低次元トポロジーセミナー
低次元トポロジーセミナー
14:00〜15:30渡邉氏, 16:00〜17:30 石井氏 | |
| 日時： | 4月12日(木) 14:00〜15:30 渡邉氏, 16:00〜17:30 石井氏 |
| 場所： | 京都大学 数理解析研究所 102号室 |
| 講演者： |
渡邉 忠之 氏
（京大数理研、機関研究員） |
| Title： | 配置空間積分の特性類について |
| Abstract： |
Maxim Kontsevichは、Chern--Simons摂動理論の高次元のアナロジーとして、
配置空間積分を用いて奇数次元の枠付きホモロジー球面をファイバーとする
ファイバーバンドル(C^\infty-smooth)の特性類を構成しました。この特性類は、
ファイバーが３次元の場合にはホモロジー３球面の位相不変量になり、
全てのQ値の有限型不変量（大槻不変量）を含む普遍的不変量であることが
Kuperberg--Thurstonにより示されています。しかし、ファイバーの次元が
３より大きい場合には、その性質は全く知られていませんでした。
この講演では、３より高い次元におけるKontsevichの特性類の非自明性に関して
得られた結果を紹介する予定です。またそれの系として、Casson不変量の高次元化
と思われる非自明な不変量が得られることなども時間があれば話す予定です。
|
| 講演者： |
石井 敦 氏
（京大数理研、日本学術振興会特別研究員） |
| Title： | A bracket polynomial and TQFT for invariants of virtual links |
| Abstract： | 仮想結び目理論において、ブラケット多項式を用いた不変量の構成方法と オペレーター不変量の構成方法がどのように働くか（働かないか）を、 宮澤多項式を例にとって、古典的な場合との違いを説明します。 |
15:00〜 Alexander Stoimenow 氏 | |
| 日時： | 7月13日(木) 15:00〜 |
| 場所： | 京都大学数理解析研究所 402号室 |
| 講演者： |
Alexander Stoimenow 氏
（京大数理研 COE研究員） |
| Title： |
結び目の交点数の評価
Estimation of crossing numbers of knots |
| Abstract： |
5月10日の談話会で講演した内容の発展として、結び目図式から
結び目の交点数を決定したり評価したりする話題を中心にお話します。
とくに Lickorish-Thistlethwaite により定義された
semiadequate 絡み目とその部分クラスに対する交点数の評価を紹介し、
さらに、時間に余裕があった場合は、
それらの Jones 多項式の非自明性についてもお話したいとおもいます。
I intend to talk about problems related to determination and estimation of crossing numbers of semiadequate knots, as defined by Lickorish-Thistlethwaite, and some of their subclasses. If time permits, I will discuss the relation to the non-triviality of their Jones polynomial. |
14:30〜15:30 Gwenael Massuyeau 氏 | |
16:00〜17:00 高瀬 将道氏 |
| 日時： | 4月20日(木) 14:30〜15:30, 16:00〜17:00 | ||||||
| 場所： | 京都大学数理解析研究所 402号室 | ||||||
| 講演者： |
Gwenael Massuyeau 氏
（CNRS - Louis Pasteur University, Strasbourg / 日本学術振興会外国人特別研究員、京大数理研） |
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| Title： | Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds. | ||||||
| Abstract： |
The Reidemeister-Turaev torsion is an invariant of a closed
oriented three-dimensional manifold equipped with an Euler structure,
with values in the ring of quotients of the group ring of the first
homology group. We will prove that its reductions by powers of the
augmentation ideal are finite-type invariants in the sense of M.
Goussarov and K. Habiro. For this, we will start off by explaining how
their theory of finite-type invariants can be refined to take into
account Euler structures (which is a joint work with F. Deloup).
| 講演者： |
高瀬 将道 氏 （京大数理研 機関研究員）
|
Title： |
Homology 3-spheres in codimension three
|
Abstract： |
For smooth embeddings of an integral homology 3-sphere in the
6-sphere, we define an integer invariant in terms of their Seifert
surfaces. Our invariant gives a bijection between the set of smooth
isotopy classes of such embeddings and the integers; and besides,
gives rise to a complete invariant for homology cobordism classes of
all embeddings of homology 3-spheres in the 6-sphere. As a
consequence, we show that two embeddings of an oriented integral
homology 3-sphere in the 6-sphere are isotopic if and only if they are
homology cobordant. We also relate our invariant to the Rohlin
invariant and accordingly characterise those embeddings which are
compressible into the 5-sphere.
|
|
15:00〜16:00 Vladimir Turaev 氏 | |
16:30〜17:30 Alexis Virelizier 氏 |
| 日時： | 1月26日(木) 15:00〜16:00, 16:30〜17:30 | ||||||
| 場所： | 京都大学数理解析研究所１０２号室 | ||||||
| 講演者： | Vladimir Turaev 氏 （CNRS - Louis Pasteur University, Strasbourg / 数理解析研究所） | ||||||
| Title： | Unoriented topological quantum field theory and link homology | ||||||
| Abstract： |
We classify 1+1-dimensional unoriented topological quantum field
theories in terms of Frobenius algebras with additional structure.
We use these theories to define Khovanov-type link homology
for link diagrams on oriented surfaces.
| 講演者： |
Alexis Virelizier 氏 (University of California, Berkeley)
|
Title： |
Hopf diagrams and quantum invariants
|
Abstract： |
I will describe a universal encoding of string links in terms of
Hopf algebraic structures. For this, I will introduce the notion of Hopf
diagrams. As a consequence, computing a quantum invariant of a 3-manifold
reduces to the purely formal computation of the associated Hopf diagram,
followed by the evaluation of this diagram in a given category. This is a
joint work with A. Bruguieres.
|
|
| 日時： | 11月17日(木) 15:00〜 |
| 場所： | 京都大学数理解析研究所１０２号室 |
| 講演者： | Alexander Stoimenow 氏 (東京大学大学院数理科学研究科、日本学術振興会特別研究員) |
| Title： | Applications of braiding sequences |
| Abstract： |
Braiding sequences describe the knot diagrams on which the
Seifert algorithm gives a surface of given genus. I'll try to
explain their relation and application to (depending on time,
some of) the following:
1. Any concordance class of knots contains only finitely many special alternating (conjecture: positive) ones. 2. An upper estimate for the Thurston-Bennequin invariant and Maslov (rotation) number of Legendrian embeddings of negative (mirror images of positive) links in the standard contact space 3. Asymptotic enumeration by crossing number of alternating knots of given genus and duals of 1-vertex triangulations of oriented surfaces 4. A conjectural relation between the maximal hyperbolic volume of alternating knots of given genus and the sl_N weight system of Vassiliev invariants of planar 3-valent graphs |
| 日時： | 7月13日(水) 15:00〜 |
| 場所： | 京都大学数理解析研究所２０２号室 |
| 講演者： | Soeren Hansen 氏 (Kansas State University) |
| Title： | Asymptotics of the quantum invariants for surgeries on the figure 8 knot |
| Abstract： | In the talk I will decsribe joint work with J.E.Andersen on the quantum SU(2)-invariants of the 3-manifolds obtained by rational surgeries on the 3-sphere along the figure 8 knot. To motivate our calculations I will first give a proof a la Kashaev of the volume conjecture for the figure 8 knot. It is well-known that this involves an asymptotic description of Faddaev's quantum dilogarithm via the Euler dilogarithm. Actually we obtain a more detailed asymptotic description of the Jones polynomial of the figure 8 knot than predicted by the volume conjecture. Next we move on to the 3-manifold invariants. Among other things we shall give a short account of the classical SU(2) Chern-Simons theory on the 3-manifolds under investigation using results of R.Riley and P.Kirk and E.Klassen. At the end of the talk we demonstrate how to link this Chern-Simons theory to the asymptotics of the quantum invariants. |
| 日時： | 6月2日(木) 15:00〜 |
| 場所： | 京都大学数理解析研究所４０２号室 |
| 講演者： | 栗屋 隆仁 氏（九州大学大学院数理学府、日本学術振興会特別研究員） |
| Title： | LMO-Aarhus extension of Kontsevich integral |
| Abstract： | It is known that the LMO invariant does not separate lens spaces. We discuss an extension of the Kontsevich integral for knots in rational homology 3-spheres. We will see the extension separate the lens spases. |
| 日時： | 4月21日(木) 15:00〜 |
| 場所： | 京都大学数理解析研究所４０２号室 |
| 講演者： | 水摩陽子氏（京大数理研 COE研究員） |
| Title： | The Casson invariant of integral homology spheres of Mazur type |
| Abstract： | We will show that for any even number n there exists an integral homology sphere of Mazur type whose Casson invariant is n. |
| 日時： | 1月24日(月) 13:30〜14:30 |
| 場所： | 京都大学数理解析研究所００９号室（地下） |
| 講演者： | Jòzef H. Przytycki 氏 (George Washington University) |
| Title： | Fox colorings, Burnside groups and Kei as obstructions to rational moves (joint with Mietek K. Dąbkowski and Makiko Ishiwata) |
| Abstract： | We search for invariants of links which are preserved by rational moves (e.g. n-moves, (2,2)-moves,...). We can test any link invariant for its behavior under a tangle move. For rational moves the Burnside groups of links are the most useful (they are nonabelian generalizations of Fox colorings). The more general objects to use are Kei, introduced by M.Takasaki in 1942, however the interesting Kei (we call them Burnside Kei) are not well understood yet and the best tool is via core Kei of Burnside group. In particular we introduce the family of Kei Q(m,n) and ask for which values of $m$ and $n$ these Kei are finite. The simplest undecided case is Q(5,3) Kei. |
| 日時： | 12月3日(金) 16:00〜 |
| 場所： | 京都大学数理解析研究所００９号室（地下） |
| 講演者： | 横田佳之 氏 （東京都立大） |
| Title： | Integral expressions of Jones polynomials |
| Abstract： | I shall explain how to express the colored Jones polynomials of knots as an integral over a torus, which is an important progress toward the volume conjecture. |
| 日時： | 11月18日(木) 14:30〜15:30 |
| 場所： | 京都大学数理解析研究所００９号室（地下） |
| 講演者： | Gregor Masbaum 氏 (Institut de Mathematiques de Jussieu) |
| Title： | 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. |
| 日時： | 11月18日(木) 16:00〜17:00 |
| 場所： | 京都大学数理解析研究所００９号室（地下） |
| 講演者： | Jørgen Ellegaard Andersen 氏 (Aarhus 大学) |
| Title： | Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups |
| Abstract： | In this talk we shall discuss our proof that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)-Verlinde bundles over Teichmuller space, is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least 3. For the genus 2 case, this intersection is exactly the order two subgroup, generated by the hyper-elliptic involution, in the case of even degree and n=2. Otherwise the intersection is also trivial in the genus 2 case. The proof makes use of the theory of Toeplitz operators and that they are asymptotically flat with respect to Hitchin's connection in the endomorphism bundle of the SU(n)-Verlinde bundle. - If time permits we will discuss further ramifications of our program of using asymptotics of Toeplitz operators to study the quantum SU(n) theories. |
| 日時： | 11月4日（木） 15時〜 |
| 場所： | 京都大学数理解析研究所００９号室（地下） |
| 講演者： | Jean-Baptiste Meilhan 氏 (RIMS) |
| Title： | Finite type invariants of homology cylinders and string links |
| Abstract： | This talk deals with the recently defined Goussarov-Habiro finite type invariant theory, in the case of homology cylinders and string links in homology balls, which are in some sense "model objects" for the theory. We study, at low degree, some conjectural results for these objects, by explicitely computing their invariants, and we investigate the correspondence between the two situations. |
| 連絡先： | 葉廣和夫 |
| 日時： | 7月15日（木） 15時〜 |
| 場所： | 京都大学数理解析研究所４０２号室 |
| 講演者： | Julien Marche 氏 (パリ第7大学) |
| Title： | Integrality and 2-loop polynomial of knots in integer homology spheres |
| Abstract： | We give a formula for 2-loop polynomial of a single clasper surgery on a knot; we relate this formula with integrality properties of 2-loop polynomials: we will show that 12 times the 2-loop polynomial of a knot is an integer rational diagram. |
| 日時： | 7月2日(金) 15時30分〜 |
| 場所： | 京都大学数理解析研究所００５号室 （地下） |
| 講演者： | Andrew Kricker 氏（Toronto 大学） |
| 題目： | Wheeling and a non-commutative Weil complex for Jacobi diagrams |
| アブストラクト： |
The Duflo isomorphism is an algebra isomorphism between the invariant pieces
of the symmetric and universal enveloping algebras of a semi-simple Lie
algebra. ``Wheeling'' refers to a diagrammatic analogue of this. Bar-Natan,
Le and Thurston's elegant proof exploited a fundamental property of the
Kontsevich integral knot invariant.
We'll discuss, in some detail, a purely combinatorial proof of wheeling based on a diagrammatic version of Alekseev and Meinrenken's non-commutative Weil complex. In this picture the algebraic core of the proof is the familiar statement that a characteristic class does not depend on the connection chosen to construct it. The relationship of this proof to the Kontsevich integral presents an interesting puzzle. |
| 日時： | 6月10日(木) 15時〜 |
| 場所： | 京都大学数理解析研究所402号室 |
| 講演者： | 樋上和弘氏（東京大学・大学院理学系研究科） |
| 題目： | Torus knot and minimal model |
| アブストラクト： |
I explain a relationship between the colored Jones polynomial
for the torus knot and the modular form which is the character
of the Virasoro minimal model.
Ref: KH & A.N.Kirillov, Phys. Lett. B 575 (2003) 343 |
| 日時： | 4月8日(木) 15時〜 |
| 場所： | 京都大学数理解析研究所402号室 |
| 講演者： | Sergei Duzhin (Steklov Institute) |
| 題目： | On the anatomy of the Drinfeld associator |
| アブストラクト： |
Looking at the explicit expression of the Drinfeld associator found by computer up to degree 12, one can make various observations and arrive at different conjectures. The talk contains no theorems, only observations, conjectures and fantasies. One of them concerns the relation between the Knizhnik-Zamolodchikov associator and the rational associator computed by Bar-Natan up to degree 7. |
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| Last modified: January 25, 2005 |