2004年10月 -- 2005年1月
[過去のプログラム]
16:30 -- 18:00 数理科学研究科棟(東京大学駒場キャンパス)
Tea: 16:00 -- 16:30 コモンルーム
Last updated November 29, 2004
世話係
河野俊丈
河澄響矢
10月12日 -- 056号室, 16:30 -- 18:00
Bob Penner (University of Southern California, Los Angeles CA)
Arc complexes
Complexes of arcs, or equivalently of suitable graphs, arise in
several related contexts in the study of Riemann's moduli space. For
instance, there is a cell decomposition of the product of moduli space
with an open simplex, where cells are indexed by appropriate families of
arcs in the underlying surface. Furthermore, the specification of an
ideal triangulation determines global coordinates on the ``decorated''
Teichm\"uller space of the underlying surface, and these coordinates
enjoy useful transformation and other properties. In this, the first of
several lectures on these topics, we shall survey these and other aspects
of the utility of arc or graph complexes in the study of moduli space.
トポロジー火曜セミナーに引き続いて，以下の日程で，Penner氏による，
詳しい内容とそれに関連する話題についての講演が予定されています。
１０月１３日（水）１５：００--１６：３０ １２２号室
１０月１５日（金）１５：００--１６：３０ １２２号室
10月19日 -- 056号室, 16:30 -- 18:00
田中 利史 (東京大学大学院数理科学研究科，COE研究員)
The colored Jones polynomial of satellite links
Skein theory を用いた Masbaum と Vogel による colored Jones polynomial
の計算手法を紹介した後，satellite link に対する計算から得られる mutation
に関した結果を与え，さらに体積予想との関わりについて説明します。
21世紀COEプログラム講演会
トポロジー火曜セミナーにおいて、 東京大学大学院数理科学研究科21世紀COEプログラム講演会が下記のように開催されます。
11月2日 -- 数理科学研究科 056号室, 16:30 〜 18:00
Jose Maria Montesinos-Amilibia（Universidad Complutense)
Representing open 3-manifolds as 3-fold branched covering spaces and some examples
abstract
11月8日〜11日 研究集会「多様体のトポロジーの未来へ」
11月16日 -- 122号室, 17:00 -- 18:30
後藤 竜司 (大阪大学大学院理学研究科)
Unobstructed deformations of topological calibrations
（位相的なキャリブレーションの非障害的な変形について)
abstract:
In this talk, I shall focus on geometric structures defined by closed
differential forms and develop a systematic approach of the deformation
problem of these structures.
Introducing suitable cohomology groups, I establish a criterion of
unobstructed deformations from the cohomological point of view.
Then as an application, I obtain a unified construction of moduli spaces of
Calabi-Yau, HyperKaehler, G_2 and Spin(7) structures.
I also apply this approach to certain geometric structures on complex
solvable manifolds, which are quotients of solvable Lie groups by discrete
subgroups.
Further (if I have enough time), I discuss the problem as to when geometric
structures with singularities can be deformed to be smooth ones.
I show that the vanishing of certain cohomology classes with compact
support is crucial
to the smoothing problem.
The first part of this talk will be based on the paper
Moduli spaces of topological calibrations, International J. Math. 15
(2004), 211-257
11月30日 -- 126号室, 16:30 〜 18:00
Alexander Stoimenow (東京大学大学院数理科学研究科， JSPS)
Properties of closed 3-braids
Abstract:
Work of Bennequin, and later Birman-Menasco, on braided
Seifert surfaces allowed to identify from 3-braid
representations their closure links. However, many
properties remain non-trivial to decide. Xu gave an
algorithm to determine the genus. I will use her algorithm
to describe the skein, Jones, and Alexander polynomial of
3-braid links and to determine which links are fibered.
The study of Jones polynomials allows to prove that no
non-trivial 3-braid links have trivial polynomial.
Finally, depending on time, I will discuss some applications
of the unitarization of the Burau representation, that
concern also braids on more strings.
12月7日 -- 056号室, 16:30 〜 18:00
Gregor Masbaum (Univ. Paris VII)
Integral lattices in TQFT
Abstract: We will describe joint work with Pat Gilmer where we
find explicit bases for naturally defined lattices in the vector
spaces associated to surfaces by the SO(3) TQFT at an odd
prime. These lattices form an "Integral TQFT" in an appropriate
sense. Some applications relating quantum invariants to classical
3-manifold topology will be given.
12月14日 -- 056号室, 16:30 〜 18:00
D. Kotschick (Ludwig-Maximillian-Univ. Munchen)
Asymptotic volumes, entropies, and some applications
Abstract: The first part of the lecture will survey the definitions and basic properties of certain topological invariants of manifolds that arise naturally from asymptotic considerations in geometry and dynamics. I will explain a chain of inequalities between such invariants which centers on the asymptotic volume, or minimal volume entropy. In the second part of the lecture we shall discuss a couple of very specific geometric problems in which the asymptotic volume and its upper and lower bounds have played a role recently.
12月21日 -- 126号室, 16:30 〜 18:00
Jozef H. Przytycki (The George Washington University)
Khovanov Homology: categorification of the Kauffman bracket relation
Abstract: We define Khovanov homology, $H_{i,j,k}$ for
links in products of surfaces and an interval and in twisted
$I$-bundles over unorientable surfaces (excluding $RP^2$).
We show how to stratify this homology so that (in the product case,
$F\times I$)
it categorifies the Kauffman bracket skein module (KBSM) of $F\times I$.
That is,
for any link $L$ in $F\times I$ we can recover coefficients of $L$ in the
standard basis $B(F)$ of the KBSM of $F\times I$.
In other words if $L = \sum_b a_b(A) b$ where
the sum is taken over all basic elements, $b\in B(F)$, then each coefficient
$a_b(A)$ can be recovered from polynomial Euler characteristics of
the stratified Khovanov homology. In the case of unorientable F we are
able to recover coefficients $a_b(A)$ only partially. We propose
another basis of the KBSM for which categorification seems to be possible
even for unorientable F (we use cores of M\"obius bands several times
even if they intersect one another).
1月18日 -- 056号室, 16:30 〜 18:30
土岐 豊嗣 (東京大学大学院数理科学研究科)
Hyperplane arrangements and
the determinant of a period Matrix
abstract:
We study multivariable hypergeometric integrals of type
$\int_{\Delta}Udx_1\cdots dx_n$, $U=\exp{(-\frac{1}{2}
(x_1^2+\cdots+x_n^2))}f_1^{\lambda_1}\cdots f_m^{\lambda_m},$
where $\lambda_i(1\le i \le m)$ is a complex number and
$f_i(1\le i \le m)$ is degree one polynomial defined in
an $n$ dimensional vector space.
The function $U$ define a local system $L$,
and we give a
twisted homology group for $L$.
Aomoto conjectured on the base of the twisted
cohomology for $L.$
陳 智 (東京大学大学院数理科学研究科)
Stratification of the complement of hyperplane arrangement
and homology of local systems
abstract:
Through some stratifications of $M(/mathcal A)$, the
complementary space of hyperplane arrangement, two kinds of finite dimensional
complexes are constructed. They can be used to compute
$H_* ^{lf}(M(/mathcal A),/mathcal L )$, the locally finite homology
groups of $M(/mathcal A)$ with coefficients in a rank 1 local system
$/mathcal L $. In addition, some description of the canonical
map from $H_* (M(/mathcal A),/mathcal L )$ to
$H_* ^{lf}(M(/mathcal A),/mathcal L )$ are given.
As an application of above results, we study the homological representations
of the braid groups. The noncommutative polynomial rings are introduced to
describe those representations.