市大数学教室

大阪市立大学大学院理学研究科数物系専攻 21世紀COEプログラム

結び目を焦点とする広角度の数学拠点の形成
(Constitution of wide-angle mathematical basis focused on knots)
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2003年度 談話会
2002年度へ
2004年度へ
講演者: Wu Yunfei(Ningbo Univ., 中国)
題 目: A new integral representation of the Riemann Zeta
日 時: 3月19日(金)16:00 〜 17:00
場 所: 数学講究室(3040)
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講演者: Zhou Songping(Zeijian Institute of Science and Technology, 中国)
題 目: A new condition in Fourier analysis
日 時: 3月18日(木)16:30 〜 17:30
場 所: 数学講究室(3040)
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講演者: Zhou Songping(Zeijian Institute of Science and Technology, 中国)
題 目: Exact Box Dimension of Weierstrass type functions
日 時: 3月18日(木)15:00 〜 16:00
場 所: 数学講究室(3040)
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講演者: M. Denker(Goettingen 大学)
題 目: Local limit theorems for stationary processes
日 時: 3月15日(月)16:00 〜 17:00
場 所: 数学講究室(3040)
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講演者: Y. Pesin(Pennsylvania 州立大学)
題 目: Pathological properties of invariant foliations
日 時: 3月9日(火)16:00 〜 17:00
場 所: 数学講究室(3040)
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講演者: 鎌田 直子(大阪市立大学)
題 目: On quasi-alternating virtual link diagrams
日 時: 2月27日(金)15:00 〜 16:00
場 所: 数学講究室(3040)
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講演者: 横田 佳之(都立大)
題 目: On the Jones polynomial and the Neumann-Zagier
function of hyperbolic knots
日 時: 2月23日(月)16:00 〜 17:00
場 所: 数学講究室(3040)
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講演者: Nanhua Xi(中国科学院)
題 目: The based ring of two-sided cells of an affine Weyl group
日 時: 2月18日(水)16:30 〜 17:30
場 所: 数学講究室(3040)
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講演者: Alexey Kulik(Institute of Math., Nat.Acad.of Sci.of Ukraine)
題 目: Malliavin calculus for Lévy processes with arbitrary
Lévy measures
日 時: 2月4日(水)16:00 〜 17:00
場 所: 数学講究室(3040)
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講演者: Nafaa Chbili(東京工業大学, JSPS)
題 目: Invariants of freely periodic knots
日 時: 1月30日(金)16:30 〜 17:30
場 所: 第3セミナー室(3153)
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講演者: 柴 雅和(広島大学大学院・工学研究科)
題 目: リーマン面上の双曲距離
(等角的埋め込み・単葉関数論・新型基本領域)
日 時: 1月29日(木)15:00 〜 16:00
場 所: 数学講究室(3040)
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講演者: Tom Schmidt(Oregon State University)
題 目: Billiards and Big Groups
日 時: 12月24日(水)16:00 〜 17:00
場 所: 数学講究室(3040)
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講演者: Myint Zaw(東京大学大学院・数理科学研究科, JSPS)
題 目: Slit domain model for moduli space of Riemann surfaces
日 時: 12月11日(木)16:30 〜 17:30
場 所: 第1セミナー室(3068)
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講演者: 斉藤 義久(東京大学大学院・数理科学研究科)
題 目: 量子群の結晶基底とquiver多様体
日 時: 12月10日(水)16:30 〜 17:30
場 所: 数学講究室(3040)
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講演者: Andrei Pajitnov(Université de Nantes)
題 目: Circle-valued Morse theory
日 時: 12月10日(水)16:30 〜 17:30
場 所: 第1セミナー室(3068)
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講演者: 今野 紀雄(横浜国立大学)
題 目: 円環上の量子ランダムウォーク
日 時: 12月3日(水)16:30 〜 17:30
場 所: 数学講究室(3040)
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講演者: Jouni Parkkonen(University of Jyvaskyla)
題 目: Unclouding the sky: geodesics and horoball packings
in negatively curved spaces
日 時: 11月27日(木)14:00 〜 15:00
場 所: 数学講究室(3040)
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講演者: Zhengyu Mao(Rutgers University-Newark)
題 目: Macdonald's formula for spherical functions on symmetric spaces
日 時: 11月26日(水)117:00 〜 18:00
場 所: 数学講究室(3040)
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講演者: Masahico Saito(University of South Florida)
題 目: Quandle homology theories and cocycle
invariants of knots and knotted surfaces
日 時: 11月21日(金)16:00〜17:00
場 所: 第3セミナー室(3153)
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講演者: Scott Carter(University of South Alabama)
題 目: Coloring knot diagrams, knotted surfaces,
and quandles
日 時: 11月21日(金)14:30〜15:30
場 所: 第3セミナー室(3153)
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講演者: Krzysztof Pawalowski(Univ. of Adam Mickiewicz)
題 目: Smith equivalence for finite Oliver group actions
日 時: 11月17日(月)16:30〜17:30
場 所: 数学講究室(3040)
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講演者: 山口 博史(奈良女子大学・理学部)
題 目: リーマン面のベルグマン計量の関数論的動きについて
日 時: 11月12日(水)16:00〜17:00
場 所: 数学講究室(3040)
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講演者: 望月 拓郎(大阪市立大学大学院・理学研究科)
題 目: Simpson の meta theorem について
日 時: 10月29日(水)15:30〜16:30
場 所: 数学講究室(3040)
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講演者: Serge Troubetzkoy(Luminy)
題 目: How complex is the game of billiards
日 時: 10月10日(水)16:00〜17:00
場 所: 第1セミナー室(3068)
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講演者: 加藤 和也(京都大学大学院・理学研究科)
題 目: Hodge 構造の退化とその分類空間
(臼井三平 氏との共同研究)
日 時: 10月8日(水)16:30 〜 17:30
場 所: 数学講究室(3040)
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講演者: Matthias Franz(Univ. of Geneva)
題 目: The homology of fibre bundles
日 時: 10月1日(水)15:30 〜 16:30
場 所: 数学講究室(3040)
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講演者: Wu Yunfei(Ningbo Univ., 中国)
題 目: A new integral representation of the Riemann Zeta Function

The series $\sum_{n=1}^{\infty}\frac{1}{n^{l+1}}e^{-z^k/n^k}$, where $k$ is any positive integer, $l$ is a positive odd number and $l\leq 2k-1$, is studied, and for each pair $(k,l)$, an integral representation of the Riemann zeta function is given. For small pairs, this provides known representations.

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講演者: Zhou Songping(Zeijian Institute of Science and Technology, 中国)
題 目: A new condition in Fourier analysis

In Fourier analysis, since Fourier coefficients are computable and applicable, people have already established many nice results by assuming monotonicty of the coefficients. One famous classical result is done by [Chaundy and Jolliffe: The uniform convergence of a certain class of trigonometric series, Proc. London Math. Soc.(2) 15(1916), 214-216] as follows: Suppose that $\{b_{n}\}$ is a non-increasing real sequence with $\lim\limits_{n\to\infty}b_{n}=0$. Then a necessary and sufficient condition for the uniform convergence of the series $ \sum_{n=1}^{\infty}b_{n}\sin nx$ is $\lim\limits_{n\to\infty}nb_{n}=0$.

Study on monotonicity gradually divided into two directions. First, people considered various quasimonotonicty, including $O$-regularly varying monotonicity. Secondly, some mathematicians such as Leindler introduced " rest bounded variation" condition. We will not mind whether quasimonotonicity (including $O$-regularly varying quasimonotonicity) and the "rest bounded variation" condition can contain each other or not, but consider to essentially generalize both conditions (more generally, in complex sense).

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講演者: Zhou Songping(Zeijian Institute of Science and Technology, 中国)
題 目: Exact Box Dimension of Weierstrass type functions

We investigate the intrinsic relationship between Box dimension of graphs of Besicovich functions and the asymptotic behavior of coefficients sequence $\{\lambda_{k}\}_{k=1}^{\infty}$. We show that the upper Box dimension does not exceed $s$ in general, and equals to $s$ while the constant $\lambda$ in Hardmard condition is sufficiently large. If the sequence $\{\lambda_{k}\}_{k=1}^{\infty}$ grows in sense faster than any geometrically rate, the lower Box dimension of $\mbox{\rm Graph}(B)$ will be strictly less than $s$. Then a necessary and sufficient condition is obtained for this type of Besicovitch functions to have exact Box dimension. We also obtain the exact Box dimensions for graphs of fractional integrals and derivatives of Weierstrass type functions. Constructive structures of such functions are also investigated.

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講演者: M. Denker(Goettingen 大学)
題 目: Local limit theorems for stationary processes

The famous de Moivre-Laplace theorem from the mid of the 18th century is one of the oldest results in probability theory. We will build on this explaining how modern dynamics and this theorem relates. We will set up a theory in which such theorems show up naturally from perturbation theory and show how they can be applied in dynamical system theory.

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講演者: Y. Pesin(Pennsylvania 州立大学)
題 目: Pathological properties of invariant foliations

I will describe some quite surprising somewhat "pathological" phenomena associated with invariant splittings for hyperbolic dynamical systems including the so-called "Fubini's nightmare". I will explain that contrary to the common belief these patalogies can happen "typically".

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講演者: 鎌田 直子(大阪市立大学)
題 目: On quasi-alternating virtual link diagrams

I introduce Manturov's work on quasi-alternating virtual link diagrams: A quasi-alternating diagram has the minimal crossing number (Thereom 1) and the span of the Jones polynomial of any virtual link diagram is less than or equal to the crossing number of the diagram (Theorem 2). Then a relation to my work on alternating virtual link diagrams is discussed.

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講演者: 横田 佳之(都立大)
題 目: On the Jones polynomial and the Neumann-Zagier function of hyperbolic knots

I shall explain how the Neumann-Zagier function, defined on the deformation space of a hyperbolic knot complement, is derived from the colored Jones polynomial of the knot.

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講演者: Nanhua Xi(中国科学院)
題 目: The based ring of two-sided cells of an affine Weyl group

Based ring of two-sided cells of certain Coxeter groups is introduced by Lusztig. The based rings are interesting and have nice relations with the corresponding Hecke algebras. In this talk I will mainly talk about a conjecture of Lusztig concerning with the structure of the based rings. We also explain some applications.

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講演者: Alexey Kulik(Institute of Math., Nat.Acad.of Sci.of Ukraine)
題 目: Malliavin calculus for Lévy processes with arbitrary Lévy measures

The new method for proving of absolute continuity of distributions of solutions of SDE's with jumps is proposed. This method is based on the "time-wise" differentiation on the space functionals from Poisson point measure, and can be applied to point measures with arbitrary Lévy measures, unlike the known methods by J.Bismut or J.Picard. Using this approach, we give sufficient conditions of absolute continuity of the solution, which don't rely on the properties of Lévy measure and are formulated only in the terms of coefficients of equation.

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講演者: Nafaa Chbili(東京工業大学, JSPS)
題 目: Invariants of freely periodic knots

Let $p \ge 2$ be an integer. A link $L$ in $S^3$ is said to be $p$-freely periodic if and only if there exists an orientation-preserving homeomorphism of order $p$, $h : (S^3, L) \to (S^3, L)$ such that $h^i$ has no fixed points for all $1 \ge i \ge p-1$. In a former work, we used the first coefficient of the Homfly polynomial to provide a necessary condition for a knot to be $p$-freely periodic. In this talk, we explain how to extend this condition to the second and third coefficient of the Homfly polynomial. We also explain how to apply this for knots with less than $9$ crossings.

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講演者: 柴 雅和(広島大学大学院・工学研究科)
題 目: リーマン面上の双曲距離
(等角的埋め込み・単葉関数論・新型基本領域)

リーマン面上の双曲計量(ポアンカレ計量)が, リーマン面の等角的埋め込みの研究,古典的な単葉関数論と幾何学的関数論,不連続群の基本領域構成など,さまざまな場面で果たす役割について述べる。

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講演者: Tom Schmidt(Oregon State University)
題 目: Billiards and Big Groups

By an unfolding process, billiard paths on a Euclidean rational angled polygonal table can be seen as straight line paths on an associated flat surface. Veech (1989) showed that unique ergodicity of the billiard flow on the table follows when the associated surface has a particularly nice group of self-maps. In work appearing in a 1992 conference proceedings from Okayama and Kyoto, Veech was lead to ask if this group of `affine diffeomorphisms' can ever be infinitely generated. In joint work with P. Hubert, we show that this does occur.

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講演者: Myint Zaw(東京大学大学院・数理科学研究科, JSPS)
題 目: Slit domain model for moduli space of Riemann surfaces

Let $F$ be a connected, compact (orientable or non-orientable) surface of some genus $g\geq 0$ with a tangent direction $\mathfrak X$ (i.e. a non-zero tangent vector up to positive multiple) specified at some base point $\mathcal O \in F$. To specify a tangent direction amounts to specify a boundary curve. Denote ${\mathfrak M}_{g,1}$ the moduli space of equivalence classes $[F, {\mathfrak X}, {\mathcal O}]$.

We will explain a homeomorphic model of ${\mathfrak M}_{g,1}$; namely the space $P(h,c)$, where $h=2g$, of slit domains of $h$ pairs of horizontal slits in the complex plane $\mathbb C$. The $P=P(h,c)$ is an open manifold embedded in a finite cell complex $\bar{P}$ such that $\bar{P}- P$ is a subcomplex of codimension 1.

Using the slit domain model, we compute the homology groups of the moduli spaces ${\mathfrak M}_{g,1}$ the moduli space of (orientable and non-orientable) Reimann surfaces for small genus. We will explain the homology computation for genus one case. Finally we will explain briefly the computation of homology of hyperelliptic moduli spaces.

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講演者: 斉藤 義久(東京大学大学院・数理科学研究科)
題 目: 量子群の結晶基底とquiver多様体

量子群とは1980年代半ばにDrinfeldと神保により可解格子模型の研究の過程で独立に発見されたKac-Moody Lie algebraの普遍包絡環の$q$-変形である。このとき量子群のパラメータ$q$は温度に対応している。

柏原はこのパラメータ$q$が$0$になった場合を考察して結晶基底を定義した。本来の物理モデルにおいて$q=0$の極限をとる操作の意味はともかくとして、結晶基底は量子群の表現論において非常に強力な手段を与える。結晶基底によって表現の複雑な構造が組合せ的な情報に翻訳されてしまう。結晶基底が持つ組合せ論的な性質は量子群の表現論のみならず、例えばある種のHecke環の表現全体のなすカテゴリーの解明に役立つなど幅広い応用を持つ。その一方で量子群そのものも1990年代以降本来の物理的意味を離れて数学的な研究対象となった。きっかけの一つはLusztigによる量子群の幾何学的実現である。Lusztigはquiverという有限有向グラフ(矢印付きDynkin図形)に付随したある代数多様体上の構成可能層を用いて量子群を再構成した。その後このような幾何学的な立場からの研究は現在でもさかんに続いている。

今回の講演ではこのような2つの流れを背景にして結晶基底をquiverに付随する代数多様体を用いて幾何学的に実現する話を紹介したい。quiverに付随する代数多様体のあるLagrange部分多様体をとり、その既約成分全体 の集合を考える。このとき既約成分全体の集合に結晶構造を定義することが出来、結晶として量子群の結晶基底と同型になることがわかる。

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講演者: Andrei Pajitnov(Université de Nantes)
題 目: Circle-valued Morse theory

One of basic constructions in Morse theory is the construction of the Morse complex. Given a real-valued Morse function on a closed manifold, the corresponding Morse complex is a chain complex of free abelian groups freely generated by the critical points of the Morse function. The homology of this complex is isomorphic to the homology of the manifold.

About 1980, S.P.Novikov introduced an analog of the Morse complex for circle-valued Morse maps. The Novikov construction associates to each Morse map of a closed manifold to a circle a chain complex of free modules over the ring of Laurent series with integral coefficients. This complex is freely generated by the critical points of the function. Its homology is a homotopy invariant of the manifold and the function in question.

The matrix entries of the boundary operators of the Novikov complex are therefore Laurent series in one variable. The Novikov exponential growth conjecture says that the convergency radius of each of these series is strictly greater than zero, at least generically.

We shall report on the progress in the direction of this conjecture, and also on the recent applications of the Morse-Novikov theory in the knot theory.

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講演者: 今野 紀雄(横浜国立大学)
題 目: 円環上の量子ランダムウォーク

最近量子コンピューターとの関連で,量子ランダムウォークの研究が世界的な規模で大変活発になされている.本講演では,特に円環上の場合について,我々の研究を中心に,その周辺の話題にもふれつつ,結果の紹介を行いたい.

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講演者: Jouni Parkkonen(University of Jyvaskyla)
題 目: Unclouding the sky: geodesics and horoball packings
in negatively curved spaces

I will discuss the geometry of a simply connected, negatively curved space and its boundary at infinity, as seen from a point in the boundary at infinity. I will concentrate on the question of finding a geodesic from the base point in the boundary which does not enter very deep in a given collection of disjoint horoballs. (Recall that a horoball is a ``ball which is tangent to the boundary at infinity'', such as the ball of Euclidean radius $1$ centered at $i$ in the upper half space model of the hyperbolic plane). I will also discuss the connections of this problem to Diophantine approximation and the heights of closed geodesics in complete, non-compact, finite volume manifolds. The research described in the talk is joint work with F. Paulin.

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講演者: Zhengyu Mao(Rutgers University-Newark)
題 目: Macdonald's formula for spherical functions on symmetric spaces

Let G be a reductive group over a p-adic field, K be its maximal compact subgroup. Let H be a subgroup such that X=G/H is a symmetric space. Anspherical function on X is an eigenfunction in S(K\X) under the action of the Hecke algebra H(G//K). We compute the spherical functions and relate them to Macdonald's polynomials. We also derive the Plancherel formula for the space S(K\X).

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講演者: Masahico Saito(University of South Florida)
題 目: Quandle homology theories and cocycle
invariants of knots and knotted surfaces

Cohomology theories have been developed for certain self-distributive groupoids called quandles. Variations of invariants of knots and knotted surfaces have been defined using quandle cocycles and the state-sum form. We review these developments, and also discuss quandle modules and their relation to generalizations of Alexander modules, and topological applications of these invariants.

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講演者: Scott Carter(University of South Alabama)
題 目: Coloring knot diagrams, knotted surfaces, and quandles

This talk is a general interest talk. We start from the point of view of Fox colorings of classical knots. This coloring concept can be used for example to show that the trefoil knot is indeed a knot. Then we generalize this notion to develop the concept of Burau matrices, and Alexander polynomials. Here we illustrate the idea on an explicit braid. Then we introduce the notion of quandles. This is a generalization of the idea of a kei which was introduced by Takasaki in the 1940s. Beyond Alexander quandles and Fox colorings, we illustrate two interesting examples that are based on symmetries of polyhedra. We introduce an $R$ matrix that can be used to count colorings, and indicate how the Yang-Baxter relations give rise to a cocycle condition. Finally, we indicate how to generalize these concepts to give invariants of knotted surfaces.

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講演者: Krzysztof Pawalowski(Univ. of Adam Mickiewicz)
題 目: Smith equivalence for finite Oliver group actions

In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we show that the Smith Isomorphism Question has a negative answer and the Laitinen number a_G of G is greater than 1 in the case when G is of odd order, or when G has a cyclic quotient of order pq for two distinct odd primes p and q. We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group G with a_G greater than 1. Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if a_G = 0 or 1.

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講演者: 山口 博史(奈良女子大学・理学部)
題 目: リーマン面のベルグマン計量の関数論的動きについて

コンパクトまたはノンコンパクトリーマン面 $R$ にはベルグマン計量 $K(z)|dz|^2$ が一意的に定まる。それらのリーマン面 $R(t)$ が複素パラメーター $t$ と共に動くとき、計量 $K(t,z)|dz|^2$ が $t$ と共にどのように動くかを調べ(一言でいえば、$R(t)$ が多変数関数論的に擬凸状に動けば $\log K(t, z)$ は劣調和に動く)その応用を述べる。

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講演者: 望月 拓郎(大阪市立大学大学院・理学研究科)
題 目: Simpson の meta theorem について

Simpson は mixed Hodge structure の一般化として mixed twistor structure を導入し, 「Hodge で成り立つことは twistor でも成り立つはずだ」という原理を提案しました. この講演では, twistor structure と Hodge structure の関係について述べ, Kashiwara の予想との関連について解説します. また, harmonic bundle の漸近挙動の研究との関連についても触れる予定です.

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講演者: Serge Troubetzkoy(Luminy)
題 目: How complex is the game of billiards

I will review various results on the metric entropy, topological entropy, word complexity and growth of generalized diagonals of polygonal billiards and then compare the results.

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講演者: Matthias Franz(Univ. of Geneva)
題 目: The homology of fibre bundles

Fibre bundles are basic objects in algebraic topology. In this talk, I want to present the three classical theorems for computing the singular (co)homology of fibre bundles: the Leray-Serre theorem, Moore's theorem and the Eilenberg-Moore theorem. Moreover, I want to show that they are all consequences of the not so well-known, but very powerful "twisted Eilenberg-Zilber theorem".

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最終更新日: 2005年1月17日
(C)大阪市大数学教室