2015年4月 -- 7月
[English]
[過去のプログラム]
17:00 -- 18:30 数理科学研究科棟(東京大学駒場キャンパス)
Tea: 16:30 -- 17:00 コモンルーム
Last updated June 22, 2015
世話係
河野 俊丈
河澄 響矢
逆井 卓也
4月7日 -- 056号室, 17:00 -- 18:30
植田 一石 (東京大学大学院数理科学研究科)
Potential functions for Grassmannians
Abstract:
Potential functions are Floer-theoretic invariants
obtained by counting Maslov index 2 disks
with Lagrangian boundary conditions.
In the talk, we will discuss our joint work
with Yanki Lekili and Yuichi Nohara
on Lagrangian torus fibrations on the Grassmannian
of 2-planes in an n-space,
the potential functions of their Lagrangian torus fibers,
and their relation with mirror symmetry for Grassmannians.
4月14日 -- 056号室, 17:00 -- 18:30
中村 信裕 (学習院大学)
Pin(2)-monopole invariants for 4-manifolds
Abstract:
The Pin(2)-monopole equations are a variant of the Seiberg-Witten equations
which can be considered as a real version of the SW equations. A Pin(2)-mono
pole version of the Seiberg-Witten invariants is defined, and a special feature of
this is that the Pin(2)-monopole invariant can be nontrivial even when all of
the Donaldson and Seiberg-Witten invariants vanish. As an application, we
construct a new series of exotic 4-manifolds.
4月21日 -- 056号室, 17:00 -- 18:30
木田 良才 (東京大学大学院数理科学研究科)
Orbit equivalence relations arising from Baumslag-Solitar groups
Abstract:
This talk is about measure-preserving actions of countable groups on probability
measure spaces and their orbit structure. Two such actions are called orbit equivalent
if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus
on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation
ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial
and geometric group theory. Whether Baumslag-Solitar groups with different p and q can
have orbit-equivalent actions is still a big open problem. I will discuss invariants under
orbit equivalence, motivating background and some results toward this problem.
4月28日 -- 056号室, 17:00 -- 18:30
正井 秀俊 (東京大学大学院数理科学研究科, JSPS)
Verify hyperbolicity of 3-manifolds by computer and its applications.
Abstract:
In this talk I will talk about the program called HIKMOT which
rigorously proves hyperbolicity of a given triangulated 3-manifold. To
prove hyperbolicity of a given triangulated 3-manifold, it suffices to
get a solution of Thurston's gluing equation. We use the notion called
interval arithmetic to overcome two types errors; round-off errors,
and truncated errors. I will also talk about its application to
exceptional surgeries along alternating knots. This talk is based on
joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and
A. Takayasu.
5月7日 -- 056号室, 17:00 -- 18:30
Patrick Dehornoy (Université de Caen)
The group of parenthesized braids
Abstract:
We describe a group B obtained by gluing in a natural way two well-known
groups, namely Artin's braid group B_infty and Thompson's group F. The
elements of B correspond to braid diagrams in which the distances
between the strands are non uniform and some rescaling operators may
change these distances. The group B shares many properties with B_infty:
as the latter, it can be realized as a subgroup of a mapping class
group, namely that of a sphere with a Cantor set removed, and as a group
of automorphisms of a free group. Technically, the key point is the
existence of a self-distributive operation on B.
このセミナーは木曜日の開催です。
5月12日 -- 056号室, 17:30 -- 18:30
浅岡 正幸 (京都大学)
genericな力学系の周期点の個数の増大度
Abstract:
双曲力学系と呼ばれる統計的によい振る舞いをする力学系に関しては，
その周期軌道の数の増大度は常に高々指数的で，増大度は系の統計的
性質と密接に関係することが知られている．一方で1999年にKaloshin
により，homoclinic接触と呼ばれる複雑な分岐現象が稠密に起きるよ
うな領域においてはgenericな力学系はその周期軌道の数の増大度は
指数的よりも速くなることが証明されている．
では，弱い双曲性を持ち，homoclinic接触からは離れている「部分双
曲系」と呼ばれる系において周期点の数の増大度がどう振る舞うだ
ろうか．双曲力学系と同様に高々指数的になるだろうか，それとも，
homoclinic 接触とは異なるメカニズムによって，指数的よりも速く
なるだろうか？
講演者は，篠原克寿氏とDimitry Turaev氏との共同研究によって，
部分双曲系のダイナミクスのある種の単純化である「区間上の反復
函数系」において，ある自然な条件の元でその周期軌道の数がgeneric
には指数的よりも速く増大することを証明した．本講演では，力学
系の周期軌道の増大度の問題の歴史の概観した後，指数的よりも速
い増大度を引き起こすメカニズムについて，Kaloshinが見つけた
homoclinic接触によるものと講演者たちが見つけたものを対比しつつ解説したい．
5月19日 -- 056号室, 17:00 -- 18:30
加藤 晃史 (東京大学大学院数理科学研究科)
Quiver mutation loops and partition q-series
Abstract:
Quivers and their mutations are ubiquitous in mathematics and
mathematical physics; they play a key role in cluster algebras,
wall-crossing phenomena, gluing of ideal tetrahedra, etc.
Recently, we introduced a partition q-series for a quiver mutation loop
(a loop in a quiver exchange graph) using the idea of state sum of statistical
mechanics. The partition q-series enjoy some nice properties such
as pentagon move invariance. We also discuss their relation with combinatorial
Donaldson-Thomas invariants, as well as fermionic character formulas of
certain conformal field theories.
This is a joint work with Yuji Terashima.
5月26日 -- 056号室, 17:00 -- 18:30
久我 健一 (千葉大学)
Introduction to formalization of topology using a proof assistant.
Abstract:
Although the program of formalization goes back to David
Hilbert, it is only recently that we can actually formalize
substantial theorems in modern mathematics. It is made possible by the
development of certain type theory and a computer software called a
proof assistant. We begin this talk by showing our formalization of
some basic geometric topology using a proof assistant COQ. Then we
introduce homotopy type theory (HoTT) of Voevodsky et al., which
interprets type theory from abstract homotopy theoretic perspective.
HoTT proposes "univalent" foundation of mathematics which is
particularly suited for computer formalization.
6月9日 -- 056号室, 17:00 -- 18:30
赤穂 まなぶ (首都大学東京)
完全ラグランジュはめ込みのシンプレクティックdisplacementエネルギーについ
て
Abstract:
この講演では完全ラグランジュはめ込みのdisplacementエネルギーと擬正則円盤
のシンプレクティック面積に関するある不等式を与える. 証明はChekanovが有理
ラグランジュ部分多様体のdisplacementエネルギーに関する不等式を示す際に用
いた技法を, ラグランジュはめ込みのFloerホモロジーに拡張して行う. また時
間が許せば, 我々の不等式とHofer--Zehnderのシンプレクティック容量に関する
考察を述べる.
6月16日 -- 056号室, 17:00 -- 18:30
石川 昌治 (東北大学)
Stable maps and branched shadows of 3-manifolds
Abstract:
We study what kind of stable map to the real plane a 3-manifold has. It
is known by O. Saeki that there exists a stable map without certain
singular fibers if and only if the 3-manifold is a graph manifold. According to
F. Costantino and D. Thurston, we identify the Stein factorization of a
stable map with a shadow of the 3-manifold under some modification,
where the above singular fibers correspond to the vertices of the shadow. We
define the notion of stable map complexity by counting the number of
such singular fibers and prove that this equals the branched shadow
complexity. With this equality, we give an estimation of the Gromov norm of the
3-manifold by the stable map complexity. This is a joint work with Yuya Koda.
6月23日 -- 056号室, 17:00 -- 18:30
松下 尚弘 (東京大学大学院数理科学研究科)
Box complexes and model structures on the category of graphs
Abstract:
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.
Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.
In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.
6月30日 -- 056号室, 17:30 -- 18:30
作間 誠 (広島大学)
The Cannon-Thurston maps and the canonical decompositions of
punctured surface bundles over the circle
Abstract:
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.
7月7日 -- 056号室, 17:00 -- 18:30
北山 貴裕 (東京工業大学)
Representation varieties detect essential surfaces
Abstract:
Extending Culler-Shalen theory, Hara and I presented a way to construct
certain kinds of branched surfaces (possibly without any branch) in a 3-
manifold from an ideal point of a curve in the SL_n-character variety.
There exists an essential surface in some 3-manifold known to be not
detected in the classical SL_2-theory. We show that every essential
surface in a 3-manifold is given by the ideal point of a line in the SL_
n-character variety for some n. The talk is partially based on joint
works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.
7月14日 -- 056号室, 17:00 -- 18:30
Carlos Moraga Ferrándiz (東京大学大学院数理科学研究科, 日本学術振興会)
How homoclinic orbits explain some algebraic relations holding in Novikov rings.
Abstract:
Given u, a de-Rham cohomology class of degree 1 of a closed manifold M, we consider the space Fu of (closed) Morse 1-forms in this class. In Morse theory, it is important to equip each α in Fu with a descending pseudo-gradient X. The case u=0 yields usual Morse theory, while u ≠ 0 yields Morse-Novikov theory, which is devoted to the understanding of the space of equipped 1-forms (α,X) with α in Fu .
Here, X is a descending pseudo-gradient, which is said to be adapted to α.
The morphism π1(M) → R induced by u (given by the integral of any α in Fu over a loop of M) determines a set of u-negative loops.
We show that for every u-negative g in π1(M), there exists a co-dimension 1 C∞-stratum Sg of Fu which is naturally co-oriented. The stratum Sg is made of elements (α, X) such that X has exactly one homoclinic orbit L whose homotopy class is g.
The goal of this talk is to show that there exists a co-dimension 1 C∞-stratum Sg (0) of Sg which lies in the closure of Sg2. This result explains geometrically an easy algebraic relation holding in the Novikov ring associated with u.
We will mention how this study generalizes to produce some non-evident symmetric formulas holding in the Novikov ring.
7月21日 -- 056号室, 17:00 -- 18:30
田神 慶士 (東京工業大学)
Ribbon concordance and 0-surgeries along knots
Abstract:
Akbulut and Kirby conjectured that two knots with
the same 0-surgery are concordant. Recently, Yasui
gave a counterexample of this conjecture.
In this talk, we introduce a technique to construct
non-ribbon concordant knots with the same 0-surgery.
Moreover, we give a potential counterexample of the
slice-ribbon conjecture. This is a joint work with
Tetsuya Abe (Osaka City University, OCAMI).