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 June 25, 2008

4月15日 -- 056号室, 16:30 -- 18:00

村上 順 (早稲田大学理工)

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.

4月22日 -- 056号室, 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.

5月13日 -- 056号室, 16:30 -- 18:00

Tamas Kalman (東京大学大学院数理科学研究科, 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.

5月20日 -- 056号室, 17:00 -- 18:30

Jerôme Petit (東京工業大学, 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.

6月3日 -- 056号室, 16:30 -- 18:00

山口 祥司 (東京大学大学院数理科学研究科)

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.

6月17日 -- 056号室, 16:30 -- 18:00

佐野 友二 (東京大学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).

6月24日 -- 056号室, 16:30 -- 18:00

Kenneth Shackleton (東京工業大学, 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.

7月1日 -- 056号室, 16:30 -- 18:00

佐藤 隆夫 (大阪大学大学院理学研究科, JSPS)


Abstract: 我々の主な研究対象は自由群の自己同型群である.端的にいえば,Jo hnson準同型とは,自由群の自己同型群の逐次近似の情報を記述するものであり,自 由群の自己同型群のホモロジー群の構造を研究する上で重要な役割を果たす.一般に, Johnson準同型の像を決定することは非常に難しい.本講演では,自由メタアーベル 群の自己同型群に対してJohnson準同型を定義し,その像が決定できたことを紹介す る.さらに,これらの結果を用いて,自由群の自己同型群の像を下から評価できるこ とを紹介する.

7月8日 -- 056号室, 16:30 -- 18:00

Otto van Koert (北海道大学大学院理学研究科, 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.

7月15日 -- 056号室, 16:30 -- 18:00

松田 浩 (広島大学大学院理学研究科)

One-step Markov Theorem on exchange classes

Abstract: 本講演で紹介する主定理は 次のことを主張します. 「L_A, L_Bを 1つの絡み目型を表す 閉a-組み紐, 閉b-組み紐とする. このとき L_Aに 高々 (a^2 b^2)/4 回のexchange操作を施すことにより 2つの閉組み紐の形が "見える"」 講演では 最初に主定理を詳しく説明し その応用をいくつか紹介します. とくに 2つの絡み目が同じであるかを判定するアルゴリズムを 構成するための 1つの戦略を紹介します.