2023年10月 -- 2024年2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)
での対面開催と Zoom でのオンライン配信,
もしくは
17:00 -- 18:00 Zoom でのオンライン開催
Last updated March 22, 2024
世話係
河澄 響矢
北山 貴裕
逆井 卓也
葉廣 和夫
10月10日 -- 現地開催 (056号室) & オンライン中継, 17:30 -- 18:30
見村 万佐人 (東北大学)
不変擬準同型と scl の粗い幾何
Abstract: 10/9〜13 の集中講義では Green--Tao
の定理とその数体への一般化について話しますが、
本講演の内容はそれとは完全に独立しています。
川崎盛通氏（北海道大学）、木村満晃氏（京都大学）、
松下尚弘氏（信州大学）、丸山修平氏（金沢大学）
との一連の共同研究の話をします。群上の擬準同型（quasimorphism）
は双曲幾何などとの関係から大変興味深いものですが、
多くの面白い群に対し擬準同型全体のなすベクトル空間がつまらないか無限次元かの二択となってしまいます。
1 つの群ではなく群と正規部分群の組の設定で不変擬準同型を考えることで、
面白い例で非ゼロな有限次元ベクトル空間を取り出すことができることをお話しします。
Bavard の双対定理はこの枠組みに拡張され、
この結果は安定交換子長（scl）の粗い幾何（coarse geometry）
への応用ももちます。
一連の理論の発展をあまり予備知識を仮定せず概観したいと思います。
10月17日 -- オンライン開催, 17:00 -- 18:00
狩野 隼輔 (東北大学 数理科学共創社会センター)
Train track combinatorics and cluster algebras
Abstract: The concepts of train track was introduced by
W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface.
In this talk, we translate some concepts of train tracks into the
language of cluster algebras using the tropicalization of
Goncharov--Shen's potential function.
Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras,
called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.
10月24日 -- オンライン開催, 17:00 -- 18:00
林 晋 (青山学院大学)
Index theory for quarter-plane Toeplitz operators via extended symbols
Abstract: We consider index theory for some Toeplitz operators on a discrete quarter-plane.
Index theory for such operators has been investigated by Simonenko, Douglas-Howe,
Park and index formulas are obtained by Coburn-Douglas-Singer, Duducava.
In this talk, we revisit Duducava’s idea and discuss an index formula for quarter-plane Toeplitz operators of two-variable rational matrix function symbols from a geometric viewpoint.
By using Gohberg-Krein theory for matrix factorizations and analytic continuation,
we see that the symbols of Fredholm quarter-plane Toeplitz operators defined originally on a two-dimensional torus can canonically be extended to some three-sphere,
and show that their Fredholm indices coincides with the three-dimensional winding number of extended symbols.
If time permits, we briefly mention a contact with a topic in condensed matter physics,
called (higher-order) topological insulators.
10月31日 -- 現地開催 (056号室) & オンライン中継, 17:30 -- 18:30
千吉良 直紀 (熊本大学)
原田予想 II について
Abstract: 有限群の指標表は非常に多くの情報を含んでいる。本講演では、
原田耕一郎氏による既約指標の次数の積と共役類の元の個数の積に関する予想
（原田予想II）についてこれまでの概要と最近の進展について講演する。
11月7日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Florent Schaffhauser (Heidelberg University)
Hodge numbers of moduli spaces of principal bundles on curves
Abstract: The Poincaré series of moduli stacks of semistable G-bundles on curves has been computed by Laumon and Rapoport.
In this joint work with Melissa Liu,
we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way.
As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé,
on the maximality on moduli spaces of vector bundles over real algebraic curves.
11月14日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
横山 知郎 (埼玉大学)
曲面上の流れの正方向と負方向の極限の振る舞いの依存性
Abstract: 曲面上の流れの正方向と負方向の極限の振る舞いの依存性について話をする．
特に，正と負の極限集合の組み合わせで起こりえる場合と起こりえない場合を決定したので，そのリストを紹介する．
また，依存のメカニズムのアイデアをトイモデルである円周上の同相写像の極限的な振る舞いの依存性を使って説明する．
予備知識をできるだけ仮定せずに概観する．
11月21日 -- 現地開催 (056号室) & オンライン中継, 17:30 -- 18:30
古宇田 悠哉 (慶應義塾大学)
Shadows, divides and hyperbolic volumes
Abstract: In 2008, Costantino and D.Thurston revealed that
the combinatorial structure of the Stein factorizations of stable maps from 3-manifolds into the real plane can be used
to describe the hyperbolic structures of the complement of the set of definite fold points, which is a link.
The key was that the Stein factorizations can naturally be embedded into 4-manifolds,
and nice ideal polyhedral decompositions become visible on their boundaries.
In this talk, we consider divides, which are the images of a proper and generic immersions of compact 1-manifolds into the 2-disk.
Due to A'Campo's theory, each divide is associated with a link in the 3-sphere.
By embedding a polyhedron induced from a given divide into the 4-ball as was done to Stein factorization,
we can read off the ideal polyhedral decompositions on the boundary.
We then show that the complement of the link of the divide can be obtained by Dehn filling a hyperbolic 3-manifold that admits a decomposition into several ideal regular hyperbolic polyhedra,
where the number of each polyhedron is determined by types of the double points of the divide.
This immediately gives an upper bound of the hyperbolic volume of the links of divides,
which is shown to be asymptotically sharp.
As in the case of Stein factorizations, an idea from the theory of Turaev's shadows plays an important role here.
This talk is based on joint work with Ryoga Furutani (Hiroshima University).
11月28日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Gwénaël Massuyeau (Université de Bourgogne)
An analogue of the Johnson-Morita theory for the handlebody group
Abstract: The Johnson-Morita theory provides an approach for the mapping class group of a surface by considering its actions on the successive nilpotent quotients of the fundamental group of the surface.
In this talk, after an outline of the original theory, we will present an analogue of the Johnson-Morita theory for the handlebody group,
i.e. the mapping class group of a handlebody.
This is joint work with Kazuo Habiro;
as we shall explain if time allows, our motivation is to recover the "tree reduction"
of a certain functor on the category of bottom tangles in handlebodies that we introduced (a few years ago) using the Kontsevich integral.
12月5日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
北野 晃朗 (創価大学)
On the Euler class for flat S1-bundles, C∞ vs Cω
Abstract: We describe low dimensional homology groups of the real analytic,
orientation preserving diffeomorphism group of S1 in terms of
BΓ1 by applying a theorem of Thurston.
It is an open problem whether some power of the rational Euler class vanishes for real analytic flat S1 bundles.
In this talk we discuss that if it does, then the homology group should contain many torsion classes that vanish in the smooth case.
Along this line we can give a new proof for the non-triviality of any power of the rational Euler class in the smooth case.
If time permits, we will mention some attempts to study a Mather-Thurston map in the analytic case.
This talk is based on a joint work with Shigeyuki Morita and Yoshihiko Mitsumatsu.
12月12日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Stavros Garoufalidis (南方科技大学)
Multivariable knot polynomials from braided Hopf algebras with automorphisms
Abstract: We will discuss a unified approach to define
multivariable polynomial invariants of knots that include the colored Jones polynomials,
the ADO polynomials and the invariants defined using the theory of quantum groups.
Our construction uses braided Hopf algebras with automorphisms.
We will give examples of 2-variable invariants,
and discuss their structural properties. Joint work with Rinat Kashaev.
12月14日（木）開催日時にご注意下さい -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
枡田 幹也 (大阪公立大学)
Torus orbit closures in the flag variety
Abstract: The study of torus orbit closures in the flag variety
was initiated by Gelfand-Serganova and Klyachko in 1980’s
but has not been studied much since then.
Recently, I have studied its geometry and topology jointly with Eunjeong Lee,
Seonjeong Park, Jongbaek Song in connection with combinatorics of polytopes,
Coxeter matroids, and polygonal triangulations.
In this talk I will report on the development of this subject.
12月19日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
河東 泰之 (東京大学大学院数理科学研究科)
Topological quantum computing, tensor networks and operator algebras
Abstract: Modular tensor categories have caught much attention in connection to topological quantum computing based on anyons recently.
Condensed matter physicists recently try to understand structures of modular tensor categories appearing in two-dimensional topological order using tensor networks.
We present understanding of their tools in terms of operator algebras.
For example, 4-tensors they use are exactly bi-unitary connections in the Jones theory of subfactors and their sequence of finite dimensional Hilbert spaces on which their gapped Hamiltonians act is given by the so-called higher relative commutants of a subfactor.
No knowledge on operator algebras are assumed.
1月9日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:00
高野 暁弘 (東京大学大学院数理科学研究科)
Stabilizer subgroups of Thompson's group F in Thompson knot theory
Abstract: Thompson knot theory, introduced by Vaughan Jones, is a study of knot theory using Thompson's group F.
More specifically, he defined a method of constructing a knot from an element of F,
and proved that any knot can be realized in his way.
This fact is called Alexander’s theorem, which is an analogy of the braid group.
In this talk, we consider Thompson knot theory in terms of a relation between subgroups of F and knots obtained from their elements.
In particular, we focus on stabilizer subgroups of F with respect to the natural action on the unit interval.
This talk is based on joint work with Yuya Kodama (Tokyo Metropolitan University).
1月16日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:00
宮澤 仁 (東京大学大学院数理科学研究科)
４次元多様体に埋め込まれた曲面の不変量とエキゾチック
P2-knot
Abstract: ４次元多様体の曲面の埋め込みがふたつ与えられたとき、これらが位相的にはアイソトピックだが滑らかにはアイソトピックでないときこれらをエキゾチック曲面対ということにする。
４次元多様体の中のエキゾチック曲面対の存在問題には多くの先行研究があるが、
S4 の中の閉曲面によるエキゾチック曲面対の先行研究は少なく、特に向き付け可能な曲面によるものは現在でも知られていない。
向き付け不可能な曲面の埋め込みについては,
初めに与えられた 1988 年の Finashin--Kreck--Viro
の例をはじめいくつか知られた例があるが、
いずれも種数は５以上である。
S4 の中のエキゾチック曲面対の検出の困難さの一因は、
滑らかにはアイソトピックでないことを示す手法の少なさにある。
特に、S4
の向き付け不可能な曲面のエキゾチック曲面対はすべて、
二重分岐被覆で得られる４次元多様体のエキゾチック性に帰着して示されている。
この手法を種数の小さい向き付け不可能曲面に適用するには「小さい４次元多様体」でエキゾチックなものを見つけねばならず、
これは困難であることが知られている。
本講演では、４次元多様体に埋め込まれた曲面の不変量を Real Seiberg--Witten 理論を用いて構成し、応用として、
実射影平面のS4
へのエキゾチックな埋め込みの無限族を与える。
1月23日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:00
王 格非 (東京大学大学院数理科学研究科)
On the rational cohomology of spin hyperelliptic mapping class groups
Abstract: Let G be the subgroup Sn−q × Sq
of the n-th symmetric group Sn for n-q ≥ q.
In this talk, we study the G-invariant part of the rational cohomology group of the pure braid group Pn.
The invariant part H*(Pn)G
includes the rational cohomology of a spin hyperelliptic mapping class group of genus g as a subalgebra when n=2g+2.
Based on the study of Lehrer-Solomon,
we prove that they are independent of n and q in degree * ≤ q-1.
We also give a formula to calculate the dimension of
H*(Pn)G and calculate it in all degree for q ≤ 3.
2月13日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Paul Norbury (The University of Melbourne)
Measures on the moduli space of curves and super volumes
Abstract: In this lecture I will define a family of finite measures on the moduli space of smooth curves with marked points.
The measures are defined via a construction analogous to that of the Weil-Petersson metric using the extra data of a spin structure.
In fact, the measures arise naturally out of the super Weil-Petersson metric defined over the moduli space of super curves.
The total measure can be identified with the volume of the moduli space of super curves.
It can be calculated in many examples,
and conjecturally satisfies a recursion analogous to Mirzakhani's recursion relations between Weil-Petersson volumes of moduli spaces of hyperbolic surfaces.
This conjecture has been verified in many cases,
including the so-called Neveu-Schwarz case where it coincides with the recursion of Stanford and Witten. The general case produces deformations of the Neveu-Schwarz volume polynomials,
satisfying the same Mirzakhani-like recursion relations.