2009年9月 -- 2010年2月
[English]
[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 January 20, 2010
世話係
河野俊丈
河澄響矢
9月29日 -- 056号室, 16:30 -- 18:00
Sergei Duzhin (Steklov Mathematical Institute, Petersburg Division)
Symbol of the Conway polynomial and Drinfeld associator
Abstract:
The Magnus expansion is a universal finite type invariant of pure braids
with values in the space of horizontal chord diagrams. The Conway polynomial
composed with the short circuit map from braids to knots gives rise to a
series of finite type invariants of pure braids and thus factors through
the Magnus map. We describe explicitly the resulting mapping from horizontal
chord diagrams on 3 strands to univariante polynomials and evaluate it on
the Drinfeld associator obtaining a beautiful generating function whose
coefficients are integer combinations of multple zeta values.
10月13日 -- 056号室, 16:30 -- 18:00
笹平 裕史 (東京大学大学院数理科学研究科)
レンズ空間に対するインスタントンFloerホモロジー
Abstract:
Y を向きのついた閉3次元多様体、PをY上のSU(2)主束とする。ある条件のもとで、Y の不変量であるインスタントン Floer ホモロジーが、Chern-Simons汎関数のMorseホモロジーとして定義することができる。ある条件とは、P上の平坦接続が全て既約であるということである。可約な平坦接続が存在するときは、（一部の場合を除いて）定義されていない。
レンズ空間の基本群は可換であるということから、レンズ空間上の平坦接続は全て可約である。今回の話で、レンズ空間に対してインスタントン Floer ホモロジーを導入する。また、いくつかの計算例を紹介する。
10月20日 -- 056号室, 16:30 -- 18:00
吉田 尚彦 (明治大学大学院理工学研究科)
Torus fibrations and localization of index
Abstract:
I will describe a localization of index of a Dirac type operator.
We make use of a structure of torus fibration, but the mechanism
of the localization does not rely on any group action. In the case of
Lagrangian fibration, we show that the index is described as a sum of
the contributions from Bohr-Sommerfeld fibers and singular fibers.
To show the localization we introduce a deformation of a Dirac type
operator for a manifold equipped with a fiber bundle structure which
satisfies a kind of acyclic condition. The deformation allows an
interpretation as an adiabatic limit or an infinite dimensional analogue
of Witten deformation.
Joint work with Hajime Fujita and Mikio Furuta.
10月27日 -- 002号室, 16:30 -- 18:00
Alex Bene (IPMU)
A new appearance of the Morita-Penner cocycle
Abstract:
In this talk, I will recall the Morita-Penner cocycle on the dual fatgraph complex for a surface with one boundary component. This cocycle, when restricted to paths representing elements of the mapping class group, represents the extended first Johnson homomorphism \tau_1, thus can be viewed as a (in some specific sense canonical) "groupoid extension" of \tau_1. There are now several different contexts in which this cocycle can be constructed, and in this talk I will briefly review several of them, including one discovered in the context of finite type invariants of homology cylinders in joint work with J.E. Andersen, J-B. Meilhan, and R.C. Penner.
11月10日 -- 056号室, 16:30 -- 18:00
Alexander Getmanenko (IPMU)
Resurgent analysis of the Witten Laplacian in one dimension
Abstract:
I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.
11月17日 -- 056号室, 16:30 -- 18:00
高田 敏恵 (新潟大学)
On the SO(N) and Sp(N) free energy of a closed oriented 3-manifold
Abstract:
We give an explicit formula of the SO(N) and Sp(N) free energy
of a lens space and show that the genus g terms of it are analytic
in a neighborhood at zero, where we can choose the neighborhood
independently of g.
Moreover, it is proved that for any closed oriented 3-manifold M
and any g, the genus g terms of SO(N) and Sp(N) free energy
of M coincide up to sign.
11月24日 -- 056号室, 16:30 -- 18:00
Adam Clay (University of British Columbia)
A topological approach to left orderable groups
Abstract:
A group G is said to be left orderable if there is a strict
total ordering of its elements such that g < h implies fg < fh for all f, g, h in G. Left orderable groups have been useful in solving many problems in topology, and now we find that topology is returning the favour: the set of all left orderings of a group is denoted by LO(G), and it admits a natural topology, under which LO(G) becomes a compact topological space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases. For example, the space of left orderings of the braid group B_n for n > 2 contains isolated points (yet it is uncountable), while the space of left orderings of the fundamental group of the Klein bottle is finite.
Twice in recent years, the space of left orderings has been used very
successfully to solve difficult open problems from the field of left
orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the newest understanding of this connection, and highlight some potential
applications of further advances.
12月1日 -- 056号室, 16:30 -- 18:00
Andrei Pajitnov (Univ. de Nantes)
Non-Abelian Novikov homology
Abstract:
Classical construction of S.P. Novikov
associates to each circle-valued Morse map
a chain complex defined over a ring
of Laurent power series in one variable.
In this survey talk we shall explain several
results related to the construction and
properties of non-Abelian generalizations of the
Novikov complex.
GCOEレクチャーズ
12月8日 -- 002号室, 17:30 -- 19:00
Giovanni Felder (ETH Zurich)
Gaudin subalgebras and stable rational curves.
Abstract:
We show that Abelian subalgebras of maximal dimensions spanned by
generators of the n-th Kohno-Drinfeld Lie algebra are classified by
the Grothendieck-Knudsen moduli space of stable rational curves with
n+1 marked points. I will explain the relation with Gaudin
integrable systems of statistical mechanics and the representation
theory of the symmetric group in the formulation of Vershik and
Okounkov. The talk is based on joint work with Leonardo Aguirre and
Alexander Veselov.
12月15日 -- 056号室, 17:00 -- 18:00 (Lie群論・表現論セミナーと合同)
砂田 利一 (明治大学)
Open Problems in Discrete Geometric Analysis
Abstract:
Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups.
12月22日 -- 056号室, 16:30 -- 18:00
寺杣 友秀 (東京大学大学院数理科学研究科)
Relative DG-category, mixed elliptic motives and elliptic polylog
Abstract:
We consider a full subcategory of
mixed motives generated by an elliptic curve
over a field, which is called the category of
mixed elliptic motives. We introduce a DG
Hopf algebra such that the categroy of
mixed elliptic motives is equal to that of
comodules over it. For the construction, we
use the notion of relative DG-category with
respect to GL(2). As an application, we construct
an mixed elliptic motif associated to
the elliptic polylog. It is a joint work with
Kenichiro Kimura.
1月5日 -- 056号室, 16:30 -- 17:30
服部 広大 (東京大学大学院数理科学研究科)
$A_{\infty}$型超ケーラー多様体の体積増大
Abstract : $A_{\infty}$型超ケーラー多様体とは、Anderson-Kronheimer-LeBrun及びGotoによって構成されたホモロジー群が無限生成となる非コンパクトな多様体上の4次元完備超ケーラー計量である。ここでは、この計量の体積増大に注目する。リーマン多様体の体積増大とは、ある点を中心とする半径$r>0$の球の体積の漸近挙動である。4次元の超ケーラー計量では、体積増大が$r^4$になる例（ALE空間）と、$r^3$になる例（Taub-NUT空間）はよく知られている。本講演では、各$3
17:30 -- 18:30
松尾 信一郎 (東京大学大学院数理科学研究科)
On the Runge theorem for instantons
Abstract:
A classical theorem of Runge in complex analysis asserts that a
meromorphic function on a domain in the Riemann sphere can be
approximated, over compact subsets, by rational functions, that is,
meromorphic functions on the Riemann sphere.
This theorem can be paraphrased by saying that any solution of the
Cauchy-Riemann equations on a domain in the Riemann sphere can be
approximated, over compact subsets, by global solutions.
In this talk we will present an analogous result in which the
Cauchy-Riemann equations on Riemann surfaces are replaced by the
Yang-Mills instanton equations on oriented 4-manifolds.
We will also mention that the Runge theorem for instantons can be
applied to develop Yang-Mills gauge theory on open 4-manifolds.
1月12日 -- 056号室, 16:30 -- 17:30
篠原 克寿 (東京大学大学院数理科学研究科)
Index problem for generically-wild homoclinic classes in dimension three
Abstract : In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a $C^1$-generic viewpoint.
17:30 -- 18:30
二木 昌宏 (東京大学大学院数理科学研究科)
On a generalized suspension theorem for directed Fukaya categories
Abstract:
The directed Fukaya category $\mathrm{Fuk} W$ of exact Lefschetz
fibration $W : X \to \mathbb{C}$ proposed by Kontsevich is a
categorification of the Milnor lattice of $W$. This is defined as the
directed $A_\infty$-category $\mathrm{Fuk} W = \mathrm{Fuk}^\to
\mathbb{V}$ generated by a distinguished basis $\mathbb{V}$ of
vanishing cycles.
Recently Seidel has proved that this is stable under the suspension $W
+ u^2$ as a consequence of his foundational work on the directed
Fukaya category. We generalize his suspension theorem to the $W + u^d$
case by considering partial tensor product $\mathrm{Fuk} W \otimes'
\mathcal{A}_{d-1}$, where $\mathcal{A}_{d-1}$ is the category
corresponding to the $A_n$-type quiver. This also generalizes a recent
work by the author with Kazushi Ueda.
1月19日 -- 056号室, 17:00 -- 18:00
小林 亮一 (名古屋大学)
群作用による局所化とその代数的極小曲面の周期条件への応用
Abstract:
「代数的極小曲面のガウス写像の除外値は高々２か３か？」
という長年の未解決問題への新しいアプローチについてお話します。
完備な曲面に対する Cohn-Vossen 不等式の類似を、基本領域の
切れ端の集団に対して考えるのが鍵です。集団 Cohn-Vossen
不等式の定式化とその証明のアイディアを述べます。証明の根拠
となるのが表題にある放物型元による集中現象です。集団 Cohn-Vossen
不等式から「effective対数微分の補題」を得ます。代数的極小曲面の
周期条件を単位円盤上のある正則関数族のふるまいに封じ込めて、その
族に「effective対数微分の補題」を適用すると、何やら面白い評価式
を抽出できます。
1月26日 -- 056号室, 17:00 -- 18:00
栗林 勝彦 (信州大学)
On the (co)chain type levels of spaces
Abstract:
Avramov, Buchweitz, Iyengar and Miller have introduced
the notion of the level for an object of a triangulated category.
The invariant measures the number of steps to build the given object
out of some fixed object with triangles.
Using this notion in the derived category of modules over a (co)chain
algebra,
we define a new topological invariant, which is called
the (co)chain type level of a space.
In this talk, after explaining fundamental properties of the invariant,
I describe the chain type level of the Borel construction
of a homogeneous space as a computational example.
I will also relate the chain type level of a space to algebraic
approximations of the L.-S. category due to Kahl and to
the original L.-S. category of a map.
2月2日 -- 056号室, 16:30 -- 18:00 (Lie群論・表現論セミナーと合同)
Fanny Kassel (Univ. Paris-Sud, Orsay)
Deformation of compact quotients of homogeneous spaces
Abstract:
Let G/H be a reductive homogeneous space. In all known examples, if
G/H admits compact Clifford-Klein forms, then it admits "standard"
ones, by uniform lattices of some reductive subgroup L of G acting
properly on G/H. In order to obtain more generic Clifford-Klein forms,
we prove that for L of real rank 1, if one slightly deforms in G a
uniform lattice of L, then its action on G/H remains properly
discontinuous. As an application, we obtain compact quotients of
SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting
properly discontinuously.
http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100202kassel
2月16日 -- 056号室, 17:30 -- 18:30
Dieter Kotschick (Univ. Munchen)
Characteristic numbers of algebraic varieties
Abstract:
The Chern numbers of n-dimensional smooth projective varieties span a vector space whose dimension is the number of partitions of n. This vector space has many natural subspaces, some of which are quite small, for example the subspace spanned by Hirzebruch--Todd numbers, the subspace of topologically invariant combinations of Chern numbers, the subspace determined by the Hodge numbers, and the subspace of Chern numbers that can be bounded in terms of Betti numbers. I shall explain the relation between these subspaces, and characterize them in several ways. This leads in particular to the solution of a long- standing open problem originally formulated by Hirzebruch in the 1950s.
講演会
2月24日 (水) -- 370号室, 15:00 -- 16:30
Robert Penner (Aarhus University / University of Southern California）
Protein Moduli Space
Abstract:
Recent joint works with J. E. Andersen and others
provide explicit discrete and continuous models
of protein geometry. These models are inspired
by corresponding constructions in the study of moduli
spaces of flat G-connections on surfaces, in particular,
for G=PSL(2,R) and G=SO(3). These models can be used
for protein classification as well as for folding prediction,
and computer experiments towards these ends will
be discussed.