of Chas-Sullivan, which uncovered the Batalin-Vilkovisky(BV) structure on homology of the free loop space of a manifold.
It is important to define chain level structures beneath the BV structure on homology, however this problem is yet to be settled.
One of difficulties is that, to define intersection products on chain level, we have to address the transversality issue.
In this talk, we introduce a notion of "de Rham chain" to bypass this trouble, and partially realize expected chain level structures.
2014/07/22
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Jesse Wolfson (Northwestern University)
The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)
Jesse Wolfson (Northwestern University)
The Index Map and Reciprocity Laws for Contou-Carrere Symbols (ENGLISH)
[ Abstract ]
In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.
In the 1960s, Atiyah and Janich constructed the families index as a natural map from the space of Fredholm operators to the classifying space of topological K-theory, and showed it to be an equivalence. In joint work with Oliver Braunling and Michael Groechenig, we construct an analogous index map in algebraic K-theory. Building on recent work of Sho Saito, we show this provides an analogue of Atiyah and Janich's equivalence. More significantly, the index map allows us to relate the Contou-Carrere symbol, a local analytic invariant of schemes, to algebraic K-theory. Using this, we provide new proofs of reciprocity laws for Contou-Carrere symbols in dimension 1 (first established by Anderson--Pablos Romo) and 2 (established recently by Osipov--Zhu). We extend these reciprocity laws to all dimensions.
2014/07/08
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Ingrid Irmer (JSPS, the University of Tokyo)
The Johnson homomorphism and a family of curve graphs (ENGLISH)
Ingrid Irmer (JSPS, the University of Tokyo)
The Johnson homomorphism and a family of curve graphs (ENGLISH)
[ Abstract ]
Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."
Abstract: A family of curve graphs of an oriented surface $S_{g,1}$ will be defined on which there exists a natural orientation, coming from the orientation of subsurfaces. Distances in these graphs represent commutator lengths in $\\pi_{1}(S_{g,1})$. The displacement of vertices in the graphs under the action of the Torelli group is used to give a combinatorial description of the Johnson homomorphism."
2014/07/01
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yohsuke Imagi (Kavli IPMU)
Singularities of special Lagrangian submanifolds (JAPANESE)
Yohsuke Imagi (Kavli IPMU)
Singularities of special Lagrangian submanifolds (JAPANESE)
[ Abstract ]
There are interesting invariants defined by "counting" geometric
objects, such as instantons in dimension 4 and pseudo-holomorphic curves
in symplectic manifolds. To do the counting in a sensible way, however,
we have to care about singularities of the geometric objects. Special
Lagrangian submanifolds seem very difficult to "count" as their
singularities may be very complicated. I'll talk about simple
singularities for which we can make an analogy with instantons in
dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do
it I'll use some techniques from geometric measure theory and Lagrangian
Floer theory, and the Floer-theoretic part is a joint work with Dominic
Joyce and Oliveira dos Santos.
There are interesting invariants defined by "counting" geometric
objects, such as instantons in dimension 4 and pseudo-holomorphic curves
in symplectic manifolds. To do the counting in a sensible way, however,
we have to care about singularities of the geometric objects. Special
Lagrangian submanifolds seem very difficult to "count" as their
singularities may be very complicated. I'll talk about simple
singularities for which we can make an analogy with instantons in
dimension 4 and pseudo-holomorphic curves in symplectic manifolds. To do
it I'll use some techniques from geometric measure theory and Lagrangian
Floer theory, and the Floer-theoretic part is a joint work with Dominic
Joyce and Oliveira dos Santos.
2014/06/24
17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)
Takefumi Nosaka (Faculty of Mathematics, Kyushu University)
On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)
Takefumi Nosaka (Faculty of Mathematics, Kyushu University)
On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)
[ Abstract ]
本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介
する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ
ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander
quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー
から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド
ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を
除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.
本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介
する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ
ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander
quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー
から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド
ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を
除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.
2014/06/17
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Yoshifumi Matsuda (Aoyama Gakuin University)
Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)
Yoshifumi Matsuda (Aoyama Gakuin University)
Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)
[ Abstract ]
Burger, Iozzi and Wienhard defined the bounded Euler number for a
continuous action of the fundamental group of a connected oriented
surface of finite type possibly with punctures on the circle. A Milnor-Wood
type inequality involving the bounded Euler number holds and its maximality
characterizes Fuchsian actions up to semiconjugacy. The definition of the
bounded Euler number can be extended to actions of 2-orbifold groups by
considering coverings. A Milnor-Wood type inequality and the characterization
of Fuchsian actions also hold in this case. In this talk, we describe when lifts
of Fuchsian actions of certain 2-orbifold groups, such as the modular group,
are characterized by its bounded Euler number.
Burger, Iozzi and Wienhard defined the bounded Euler number for a
continuous action of the fundamental group of a connected oriented
surface of finite type possibly with punctures on the circle. A Milnor-Wood
type inequality involving the bounded Euler number holds and its maximality
characterizes Fuchsian actions up to semiconjugacy. The definition of the
bounded Euler number can be extended to actions of 2-orbifold groups by
considering coverings. A Milnor-Wood type inequality and the characterization
of Fuchsian actions also hold in this case. In this talk, we describe when lifts
of Fuchsian actions of certain 2-orbifold groups, such as the modular group,
are characterized by its bounded Euler number.
