Floer

**安德烈斯・弗洛爾**（**Andreas Floer**，1956 年 8 月 23 日 —1991 年 5 月 15 日)，德國數學家，為幾何學、拓撲學以及數學物理等領域作出很可貴的開創性貢獻，並提出弗洛爾同調理論（Floer homology），一種非常實用的數學工具。

## 生平

1982 年波鴻魯爾大學數學系碩士，赴美國柏克萊加州大學深造，在 Clifford Taubes 教授之指導下攻讀博士學位，研究範圍為「monopoles on 3-manifolds」，只因必須回德服社會役，故未完成此學業，後來於 1984 年在波鴻 Eduard Zehnder 教授之指導下獲得博士學位。

弗洛爾首次的關鍵貢獻，是解決若干關於弗拉基米爾・阿諾爾德 conjecture 與 symplectomorphism 的問題。對阿諾爾德數學之研究和自己所提出的同調理論（instanton homology）之成果，使弗洛爾深得世人之肯定，並於 1990 年八月赴京都國際數學家大會發表演說。弗洛爾於 1989 年領取 Sloan Fellowship 獎學金。

在 1988 年任柏克萊加州大學助理教授，於 1990 年晉升為數學系正教授。從 1990 年起任波鴻魯爾大學數學教授，直到突然於 1991 年出人意料地自盡為止。

## 評價

"Andreas Floer's life was tragically interrupted, but his mathematical visions and striking contributions have provided powerful methods which are being applied to problems which seemed to be intractable only a few years ago." [1]（安德烈斯・弗洛爾的生活雖然很不幸地終止，但是他對數學之遠見與卓越貢獻，提供了一些極為有效的方法，可以解決我們幾年前還以為無法解決的問題。）

Simon Donaldson: "The concept of Floer homology is one of the most striking developments in differential geometry over the past 20 years. ... The ideas have led to great advances in the areas of low-dimensional topology and symplectic geometry and are intimately related to developments in Quantum Field Theory"[2] ... "the full richness of Floer's theory is only beginning to be explored".[3]（弗洛爾的同調理論是這二十年以來微分幾何上最令人注目的發現之一…… 他的創見在幾何拓撲學與辛拓撲領域造成很大的進步，並與量子場論有甚密切的關係。）……（我們目前才開始了解弗洛爾理論的整個價值。）

"Since its introduction by Andreas Floer in the late nineteen eighties, Floer theory has had a tremendous influence on many branches of mathematics including geometry, topology and dynamical systems. The development of new Floer theoretic tools continues at a remarkable pace and underlies many of the recent breakthroughs in these diverse fields."[4]（自從安德烈斯・弗洛爾於 1980 年代末提出弗洛爾理論，其對數學許多分支之影響很大，包括幾何、拓撲學以及動力系統。以弗洛爾理論為基礎的新方法以很快的速度繼續產生，不少最近的突破都是從它而出發的。）

## 重要著作

* Floer, Andreas. An instanton-invariant for 3-manifolds. Comm. Math. Phys. 118 (1988), no. 2, 215–240. Project Euclid
* Floer, Andreas. Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.
* Floer, Andreas. Cuplength estimates on Lagrangian intersections. Comm. Pure Appl. Math. 42 (1989), no. 4, 335–356.

## 參考文獻

* Simon Donaldson, _On the work of Andreas Floer_, Jahresber. Deutsch. Math.-Verein. 95 (3) (1993) （頁面存檔備份，存於網際網路檔案館）, 103-120.
* The Floer Memorial Volume (H. Hofer, C. Taubes, A. Weinstein, and E. Zehnder, eds.), Progress in Mathematics, vol. 133, Birkhauser Verlag, 1995.
* Simon Donaldson, _Floer Homology Groups in Yang-Mills Theory_, With the assistance of M. Furuta and D. Kotschick. Cambridge Tracts in Mathematics, 147. Cambridge University Press, Cambridge, 2002. viii+236 pp. ISBN 0-521-80803-0

## 身後發表的著作

* Hofer, Helmut. Coherent orientation for periodic orbit problems in symplectic geometry (**jointly with A. Floer**) Math. Zeit. 212, 13–38, 1993.
* Hofer, Helmut. Symplectic homology I: Open sets in C^n (**jointly with A. Floer**) Math. Zeit. 215, 37–88, 1994.
* Hofer, Helmut. Applications of symplectic homology I (**jointly with A. Floer and K. Wysocki**) Math. Zeit. 217, 577–606, 1994.
* Hofer, Helmut. Symplectic homology II: A General Construction (**jointly with K. Cieliebak and A. Floer**) Math. Zeit. 218, 103–122, 1995.
* Hofer, Helmut. Transversality results in the elliptic Morse theory of the action functional (**jointly with A. Floer and D. Salamon**) Duke Mathematical Journal, Vol. 80 No. 1 , 251–292, 1995. Download from H. Hofer's homepage at NYU （頁面存檔備份，存於網際網路檔案館）
* Hofer, Helmut. Applications of symplectic homology II (**jointly with K. Cieliebak, A. Floer and K. Wysocki**) Math. Zeit. 223, 27–45, 1996.

## 註釋

1. **^** Hofer, Weinstein, and Zehnder, _Andreas Floer: 1956-1991,_ Notices Amer. Math. Soc. **38** (8) , 910-911
2. **^** Simon Donaldson, _Floer Homology Groups in Yang-Mills Theory_, With the assistance of M. Furuta and D. Kotschick. Cambridge Tracts in Mathematics, 147. Cambridge University Press, Cambridge, 2002. viii+236 pp. ISBN 0-521-80803-0 (The above citation is from the front flap.) （頁面存檔備份，存於網際網路檔案館）
3. **^** _Mathematics: frontiers and perspectives_. Edited by V. Arnold, M. Atiyah, P. Lax and B. Mazur. American Mathematical Society, Providence, RI, 2000. xii+459 pp. ISBN 0-8218-2070-2 (Amazon search)
4. **^** From the Press Release to the Workshop _New Applications and Generalizations of Floer Theory_ of the Banff International Research Station (BIRS), May 2007 ([5])

## 外部連結

* 約翰・J・奧康納； 埃德蒙・F・羅伯遜, Floer, MacTutor 數學史檔案 （英語）
* The In Memoriam website of the Department of Mathematics at the University of California, Berkeley （頁面存檔備份，存於網際網路檔案館）
* The In Memoriam website of the University of California, Berkeley
* Elaborate obituary by Addison/Casson/Weinstein at OAC, Online Archive of California, 1992
* 照片看 Wikimedia Commons Category:Andreas Floer （頁面存檔備份，存於網際網路檔案館） 與 a public Andreas Floer photo album on Facebook