スタート いま 数学の多変数複素関数論で 話題になっていることは 何ですか ↓ （google英訳（を若干手直し）） What is the hot topic right now in the theory of functions of several complex variables ?
Copilotさんの回答1 One of the current hot topics in the theory of functions of several complex variables is the (L^2)-theory and its applications. Recent research has focused on the (L^2) existence theorem and (L^2) extension theorem, which play a crucial role in understanding positivity conditions in complex geometry. These results are being used to develop criteria for Griffiths positivity and Nakano positivity of Hermitian metrics on holomorphic vector bundles. Another active area of research is the study of plurisubharmonic functions, which are essential in complex analysis and geometry. Researchers are investigating the strong openness property and stability of multiplier submodule sheaves associated with singular Nakano semipositive Hermitian metrics.
Would you like more details on any of these topics? I can help break them down further!
参考Link（殆ど 下記 Abstract通り？） https://link.springer.com/article/10.1134/S0040577924010112 Home Theoretical and Mathematical Physics Article Recent progress in the theory of functions of several complex variables and complex geometry Research Articles Published: 01 February 2024 Volume 218, pages 163–176, (2024) Cite this article Abstract We give a survey on recent progress on converses of L^2 existence theorem and L^2 extension theorem which are two main parts in L^2 -theory, and their applications in getting criteria of Griffiths positivity and characterizations of Nakano positivity of (singular) Hermitian metrics of holomorphic vector bundles, as well as the strong openness property and stability property of multiplier submodule sheaves associated to singular Nakano semipositive Hermitian metrics on holomorphic vector bundles.
Copilotさんの回答2 Absolutely! Let's dive deeper. (L^2)-Theory in Several Complex Variables The (L^2)-method is a powerful tool in complex analysis, particularly in solving the (\bar{\partial})-equation. This approach, pioneered by Lars Hörmander, provides existence theorems for holomorphic functions under certain curvature conditions. 1) Recent research has refined the Oka–Cartan theory using (L2)-methods, leading to precise on hlomorphic extension. 2) The Bergman kerenel, a fundamental object in complex analysis, has olso been extensively using L2-techniques.2)
Plurisubharmonic Functions Plurisubharmonic functions are crucial in complex analysis and geometry. They generalize subharmonic functions to multiple complex variables and play a key role in defining pseudoconvex domains.3) These functions are used to study multiplier ideal sheaves, which have applications in algebraic geometry and singularity theory. 4) The Levi problem, which characterizes domains of holomorphy, was historically solved using plurisubharmonic functions.3) Would you like to explore specific applications or recent breakthroughs in these areas? I can help break them down further!
参考 1) https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hormander_l2_estimates.pdf Lars H¨ormander and the theory of L2 estimates for the ∂ operator Jean-Pierre
466 名前：Demailly
2) https://link.springer.com/book/10.1007/978-4-431-56852-0 L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Authors: Takeo Ohsawa Book 2018
3) https://en.wikipedia.org/wiki/Plurisubharmonic_function Plurisubharmonic function
4) https://www.math.ucdavis.edu/~hunter/m205b_18/projects/Plurisubharmonic_Functions.pdf Plurisubharmonic Functions and Pseudoconvex Domains Thomas Jackson June 8, 2018 (引用終り) 以上 []
>>432-433 "Would you like to explore specific applications or recent breakthroughs in these areas? I can help break them down further!" とお誘いがありますが まあ、ここらでお茶濁す ；ｐ）
”2) https://link.springer.com/book/10.1007/978-4-431-56852-0 L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Authors: Takeo Ohsawa Book 2018” か、なかなか物知りです （＾＾
AI Copilot さんを、多変数複素関数論 で試してみました >>432-433 まあ、合格点ですかね 優良可不可でいえば、良くらいにはいくかもですね ”Recent progress in the theory of functions of several complex variables and complex geometry”>>432 などは、なかなかよさそうな文献を上げていますね
”2) https://link.springer.com/book/10.1007/978-4-431-56852-0 L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Authors: Takeo Ohsawa Book 2018” などは、こちらの意図を読んだのか
そこから 下記へ行ける https://direct.mit.edu/books/oa-monograph/5460/Sheaf-Theory-through-Examples Sheaf Theory through ExamplesOpen Access By Daniel Rosiak The MIT Press DOI: https://doi.org/10.7551/mitpress/12581.001.0001 ISBN electronic: 9780262370424 Publication date: 2022
ここに、Book PDF というリンクアイコンがあって（リンクは貼らないので自分で訪問すること） そこから、PDF図書が落とせる
Grokさん偉いね Grokさんと、一つ対局してみるかな？ｗ ；ｐ） ＜目次＞ 1 Categories 21 1.1 Categorical Preliminaries 21 1.2 A Few More Examples 35 1.3 Returning to the Definition and Distinctions of Size 36 1.4 Some New Categories from Old 40 1.5 Aside on “No Objects” 43
2 Prelude to Sheaves: Presheaves 45 2.1 Functors 46 2.2 Natural Transformations 61 2.3 Seeing Structures as Presheaves 65 2.4 The Presheaf Action 71 2.5 Philosophical Pass: The Four Action Perspectives 85
3 Universal Constructions 89 3.1 Limits and Colimits 89 3.2 Philosophical Pass: Universality and Mediation 104
4 Topology: A First Pass at Space 109 4.1 Motivation 109 4.2 A Dialogue Introducing the Key Notions of Topology 111 4.3 Topology and Topological Spaces More Formally 123 4.4 Philosophical Pass: Open Questions 144
5 First Look at Sheaves 147 5.1 Sheaves: The Topological Definition 147 5.2 Examples 151 5.3 Philosophical Pass: Sheaf as Local-Global Passage 168
6 There’s a Yoneda Lemma for That 171 6.1 First, Enrichment! 171 6.2 Downsets and Yoneda in the Miniature 175 6.3 Representability Simplified 179 6.4 More on Representability, Fixed Points, and a Paradox 183 以下略
>>470 追加 Sheaf Theory through ExamplesOpen Access Daniel Rosiak
Introduction Open Access https://direct.mit.edu/books/oa-monograph/chapter-pdf/2368031/c000400_9780262370424.pdf
P11より This system of mutually compatible local data assignments or “measurements” of the happenings on the space—where the various data assignments are, piece by piece, constrained by one another, and thereby patched together to supply an assignment over the entire space covered by the individual regions—is, in essence, what constitutes our sheaf. The idea is that the data assignments are being “tied together” in a natural way
where this last picture is meant to serve as motivation or clarification regarding the agricultural terminology of “sheaf”:
Here one thinks of various regions as the parcels of an overall space covered by those pieces, the collection of which then serves as a site where certain happenings are held to take place, and the abstract sensors capturing local snapshots or measurements of all that is going on in each parcel are then regarded as being collected together into “stalks” of data, regarded as sitting over (or growing out of) the various parts of the ground space
506 名前：to which they are attached. A selection of a particular snapshot made from each of the individual stalks (collections of snapshots) amounts to a cross-section and the process of restriction (along intersecting regions) and collation (along unions of regions) of these sections captures how the various stalks of data are bound together. (引用終り)
(参考) https://ja.wikipedia.org/wiki/%E6%9D%9F_(%E6%9D%9F%E8%AB%96) 束 (束論) 出典 1^ Dedekind, Richard (1897), “Ueber Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler”, Braunschweiger Festschrift: 1–40
https://de.wikipedia.org/wiki/Verband_(Mathematik) Verband (Mathematik) google英訳 Association (mathematics) In mathematics, a lattice is a structure that can be completely described as both an order structure and an algebraic structure . Hasse diagrams for some examples → Main article : Hasse diagram
https://context.reverso.net/%E7%BF%BB%E8%A8%B3/%E3%83%89%E3%82%A4%E3%83%84%E8%AA%9E-%E6%97%A5%E6%9C%AC%E8%AA%9E/Verband?d=0 reverso.net 独語辞書 der Verbandnm Ich hab 'n Verband gefunden. なぜか包帯が巻かれてた Warum nimmst du den Verband nicht ab? 彼の包帯を取ってみたらいい？ Der Verband bietet Verbrauchern und Kunden Rückgriff auf rohe Nachlassforschungsunternehmen. この協会は、消費者および顧客に不正な検定調査会社に対する訴求を提供する。
https://fr.wikipedia.org/wiki/Treillis_(ensemble_ordonn%C3%A9) Treillis (ensemble ordonné) ( google訳：格子（順序集合）) En mathématiques, un treillis[1] (en anglais : lattice) est une des structures algébriques utilisées en algèbre générale. Notes et références [1]N. Bourbaki, Éléments de mathématique : Théorie des ensembles [détail des éditions], p. ER.28, aperçu sur Google Livres, parle d'« ensemble réticulé, ou réseau ordonné (ou lattis) ».
>S先生はバスの入り口で >「入口」を見て >What is this lambda Laplacian? >と尋ねられた。
Copilot に入れてみると Q: What is this lambda Laplacian? A: The Laplacian is a differential operator that plays a crucial role in mathematics and physics, particularly in areas like potential theory, differential equations, and quantum mechanics. It is often denoted as Δ or ∇² and is defined as the sum of second partial derivatives: [ \Delta f = \frac{\partial2} + \frac{\partial2} + \frac{\partial2} ] If you're referring to a lambda Laplacian, it could relate to eigenvalues of the Laplacian operator in spectral theory. The eigenvalue problem for the Laplacian is: [ \Delta f = \lambda f ] where λ represents an eigenvalue associated with the Laplacian. This arises in various contexts, such as solving the Helmholtz equation, studying heat diffusion, and analyzing vibrational modes. Would you like to explore its applications in functional analysis or spectral geometry? (引用終り)
なので、”λ represents an eigenvalue associated with the Laplacian”と 固有値の連想ゲームです バスの入り口に、Δ or ∇ が有ったのでしょうか？ （＾＾
(参考) https://en.wikipedia.org/wiki/DBAR_problem DBAR problem The DBAR problem is of key importance in the theory of integrable systems, Schrödinger operators and generalizes the Riemann–Hilbert problem.[1][2][3]
References [2]Haslinger, Friedrich (2014). The d-bar Neumann Problem and Schrödinger Operators. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-031535-6. PDF https://www.mat.univie.ac.at/~has/dbar/dbar1.pdf Preface The rst chapters contain a discussion of Bergman spaces and of the solution operator to @ restricted to holomorphic L2 -functions in one complex variable, pointing out that the Bergman kernel of the associated Hilbert space of holomorphic functions plays an important role.
The next chapter contains a detailed account of the application of the @-methods to Schr odinger operators, Pauli and Dirac operators and to Witten-Laplacians.
In this way one obtains a rather general basic estimate, from which one gets H ormander's L2 -estimates for the solution of the CauchyRiemann equation together with results on related weighted spaces of entire functions, such as that these spaces are in nite-dimensional if the eigenvalues of the Levi-matrix of the weight function show a certain behavior at in nity. In addition, it is pointed out that some L2 -estimates for @ can be interpreted in the sense of a general Brascamp-Lieb inequality.
Contents Preface iii 1. Bergman spaces 2 2. The canonical solution operator to @ restricted to spaces of holomorphic functions 10 3. Spectral properties of the canonical solution operator to @ 21 4. The @-complex 33 5. The weighted @-complex 50 6. The twisted @-complex 58 7. Applications 62 8. Schr odinger operators 69 9. Compactness 74 10. The @-Neumann operator and commutators of the Bergman projection and multiplication operators. 85
https://www.mat.univie.ac.at/~has/ Friedrich Haslinger Faculty of Mathematics University of Vienna Research interests d-bar Neumann problem Hardy and Bergman spaces in several complex variables Bergman and Szegö kernels Spectral analysis of Schrödinger operators
Liebさん In 2022 the Carl Friedrich Gauss Prize at the International Congress of Mathematicians In 2023 Lieb received Kyoto Prize in Basic Sciences for his achievements in many-body physics.[23]
https://en.wikipedia.org/wiki/Elliott_H._Lieb Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist. He is a professor of mathematics and physics at Princeton University. Lieb's works pertain to quantum and class
532 名前：ical many-body problem,[1][2][3] atomic structure,[3] the stability of matter,[3] functional inequalities,[4] the theory of magnetism,[2] and the Hubbard model.[2] Awards In 2022 the Carl Friedrich Gauss Prize at the International Congress of Mathematicians ″for deep mathematical contributions of exceptional breadth which have shaped the fields of quantum mechanics, statistical mechanics, computational chemistry, and quantum information theory.″[16] In 2023 Lieb received Kyoto Prize in Basic Sciences for his achievements in many-body physics.[23] []
下記ですね ”Dedicated to the 100th anniversary of the creation of the Bergman kernel ”か ”Haslinger”さん 二人います Professor Fritz Haslinger さんと Friedrich Haslinger (Universität Wien) さんと 同一人物か あるいは ご親戚か？
(参考) https://sites.google.com/view/hayama-scv/2022 HAYAMA Symposium on Complex Analysis in Several Variables XXIII Dedicated to the 100th anniversary of the creation of the Bergman kernel July 23(Sat) – July 26(Tue), 2022 uly 27 and 28(Wed and Thu), 2022
Stefan Bergman publised his first paper on the reproducing kernel in 1922, which was his doctoral dissertation under the supervision of v. Mises and Erh. Schmidt. On the occasion of 100th anniversary, we plan to review recent progresses of the Bergman kernel and related topics. The meeting will be held on site but includes some online talks.
https://lh6.googleusercontent.com/BI_CyAfBu2DJC--g1CaNGsDe9pBVVbznfV47wICATIagy74a-ldg1yGaruXIxE4gjLo0l8-2UgGtflP4y__XQN4XBR-aLr0mLVIthqLiCENSVdAi01wwgHG54Cz6zSpZK-VGDe5RBZSh9dRYaGkN2gw8P5gC5uAWNAoTh2OsD3lHWV2N2_qGQw=w1280 A birthday cake made in Vienna by courtesy of Professor Fritz Haslinger.
Ph.D. Eberhard-Karls-Universität Tübingen 1953 Germany Dissertation: Several Complex Variables and Manifolds for the Cartan Conjecture Advisor: Hellmuth Kneser
(参考) https://en.wikipedia.org/wiki/Karl_Stein_(mathematician) Karl Stein (1 January 1913 in Hamm, Westphalia – 19 October 2000) was a German mathematician. He is well known for complex analysis and cryptography. Stein manifolds and Stein factorization are named after him.
google検索 K. Stein 多変数関数論 AI による概要 （AI の回答には間違いが含まれている場合があります。） K. Steinは、多変数複素関数論の重要な人物であり、特に岡潔の多変数関数論の理論に強い影響を与えました。Steinは、岡潔の理論をさらに発展させ、多変数の解析関数の性質を詳しく調べました。
