[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 December 20, 2024
世話係
河澄 響矢
北山 貴裕
逆井 卓也
葉廣 和夫
2024 年度の冬学期も
「対面開催 & オンライン中継形式」(90 分 or 60 分) と「完全オンライン形式」(60 分)
を併用してセミナーを行います.
各回の開催の形式については本ページにてご案内いたします.
参加にあたっては, 事前登録が必要です.
こちら
のページにおいて必要事項をすべてご記入の上, 登録を行って下さい.
参加者の把握のため, 対面参加の場合であっても
事前登録が無い方の参加はお断りさせていただきます.
Zoom ミーティングの開始はセミナー開始時刻 15 分ほど前の予定です.
それまでは「待機室」にてお待ち下さい.
講演の録画・録音は禁止といたします.
セミナー中にアクセスが遮断された場合など, 緊急時はこのページに代替の接続アドレスを掲示します.
10月8日 開催場所にご注意下さい -- 現地開催 (123号室) & オンライン中継, 17:00 -- 18:30
今野 北斗 (東京大学大学院数理科学研究科)
Dehn twists on 4-manifolds
Abstract: Dehn twists on surfaces form a basic class of diffeomorphisms.
On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds.
In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology,
and occasionally from the viewpoint of symplectic geometry as well.
The proof involves gauge theory for families.
This talk includes joint work with Abhishek Mallick and Masaki Taniguchi,
as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.
10月17日（木）開催日時と場所にご注意下さい -- 現地開催 (123号室) & オンライン中継, 17:00 -- 18:30
榎園 誠 (東京大学大学院数理科学研究科)
Slope inequalities for fibered complex surfaces
Abstract: Slope inequalities of fibered surfaces are important in relation to the classification of algebraic surfaces and the complex structure of Lefschetz fibrations in four-dimensional topology.
It is also known that many slope inequalities for semi-stable fibered surfaces can be derived from the intersection theory on the moduli space of stable curves.
In this talk, after outlining the background of these studies,
I will explain how various slope inequalities can be obtained for fibered surfaces that are not necessarily semi-stable by extending the discussion of the moduli space.
10月22日 -- オンライン開催, 17:00 -- 18:00
桑垣 樹 (京都大学)
On the generic existence of WKB spectral networks/Stokes graphs
Abstract: リーマン面上の二次微分の定める軌道（葉層）は、古典的な研究対象である。
特に、零点を通る軌道はWKB解析、タイヒミュラー理論、場の量子論などの関係から興味を持たれてきた。
WKBスペクトルネットワーク（もしくはストークスグラフ）とは、その高階微分版である。
この講演では、WKBスペクトルネットワークの存在証明について説明する。
時間があれば、ラグランジュ交差フレアー理論との関係についても説明する。
10月29日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
内藤 貴仁 (日本工業大学)
Cartan calculus in string topology
Abstract: The homology of the free loop space of a closed oriented manifold
(called the loop homology) has rich algebraic structures.
In the theory of string topology due to Chas and Sullivan,
it is well known that the loop homology has a structure of Gerstenhabar algebras with a multiplication called the loop product and a Lie bracket called the loop bracket.
On the other hand, Kuribayashi, Wakatsuki, Yamaguchi and the speaker gave a Cartan calculus on the loop homology,
which is a geometric description of a homotopy Cartan calculus in the sense of Fiorenza and Kowalzig on the Hochschild homology.
In this talk, we will investigate a relationship between the string topology operations and the Cartan calculus.
Especially, we will show that the Cartan calculus can be described by using the loop product and the loop bracket with rational coefficients.
As an application, the nilpotency of some loop homology classes are determined.
11月5日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
村上 順 (早稲田大学)
ダブルツイスト結び目に対応する複素化された四面体とその体積予想への応用
Abstract: まずダブルツイスト結び目の補空間を分割する複素化された四面体について紹介し，
その後で体積予想への応用について説明する．
複素化された四面体は，ダブルツイスト結び目の補空間の基本群から構成される．
これはボロミアン環の補空間の半分をなす理想正８面体の変形であり，
ここでの変形はボロミアン環の手術によるダブルツイスト結び目の構成に対応している．
一方で，ダブルツイスト結び目のカラードジョーンズ多項式は量子 6j 記号に少し項を加えたもので表わすことができる．
Neumann-Zagier 理論により量子 6j 記号と複素化された四面体の体積とが対応することがわかるので，
これをダブルツイスト結び目の体積予想の証明に応用する．
そのために，カラードジョーンズ多項式の代わりに ADO 不変量を用いて，ロピタルの定理によりこれを求める．
また，部分積分を用いることで，big cancellation problem と呼んでいる困難さを解消する．
最後に，このケースでは鞍点法が比較的容易に適用できることを紹介する．
11月12日 [Lie群論・表現論セミナーと合同]
-- 現地開催 (056号室) & オンライン中継, 17:30 -- 18:30
井上 順子 (鳥取大学)
Holomorphically induced representations of some solvable Lie groups
Abstract: From a viewpoint of the orbit method,
holomorphic induction is originally based on the idea of realizing an irreducible unitary
representation of a Lie group
G in an L2-space of some holomorphic sections of some line bundle over a
G-homogeneous space associated with a polarization for a linear form of the Lie algebra of G.
It is a generalization of ordinary induction from a unitary character;
Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1,
connected, simply connected solvable Lie groups.
In this talk, focusing on the class of exponential solvable Lie groups,
we are concerned with holomorphically induced representations ρ in some general settings.
We would like to discuss the following problems:
(1) conditions of non-vanishing of ρ,
(2) decomposition of ρ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.
11月19日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Bruno Scárdua (Federal University of Rio de Janeiro)
On real center singularities of complex vector fields on surfaces
Abstract: One of the various versions of the classical Lyapunov-Poincaré
center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral.
In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations
([2]).
In this paper we consider generalizations for two main frameworks:
(i) planar real analytic vector fields with "many" periodic orbits near the singularity and
(ii) germs of holomorphic foliations having a suitable singularity in dimension two.
In this talk we discuss some versions of Poincaré-Lyapunov center theorem,
including for the case of holomorphic vector fields.
We also give some applications, hinting that there is much more to be explored in this framework.
References
[1] V. León, B. Scárdua, On a Theorem of Lyapunov-Poincaré in Higher Dimensions, July 2021, Arnold Mathematical Journal 7(3) DOI:10.1007/s40598-021-00183-x.
[2] R. Moussu: Une démonstration géométrique d’un théorème de Lyapunov-Poincaré. Astérisque, tome 98-99 (1982), p. 216-223.
[3] A. Lyapunov: Etude d’un cas particulier du problème de la stabilité du mouvement. Mat. Sbornik 17 (1893) pages 252-333 (Russe).
[4] H. Poincaré: Mémoire sur les courbes définies par une équation différentielle (I), Journal de mathématiques pures et appliquées 3e série, tome 7 (1881), p. 375-422.
[5] Minoru Urabe and Yasutaka Sibuya; On Center of Higher Dimensions; Journal of Science of the Hiroshima University, Ser. A, . Vol. 19, No. I, July, 1955.
11月26日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
名取 雅生 (東京大学大学院数理科学研究科)
A proof of Bott periodicity via Quot schemes and bulk-edge correspondence
Abstract: The bulk-edge correspondence refers to the phenomenon typically found in topological insulators,
where the topological restriction of the bulk (interior) determines the physical state, such as electric currents, at the edge (boundary).
In this talk, we focus on the formulation by G. M. Graf and M. Porta and later by S. Hayashi and provide a new proof of bulk-edge correspondence.
It is more direct compared to previous approaches. Behind the proof lies the Bott periodicity of K-theory.
The proof of Bott periodicity has been approached from various perspectives.
We provide a new proof of Bott periodicity. In the proof, we use Quot schemes in algebraic geometry as configuration spaces.
12月3日 -- 現地開催 (056号室) & オンライン中継, 17:30 -- 18:30
井ノ口 順一 (北海道大学)
３次元空間内の曲面と可積分系
Abstract: 3次元双曲空間の平均曲率一定曲面は平均曲率の値により様相が異なる．
とくに平均曲率の値が１未満の場合はユークリッド空間や球面に類似物をもたない双曲幾何特有のクラスを与える．
本講演では平均曲率の値が１未満の平均曲率一定曲面の可積分系理論的構成法について解説する
(Josef F. Dorfmeister氏, 小林真平氏との共同研究)．
12月10日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
若月 駿 (名古屋大学)
Computation of the magnitude homology as a derived functor
Abstract: Asao-Ivanov showed that the magnitude homology of a finite metric space is isomorphic to the derived functor Tor over some ring.
In this talk, I will explain an application of the theory of minimal projective resolution to this derived functor.
Especially in the case of a geodetic graph, torsion-freeness and a criterion for diagonality of the magnitude homology are established.
Moreover, I will give computational examples including cyclic graphs. This is a joint work with Yasuhiko Asao.
12月17日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:30
Emmanuel Graff (東京大学大学院数理科学研究科)
Is there torsion in the homotopy braid group?
Abstract: In the 'Kourovka notebook,' V. Lin questions the existence of a non-trivial epimorphism from the braid group onto a non-abelian torsion-free group.
The homotopy braid group, studied by Goldsmith in 1974,
naturally appears as a potential candidate.
In 2001, Humphries showed that this homotopy braid group is torsion-free for less than six strands.
In this presentation, we will see a new approach based on the broader concept of welded braids,
along with algebraic techniques,
to determine whether the homotopy braid group provides a complete answer to Lin’s question.
1月14日 -- 現地開催 (056号室) & オンライン中継, 17:00 -- 18:00
吉岡 玲音 (東京大学大学院数理科学研究科)
Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop graphs
Abstract: In this talk, we give some non-trivial cocycles and cycles of the space of long embeddings R^j --> R^n modulo immersions.
First, we construct a cocycle through configuration space integrals with the simplest
2-loop graph cocycle of the Bott-Cattaneo-Rossi graph complex for odd n and j.
Then, we give a cycle from a chord diagram on oriented lines,
which is associated with the simplest 2-loop hairy graph.
We show the non-triviality of this (co)cycle by pairing argument,
which is reduced to pairing of graphs with the chord diagram.
This construction of cycles and the pairing argument to show the non-triviality is also applied to general
2-loop (co)cycles of top degree. If time permits,
we introduce a modified graph complex and configuration space integrals to give more general cocycles.
This new graph complex is quasi-isomorphic to both the hairy graph complex and the graph complex introduced in embedding calculus by Arone and Turchin. With these modified cocycles,
our pairing argument provides an alternative proof of the non-finite generation of the
(j-1)-th rational homotopy group of the space of long j-knots R^j -->R^{j+2},
which Budney-Gabai and Watanabe first established.