Yuichi YAMADA (山田 裕一)

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3100000, indirect: 930000)
デーン手術によって双曲的な結び目から“例外的に”双曲的でない３次元多様体が生じる現象は「例外的手術」と呼ばれる低次元多様体論の１つの課題である。筆者はこの現象に関連して特殊な４次元多様体を構成・分析することを研究目標としている。研究開始の令和３年度はコロナ禍がより深刻にかつ長引き、ほぼすべての研究集会がオンライン開催となった。本務先では数学部会長として採用人事に取り組み、３年間続いた減員状態をようやく脱した。これらにより研究の遂行には辛い１年間であったが、いくつかの研究活動を行うことができた。以下、それらを具体的に述べる：1. レンズ空間を生じる結び目のディバイド曲線表示のうち最後まで残っていた課題（VIII型と呼ばれる結び目族の具体的表示）について、計算機を利用した実験で、当初推測していた形状は正しくないことが判明した。この成果を研究集会「４次元トポロジー」および国際研究集会「The 17th East Asian Conference of Geometric Topology」で講演した。2. 丹下氏（筑波大）と安部氏（立命館大）が主催したオンライン研究集会「微分トポロジー22」のテーマは「デーン手術」であった。筆者は最終講演の機会を与えられ、VII型，VIII型のレンズ空間手術に関連する研究を、特に４次元多様体からの興味に主眼をおいて、自分の過去の成果を軸にしつつ最新の研究動向についても勉強して、講演した。
筆者は元々自宅より研究室で研究する様式で、在宅勤務の増えた現在の研究活動に慣れないが、これからはコロナ禍を乗り越える新しい研究生活様式を模索する必要があると考えて努力した。上記の1.は研究の進展としては新たな課題の発見である。２成分絡み目の例外的デーン手術の分布に関する執筆準備中の論文もある。これらの課題を中心に本研究課題に取り組みたい。

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 2800000, indirect: 840000)
For complex manifolds X and Y (resp. real algebraic varieties X and Y), let Hol(X,Y) (resp. Alg(X,Y)) denote the space of all holomorphic maps (resp. regular maps) from X to Y. When we denote by Map(X,Y) the space of continuous maps from X to Y, we consider what dimension the finite dimensional subspace Hol(X,Y) (resp. Alg(X,Y)) approximates the homotopy type of the infinite dimensional space Map(X,Y). This problem is usually called the Atiyah-Jones-Segal conjecture. In this research we mainly consider the case for the Riemann surface X (resp. 1 dimensional sphere) and a toric variety Y. We also investigate the analogues problem for several related spaces defined from the resultants.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3500000, indirect: 1050000)
Phenomenon that a Dehn surgery along a hyperbolic knot in the 3-shpere yields a non-hyperbolic manifold is called "exceptional surgery" and is a subject in the topology of low-dimensional manifolds. We are interested in construction and surgery on 4-manifolds related to exceptional Dehn surgeries. Results : Distributions of integral coefficient exceptional surgeries along Akbulut-Yasui link, including the Mazur link, are decided. Divide presentation of knots in the minor subfamily is considered, and is published in a paper. As research action, I attended almost all workshops "Differential topology” and "handle seminor".

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3400000, indirect: 1020000)
For complex manifolds X and Y (resp. real algebraic varieties X and Y), let Hol(X,Y) (resp. Alg(X,Y)) denote the space of holomorphic maps (resp. algebraic maps represented by polynomials) from X to Y. In this situation, we consider the inclusion map from Hol(X,Y) or Alg(X,Y) into the space Map(X,Y) of all continuous maps from X to Y, and we would like to investigate what dimension this inclusion map approximates the infinite dimensional space Map(X,Y). This problem is called the Atiyah-Jones-Segal conjecture. In particular, in this research we generalize the result of G. Segal concerning to the space of rational functions.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3400000, indirect: 1020000)
一般に、写像空間は無限次元位相空間であり、その空間の位相的性質を研究するのは難しい。本研究では、無限次元モース理論の原理を利用して、特に空間X,Yが実代数的多様体(実数係数の多変数多項式の零点集合で表現される特異点のない空間)の間の写像空間のホモトピー型を研究した。とくに、空間Yがグラスマン多様体で、空間Xが、その上のベクトル束がある条件を満足するとき、写像空間Map(X,Y)をその間の代数的写像のなす部分空間Alg(X,Y)でホモトピー的に近似できるという結果を証明できた。このことにより、Gromovのホモトピー原理が成り立つことを証明できた。さらに、空間X,Yが実射影空間の場合にその有限次元近似の次元を多項式の次数と関連した公式で表すこと(Atiyah-Jones型定理)にも成功した。

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3400000, indirect: -)
Previously, Professors M.Guest, A.Kozlowski and the author showed that the Atiyah-Jones-Segal type Theorem holds for spaces of holomorphic maps from the 1 dimensional complex projective space to certain family of complex projective varieties. Now he showed that a similar result holds for certain subspaces of them which are defined by using the concept of multiplicities induced from the representations of polynomials of holomorphic maps. Furthermore, he computed the fundamental groups for spaces of self-holomorphic maps on the n dimensional complex Projective spaces.
Until now, we usually investigate whether AJS type Theorem holds or not for spaces of holomorphic (or algebraic) maps from one real dimensional (or complex one dimensional) spaces. In our investigation, now we can investigate whether such a problem for spaces of holomorphic or algebraic maps from high dimensional spaces. As one example, we can show that the spaces of regular maps from certain compact affine spaces into complex or real Grassmanian manifolds are homotopy equivalent of spaces of continuous maps between these spaces if these varieties Affine spaces satisfy certain conditions of vector bundles, which is one of joint works with Professor A. Kozlowski. To prove these results, we use the technique of real algebraic geometry. Moreover, we can prove that AJS type Theorem holds for such spaces by using the above Theorem. In particular, we also determine the fundamental groups of spaces of maps from m dimensional real projective space into n dimensional one when m=n-1, or m=n. Such a result can be regarded as a real version of the study investigated in the above first case.
We also study the exceptional surgery from the new point view of singularity theory by using the divide theory. In particular, we study the mechanism of such surgeries and the structure of the set of exceptional surgeries.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3400000, indirect: -)
Consider the energy functionals E on spaces consisting of all smooth maps from a Riemann surface to complex projective spaces. In this case, it is very important to study the spaces consisting of all critical points of E.K.Yamaguchi suceeds to define a finite dimensional homotopy configuration space models from a Riemann surface of genus g into a complex projective space for g>O. He also obtains a similar result for the space of algebraic maps between real projective spaces. Moreover, he shows that a homotopy asymptic stability theorem holds for such spaces of algebraic maps. Kida studies elliptic curves and algebraic field extensions associated to certain maps on algebraic torus. As an application he obtains an easy method for checking prime numbers. M.Ohno studied the vector bundles over non-singular projective varieties and investigated them from the point of view of "nef value". Y.Yamada studied the topology of 4-manifolds and obtained several results related to Gluck surgery.

Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (C), Fund Type: -, Overall Grant Amount: - (direct: 3000000, indirect: -)
The main purpose of K.Yamaguchi is to study the topologies of labelled configuration spaces. Nowdays he and Kozlowski found that the Morse theoretic principle holds for the space P^d_n(C), where P^d_n(C) denotes the space consisting of all monic polynomials f(z) ∈ C [z] of dgree d without real roots of multiplicity 【greater than or equal】 n. It follows from the above results that we knew that Morse theoretic principle (which is also called as Smale-Hirsh principle) holds for these cases and that it also sometimes holds even in the infinite dimensional cases. Similarly, we investigated the topology of spaces of holomorphic maps from Riemann surface to complex projective space with bounded multiplicity case. In this case, we found that similar Morse theoretic principle also holds. Finally, concerning to the latter subject, he noticed the group structure of the group of self- homotopy equivalences of SO(4) and published it too. M.Ohno studied the vector bundles over non-singular projective varieties and investigated them from the point of view of "nef value". Y.Yamada studied the topology of 4-manifolds and obtained several results related to Gluck surgery. M.Misawa studied the valation principle related to harmonic maps from the point of view of partial differential equation. In particular, he found the existence and regurality of p-harmonic maps (weak solution).