2019年10月 -- 2020年1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 March 13, 2020
世話係
河野 俊丈
河澄 響矢
北山 貴裕
逆井 卓也
10月1日 -- 056号室, 17:00 -- 18:30
村上 順 (早稲田大学)
Quantized SL(2) representations of knot groups
Abstract: Let K be a knot and G be a group.
The representation space of K for the group G means the space of homomorphisms from the knot group to G
and is defined by using the group ring C[G],
where C[G] is the ring of functions on G and has a commutative Hopf algebra structure.
This construction can be generalized to any commutative Hopf algebras.
In this talk, we extend this construction to any braided Hopf algebras with braided commutativity.
A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid.
Applying the above construction to BSL(2), we get the space of BSL(2) representations,
which provides a quantization of SL(2) representations of a knot.
This is joint work with Roloand van der Veen.
10月8日 -- 056号室, 17:30 -- 18:30
塚本 真輝 (九州大学)
いかにして双曲的力学系を群作用に拡張するか？
Abstract: 双曲性は通常の力学系（1パラメータ群作用の研究）において最も基本的な重要性を持つ概念です．
それは，十分な豊かさ（拡大性や正エントロピー）を持ちながらも，
同時に制御可能（安定性や適切な意味での良い測度の一意性）な力学系の例を与えます．
ではこれを群作用に拡張できるでしょうか？
ナイーブには困難です．例えば Z2 の作用を考えましょう（つまり可換な2パラメータ作用）・
簡単にわかるのは，有限次元のコンパクト多様体に Z2 が可微分に作用するとき，
その Z2 作用としてのエントロピーはゼロになります．
つまり，通常の有限次元の状況には，豊かな Z2 作用は存在しません．
言い換えると，十分に豊かな群作用を得るためには無限次元の世界に行かざるを得ません．
しかし，無限次元の世界でどのような構造を見出せばよいのでしょうか？
この講演では，このような方向性にアプローチする際に，
平均次元と呼ばれる量が大きな役割を果たす可能性を説明します．
特に，次のような原理についてお話します：
Zk（可換な k パラメータ群）が空間 X に何らかの「双曲性」を持って作用するとき，
Zk のランク k-1 の部分群 G の部分作用に対する平均次元が制御できる．
この講演はTom Meyerovitch，篠田万穂との共同研究に基づきます．
10月15日 -- 056号室, 17:00 -- 18:30
Gwénaël Massuyeau (Université de Bourgogne)
Generalized Dehn twists on surfaces and surgeries in 3-manifolds
Abstract: (Joint work with Yusuke Kuno.)
Given an oriented surface S and a simple closed curve C in S,
the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist.
If the curve C is not simple, this transformation of S does not make sense anymore,
but one can consider two possible generalizations:
one possibility is to use the homotopy intersection form of S to
"simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S;
another possibility is to view C as a curve on the top boundary of the cylinder S×[0,1],
to push it arbitrarily into the interior so as to obtain,
by surgery along the resulting knot, a new 3-manifold.
In this talk, we will relate two those possible generalizations of a Dehn twist
and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.
10月29日 -- 056号室, 17:00 -- 18:30
Chung-Jun Tsai (National Taiwan University)
Strong stability of minimal submanifolds
Abstract: It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function.
One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario.
Besides a strong local uniqueness property,
a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow.
We will also discuss some famous local (complete, non-compact) models.
This is based on a joint work with Mu-Tao Wang.
11月5日 -- 056号室, 17:00 -- 18:30
五味 清紀 (東京工業大学)
Magnitude homology of geodesic space
Abstract: Magnitude is an invariant which counts `effective number of points' on a metric space.
Its categorification is magnitude homology.
This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton,
and then for any metric spaces by Leinster and Shulman.
The definition of the magnitude homology is easy, but its calculation is rather difficult.
For example, the magnitude homology of the circle with geodesic metric was known partially.
In my talk,
I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption.
In this result, the magnitude homology is described in terms of geodesics.
Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.
11月19日 -- 056号室, 17:00 -- 18:30
Ramón Barral Lijó (立命館大学)
The smooth Gromov space and the realization problem
Abstract: The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds.
In this talk we will present the definition and basic properties of this space as well as two different applications:
The first addresses the following classical problem in foliation theory.
Realization problem: what kind of manifolds can be leaves of compact foliations?
Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.
Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor.
Moreover, we can assume that X has trivial holonomy and is transversely Cantor.
The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.
11月26日 -- 056号室, 17:00 -- 18:30
Marco De Renzi (早稲田大学)
2+1-TQFTs from non-semisimple modular categories
Abstract: Non-semisimple constructions have substantially generalized the standard approach of Witten,
Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties.
We will explain how to use the theory of modified traces
to renormalize Lyubashenko’s closed 3-manifold invariants coming from
finite twist non-degenerate unimodular ribbon categories.
Under the additional assumption of factorizability, our renormalized invariants extend to 2+1-TQFTs,
unlike Lyubashenko’s original ones.
This general framework encompasses important examples of non-semisimple modular categories
which were left out of previous non-semisimple TQFT constructions.
Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.
12月3日 -- 056号室, 17:00 -- 18:30
Anton Zeitlin (Louisiana State University)
Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations
Abstract: I will talk about the underlying homotopical structures within field equations,
which emerge in string theory as conformal invariance conditions for sigma models.
I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids,
thus making all such equations the natural objects within vertex algebra theory.
12月10日 -- 056号室, 17:00 -- 18:30
小木曽 岳義 (城西大学)
q-Deformation of a continued fraction and its applications
Abstract: Morier-Genoud と Ovsienko によって連分数のある種の q-変形が導入された。
このq-変形の最大の応用はそれを用いて向きづけられた有理絡み目の
Jones 多項式がそれから直接求めることができることである。
またこの連分数のq-変形は結び目理論への応用以外にも、
2次無理数論、組み合わせ論への応用もあり、それについても紹介する。
一方、Lee-Schiffler の snake graph を用いる方法や Kogiso-Wakui による
Conway-Coxeter frieze を持ちいる方法で Jones 多項式を計算するレシピが与えられている。
そのことから、Morier-Genoud and Ovsienko の結果のそれらの観点からの別証明が考えられるが、
それについて紹介し、さらに,
Kogiso-Wakui の研究で用いた Ancestoral triangles の観点から連分数のq-変形をさらに一般化でき、
連分数の cluster-variable 変形が出来ることを紹介する。
12月17日 -- 056号室, 17:00 -- 18:30
入江 慶 (東京大学大学院数理科学研究科)
Symplectic homology of fiberwise convex sets and homology of loop spaces
Abstract: シンプレクティック・ベクトル空間の（コンパクト）部分集合に対して、
シンプレクティック・ホモロジー（Floer ホモロジーの一種）
を用いてそのシンプレクティック容量（capacity）を定義することができる。
一般に、Floerホモロジーの定義には非線形偏微分方程式（いわゆるFloer方程式）
の解の数え上げが関わるため、容量を定義から直接計算したり評価したりするのは難しい。
この講演では（シンプレクティック・ベクトル空間をEuclid空間の余接空間とみなしたとき）
fiberwiseに凸な集合のシンプレクティック・ホモロジーおよび容量をループ空間のホモロジーから計算する公式を示し、
その応用を二つ与える。
1月7日 -- 056号室
17:00 -- 18:00
浅尾 泰彦 (東京大学大学院数理科学研究科)
Magnitude homology of crushable spaces
Abstract: The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al.
They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of
"persistence of points clouds".
However, little property of them has been revealed.
In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy.
We use techniques from singular homology theory.
18:00 -- 19:00
浅野 知紘 (東京大学大学院数理科学研究科)
Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory
Abstract: The microlocal sheaf theory due to Kashiwara-Schapira has been applied to symplectic geometry for these ten years.
Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds.
In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory.
This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory.
This is joint work with Yuichi Ike.
1月14日 -- 056号室
17:00 -- 18:00
茅原 涼平 (東京大学大学院数理科学研究科)
SO(3)-invariant G2-geometry
Abstract: Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G2 and Spin(7).
Many authors have studied G2- and Spin(7)-manifolds with torus symmetry.
In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G2-manifolds with SO(3)-symmetry.
Such torsion-free G2-structures are described as a dynamical system of SU(3)-structures on an SO(3)-fibration over a 3-manifold.
As a main result, we reduce this system into a constrained Hamiltonian dynamical system on the cotangent bundle over the space of all Riemannian metrics on the 3-manifold.
The Hamiltonian function is very similar to that of the Hamiltonian formulation of general relativity.
18:00 -- 19:00
石橋 典 (東京大学大学院数理科学研究科)
Algebraic entropy of sign-stable mutation loops
Abstract: Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics,
unifying several combinatorial operations as well as their positivity notions.
A mutation loop induces several dynamical systems via cluster transformations,
and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.
We introduce a new property of mutation loops called the sign stability,
with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation.
A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor",
in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface.
We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop,
and conclude that these two coincide with the logarithm of the cluster stretch factor.
This talk is based on a joint work with Shunsuke Kano.
1月28日 -- 056号室
17:00 -- 18:00
関野 希望 (東京大学大学院数理科学研究科)
Existence problems for fibered links
Abstract: It is known that every connected orientable closed 3-manifold has a fibered knot. However,
finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general.
We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram.
As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.
There is one generalization of fibered links, homologically fibered links.
This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval.
We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.
18:00 -- 19:00
渡部 淳 (東京大学大学院数理科学研究科)
Fibred cusp b-pseudodifferential operators and its applications
Abstract: Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities.
In this talk, we introduce a new variant "fibred cusp b-calculus",
which isa generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose.
We discuss the basic property of this calculus and give a relative index formula.
As its application, we prove the index theorem for a Z/k manifold with boundary,
which is a generalization of the mod k index theorem of Freed-Melrose.