Tuesday Seminar on Topology (April -- July, 2008)
[Japanese]
[Past Programs]
16:30 -- 18:00 Graduate School of Mathematical Sciences,
The University of Tokyo
Tea: 16:00 -- 16:30 Common Room
Last updated June 25, 2008
Information :
Toshitake Kohno
Nariya Kawazumi
April 15 -- Room 056, 16:30 -- 18:00
Jun Murakami (Waseda University)
On the invariants of knots and 3-manifolds related to the restricted quantum group
Abstract:
I would like to talk about the colored Alexander invariant and the logarithmic
invariant of knots and links. They are constructed from the universal R-matrices
of the semi-resetricted and restricted quantum groups of sl(2) respectively,
and they are related to the hyperbolic volumes of the cone manifolds along
the knot. I also would like to explain an attempt to generalize these invariants to
a three manifold invariant which relates to the volume of the manifold actually.
April 22 -- Room 056, 16:30 -- 18:00
Sergey Yuzvinsky (University of Oregon)
Special fibers of pencils of hypersurfaces
Abstract:
We consider pencils of hypersurfaces of degree d>1 in the complex
n-dimensional projective space subject to the condition that the
generic fiber is irreducible. We study the set of completely reducible
fibers, i.e., the unions of hyperplanes. The first surprinsing result is
that the cardinality of thie set has very strict uniformed upper bound
(not depending on d or n). The other one gives a characterization
of this set in terms of either topology of its complement or combinatorics
of hyperplanes. We also include into consideration more general special
fibers are iimportant for characteristic varieties of the hyperplane
complements.
May 13 -- Room 056, 16:30 -- 18:00
Tamas Kalman (The University of Tokyo, JSPS)
The problem of maximum Thurston--Bennequin number for knots
Abstract:
Legendrian submanifolds of contact 3-manifolds are
one-dimensional, just like knots. This "coincidence'' gives rise to an
interesting and expanding intersection of contact and symplectic geometry
on the one hand and classical knot theory on the other. As an
illustration, we will survey recent results on maximizing the
Thurston--Bennequin number (which is a measure of the twisting of the
contact structure along a Legendrian) within a smooth knot type. In
particular, we will show how Kauffman's state circles can be used to solve
the maximization problem for so-called +adequate (among them, alternating
and positive) knots and links.
May 20 -- Room 056, 17:00 -- 18:30
Jerôme Petit (Tokyo Institute of Technology, JSPS)
Turaev-Viro TQFT splitting.
Abstract:
The Turaev-Viro invariant is a 3-manifolds invariant. It is obtained in this way :
1) we use a combinatorial description of 3-manifolds, in this case it is : triangulation / Pachner moves
2) we define a scalar thanks to a categorical data (spherical category) and a topological data (triangulation)
3)we verify that the scalar is invariant under Pachner moves, then we obtain a 3-manifolds invariant.
The Turaev-Viro invariant can also be extended to a TQFT. Roughly speaking a TQFT is a data which assigns a finite dimensional vector space to every closed surface and a linear application to every 3-manifold with boundary.
In this talk, we will give a decomposition of the Turaev-Viro TQFT. More precisely, we decompose it into blocks. These blocks are given by a group associated to the spherical category which was used to construct the Turaev-Viro invariant. We will show that these blocks define a HQFT (roughly speaking a TQFT with an "homotopical data"). This HQFT is obtained from an homotopical invariant, which is the homotopical version of the Turaev-Viro invariant. Moreover this invariant can be used to obtain the homological Turaev-Viro invariant defined by Yetter.
June 3 -- Room 056, 16:30 -- 18:00
Yoshikazu Yamaguchi (The University of Tokyo)
On the geometry of certain slices of the character variety of a knot group
Abstract:
joint work with Fumikazu Nagasato (Meijo University)
This talk is concerned with certain subsets in the character variety of a knot group.
These subsets are called '"slices", which are defined as a level set of a regular function associated to a meridian of a knot.
They are related to character varieties for branched covers along the knot.
Some investigations indicate that an equivariant theory for a knot is connected to a theory for branched covers via slices, for example, the equivariant signature of a knot and the equivariant Casson invariant.
In this talk, we will construct a map from slices into the character varieties for branched covers and investigate the properties.
In particular, we focus on slices called "trace-free", which are used to define the Casson-Lin invariant, and the relation to the character variety for two--fold branched cover.
June 17 -- Room 056, 16:30 -- 18:00
Yuji Sano (The University of Tokyo, IPMU)
Multiplier ideal sheaves and Futaki invariant on toric Fano manifolds.
Abstract:
I would like to discuss the subvarieties cut off by the multiplier
ideal sheaves (MIS)
and Futaki invariant on toric Fano manifolds. Futaki invariant is one
of the necessary
conditions for the existence of Kahler-Einstein metrics on Fano manifolds,
on the other hand MIS is one of the sufficient conditions introduced by Nadel.
Especially I would like to focus on the MIS related to the Monge-Ampere equation
for Kahler-Einstein metrics on non-KE toric Fano manifolds.
The motivation of this work comes from the investigation of the
relationship with slope stability
of polarized manifolds introduced by Ross and Thomas.
This talk will be based on a part of the joint work with Akito Futaki
(arXiv:0711.0614).
June 24 -- Room 056, 16:30 -- 18:00
Kenneth Shackleton (Tokyo Institute of Technology, JSPS)
On computing distances in the pants complex
Abstract:
The pants complex is an accurate combinatorial
model for the Weil-Petersson metric (WP) on Teichmueller space
(Brock). One hopes that many of the geometric properties
of WP are accurately replicated in the pants complex, and
this is the source of many open questions. We compare these
in general, and then focus on the 5-holed sphere and the
2-holed torus, the first non-trivial surfaces. We arrive at
an algorithm for computing distances in the (1-skeleton of the)
pants complex of either surface.
http://www.is.titech.ac.jp/~kjshack5/FYEO.pdf
July 1 -- Room 056, 16:30 -- 18:00
Takao Satoh (Osaka University, JSPS)
On the Johnson homomorphisms of the automorphism group of
a free metabelian group
Abstract:
The main object of our research is the automorphism group of a
free group. To be brief, the Johnson homomorphisms are studied in order to
describe one by one approximations of the automorphism group of a free group
. They play important roles on the study of the homology groups of the autom
orphism group of a free group. In general, to determine their images are ver
y difficult problem. In this talk, we define the Johnson homomorphisms of th
e automorphism group of a free metabelian group, and determine their images.
Using these results, we can give a lower bound on the image of the Johnson
homomorphisms of the automorphism group of a free group.
July 8 -- Room 056, 16:30 -- 18:00
Otto van Koert (Hokkaido University, JSPS)
Contact homology of left-handed stabilizations and connected sums
Abstract:
In this talk, we will give a brief introduction to contact homology, an invariant of contact manifolds defined by counting holomorphic curves in the symplectization of a contact manifold. We shall show that certain left-handed stabilizations of contact open books have vanishing contact homology.
This is done by finding restrictions on the behavior of holomorphic curves in connected sums. An additional application of this behavior of holomorphic curves is a long exact sequence for linearized contact homology.
July 15 -- Room 056, 16:30 -- 18:00
Hiroshi Matsuda (Hiroshima University)
One-step Markov Theorem on exchange classes
Abstract:
The main theorem in this talk claims the following.
"Let L_A and L_B denote a closed a-braid and a closed b-braid,
respectively, that represent one link type.
After at most (a^2 b^2)/4 exchange moves on L_A,
we can 'see' the pair of closed braids."
In this talk, we explain the main theorem in details, and
we present some applications.
In particular, we propose a strategy to construct an algorithm
that determines whether two links are ambient isotopic.