马克斯·登Max Dehn

马克斯·登 撰写了最早的 拓扑学 系统阐述之一（1907 年），后来在 群的表现 上提出了重要问题，即字问题和同构问题。

让我们先给出一些关于 登 家庭的细节。他的父亲 Maximillian Moses 登（1841 年 3 月 15 日 - 1897 年 4 月 16 日）是一名医生。Maximillian 是一个有九个兄弟姐妹的大家庭中的一员：弗拉基米尔·阿诺尔德、Henne Hannah、Gustav、Rudolf Joseph、Otto Carl Isaak、Hannah Bertha、Martin、Elizabeth Charlotte 和 Moses Wilhelm。Maximillian 于 1867 年 5 月 2 日在德国汉堡与 Berta Raf Hirsch（1845 年 4 月 15 日 - 1926 年 2 月 2 日）结婚。他们有九个孩子：Rudolph Bernhard、Elizabeth Hanne Adeline、Henriette Marianne、登 Wilhelm（本传记的主角）、Berta Meta、Karl Arnold、George Ernst、Charlotte 和 Marie Auguste。这个家庭在种族上是犹太人，但并不认为自己是犹太人，而是德国人。登 在汉堡长大，就读于 Wilhelm 文理中学（Gymnasium）。这所学校成立于 1881 年，以皇帝威廉一世命名，当 登 就读时，它位于 Moorweidenstrasse。这所学校为学生进入大学做准备，毕业后，登 进入了弗莱堡大学。

登学习数学，但正如当时学生常见的情况，他并未将所有时间都花在一所大学。他还在哥廷根学习，在那里，在大卫·希尔伯特的指导下，他于1900年获得博士学位，学位论文题为Die Legendreschen Sätze über die Winkelsumme im Dreieck Ⓣ（三角形内角和上的阿德里安-马里·勒让德集）。在这篇论文中，他证明了Saccheri-阿德里安-马里·勒让德定理，该定理指出在绝对几何中，三角形内角和至多为180°。所谓绝对几何，我们指的是满足欧几里得几何公理但平行公设除外的几何。1900年8月，大卫·希尔伯特在巴黎举行的国际数学家大会上发表了他著名的演讲。在演讲中，他提出了十个他认为对数学发展很重要的问题。事实上，他演讲的出版版本包含23个这样的问题。1900年，登解决了其中第3个问题，即：-

给定任意两个体积相等的多面体，是否总能将第一个切割成有限多个多面体碎片，这些碎片可以重新组装成第二个？

登表明这个问题的答案是“否”，他使用今天所谓的‘登不变量’构造了一个反例。这是大卫·希尔伯特的问题中第一个被解决的。他在1900年底之前将解决第3个大卫·希尔伯特问题的论文作为他的教授资格论文（Habilitation）学位论文提交给明斯特大学，并在次年受聘为明斯特的Privatdocent。他担任此职位直到1911年。1907年，登与Poul Heegaard合作，为Enzyklopädie der mathematischen Wissenschaften [2]撰写了最早的拓扑学系统阐述之一：-

它首次系统地呈现了一个极具深度与美感的研究方向——可追溯至莱昂哈德·欧拉、卡尔·弗里德里希·高斯、利斯廷、波恩哈德·黎曼、奥古斯特·费迪南德·莫比乌斯、恩里科·贝蒂和儒勒·昂利·庞加莱——当时称为“位置分析”，今天大概会被称为几何拓扑学。

登随后试图解决儒勒·昂利·庞加莱猜想，但当然失败了。然而，他的尝试促使他发表了论文Über die Topologie des dreidimensionalen Raumes Ⓣ（《论三维空间的拓扑学》）（1910），其中他讨论了具有平凡同调的三维流形，这些流形由三叶结及其他各种纽结的补集构造而成。为了研究这类流形，他构造了它们基本群的阿瑟·凯莱图。1911年，登被任命为基尔大学的编外教授，1913年至1921年，他是布雷斯劳大学的正式教授。他于1912年8月23日与Antonie Landau结婚；他们有三个孩子Helmut（1914-2007）、Maria（1915-2013）、Eva Agathe（1919-2008），均出生于布雷斯劳。

1914年，登发表了论文Die beiden Kleeblattschlingen Ⓣ（《两个三叶结》）。琼·蕾特·比尔曼在[5]中描述了这篇论文：-

写于1914年，即在拓扑空间基本群发现后不久，它处理了一个简单而美丽的问题：确认最简单的纽结的一个性质，这个性质由五分钟的实验即可提示：即右三叶结和左三叶结不是同痕的。基本群在1914年非常新，登以精巧的方式使用了它，通过研究其外自同构群。用它证明了三叶结不是可逆的之后，登的论文以对八字结基本群的外自同构群的简要研究结束，八字结是可逆的。他确定了（如后来所证明的）其生成元。……在1984年之前，我们确实没有任何简单的测试来判断非可逆性，因此登的工作（乍看之下像是用大炮打麻雀）在近70年里一直是该领域的核心。

当然，1914年标志着第一次世界大战的开始，登的职业生涯因1915年至1918年在军队服役而中断。服役结束后，他回到布雷斯劳大学的教授职位，但1921年他被任命为法兰克福大学纯粹与应用数学讲席，接替路德维希·比贝尔巴赫。在那里，他组织了一个著名的数学史讨论班。安德烈·韦伊写道[4]：-

一位人文主义数学家，将数学视为人类思想史中的一章——当然不是最不重要的一章——登不可能不对数学的历史研究做出原创性贡献，并让他的同事和学生参与这一项目。这一贡献，或者说这一创造，就是法兰克福数学研究所的历史讨论班。没有什么比这更简单或更不张扬的了。选择一篇文本，阅读原文，努力不仅紧跟表面线索，还要把握深层思想的推动力。……直到后来，我在随后访问法兰克福时才参加，那是我尽可能经常访问的地方。我不确定是否已经在1926年夏季学期，在一次专门讨论博纳文图拉·卡瓦列里的讨论班上，登展示了如何从作者的角度阅读这篇文本，同时考虑到他生前普遍接受的内容以及博纳文图拉·卡瓦列里正尽力实施的新思想。每个人都参与了讨论，为集体努力贡献自己所能。

1932年，登撰写了文章Das Mathematische im Menschen Ⓣ（人类的数学能力），并将其发表在Scientia上，这是一本由费代里戈·恩里克斯[2]在博洛尼亚出版的意大利期刊：-

在这篇文章中，登试图如他所写的那样“将数学科学与人类活动的整体联系起来”。根据登的观点，产生数学只需要少数几个前提条件。他声称，每当数学情感与逻辑结论相结合时，数学就会自行产生——而所有人都具备这两种能力。因此，登将“数学能力”置于基本的人类学层面。他相信数学深深植根于人性之中，就像音乐基于天生的音乐能力一样，这使他确信非专业的数学家也能被培养出对这种科学的欣赏。

参见Dehn on Mathematical abilities在THIS LINK的前几段。

1933年1月30日，希特勒在德国上台，4月1日出现了所谓的“抵制日”，犹太商店遭到抵制，犹太讲师不被允许进入大学。1933年4月7日通过了《公务员法》，该法提供了将犹太教师从大学中清除的手段，当然也提供了将犹太血统者从其他角色中清除的手段。所有非雅利安血统的公务员（有一个祖父母信奉犹太教就使某人成为非雅利安人）都应退休，但对在第一次世界大战中为德国作战的人有豁免。I. Dehn符合这一豁免条款，因此继续领导法兰克福大学的数学讨论班。然而，1935年秋纽伦堡党代会的决定明确表示，非雅利安人即使曾在第一次世界大战中服役，也不再能保留其职位。I. Dehn被迫辞职，但仍留在法兰克福。1936年，他把孩子们送出德国。他的儿子赫尔穆特去了美国，他的女儿玛丽亚和伊娃被送到英国继续学业。在接下来的几年里，登在欧洲多个国家讲学，并于1938年1月至4月与女儿们一起在英国度过。他继续发表文章。1936年，他在Scientia上发表了Raum, Zeit, Zahl bei Aristoteles vom mathematischen Standpunkt ausⓉ（《从数学观点看亚里士多德中的空间、时间、数》）。

参见Max Dehn on Space, Time and Number in Aristotle在THIS LINK的前几段。

1938年，他发表了Die Gruppe der AbbildungsklassenⓉ（映射类群），其中包含了今天被称为“登扭转”的重要思想。1938年11月9日至10日的水晶之夜（因次日早晨街道上的碎玻璃而得名）期间，登和他的妻子正在法兰克福。许多犹太人被谋杀，数百人受重伤，数千人遭受了可怕的经历。数千家犹太企业被烧毁，超过150座犹太教堂也被毁。盖世太保逮捕了3万名富裕的犹太人，释放的条件是他们必须移居国外。1938年11月11日，登被捕，但由于关押的犹太人太多，监狱已满，他当天晚些时候就被释放了。登和他的妻子因害怕再次被捕，立即逃往法兰克福以北巴特洪堡的朋友Willi Hartner家中。Hartner写道[11]：-

对于当时见过他的人来说，难忘的是他的冷静，他的哲学沉着。因为谈话的中心不是当天的事件，而是数学与艺术的关系、考古学问题，最后是孔子的仁爱概念。

几周后，盖世太保对犹太人的迫害活动有所减弱，登和他的妻子冒险逃往汉堡，在那里他们暂时住在登的一位姐姐家中，随后于1939年1月先逃到丹麦，再从那里前往挪威。登很快在特隆赫姆技术学院获得了一个临时职位，接替正在休假的维戈·布朗（1885-1978）。然而，1940年3月1日德国入侵挪威，特隆赫姆于4月9日落入德国人手中。登逃离了这座城市，但尽管风险巨大，他还是在6月前回到了城里，并开始计划移居美国。

1940年10月，德恩一家移居美国，从挪威前往瑞典斯德哥尔摩，然后前往俄罗斯莫斯科，再从那里乘坐跨西伯利亚火车到符拉迪沃斯托克。从那里他们乘船到日本神户，然后另一艘船到旧金山，于1941年1月1日抵达。到达美国后，登在几所大学和学院任教，例如在波卡特洛的爱达荷大学、伊利诺伊理工学院和马里兰州安纳波利斯的圣弗瑞兹·约翰学院。保罗·L·切辛描述了登在威斯康星州麦迪逊时的一些有趣的个人回忆：-

1945年，[登]接替了鲁道夫·兰格，当时他是威斯康星州麦迪逊的数学系主任，后者休假去了德克萨斯大学。登教授来教授几门研究生课程。我参加了他关于“非线性偏微分方程”的课程。我们很高兴有一位讲德语的讲师，因为我们要参加德语语言考试（博士学位要求）。但他拒绝了，因为那个学期他要参加美国公民身份考试！登会在Rathskellar（十大大学中唯一的校园啤酒场所）高谈阔论。我们在啤酒馆里学到了更多——比如他的个人生活。他是家里的败家子。只要他留在大学里，他就得到支持。因此他获得了许多博士学位。他会手持啤酒杯，用希腊语朗诵一些经典段落。在设定最终成绩时，他只是点名，然后基本上一个接一个地询问教科书章节的标题。显然我们可以提前瞥一眼以确保正确答案。我相信他给了大约18个A和3-4个B。在整个课堂中，他会打断，表达一些关于通过公民身份考试的担忧。

登无法找到全职职位，桑德斯·麦克兰恩最近写道：-

……大多数逃离欧洲的数学家在美国得到了某种职位……我记得两个失败的案例：登，以拓扑学工作闻名，只得到了一个弱势职位。

桑德斯·麦克兰恩提到的弱势职位是在黑山学院。这所学院没有认证学位，主要教授创意艺术。1944年登被邀请在那里做两次讲座时，教职员中没有受过训练的数​​学家。他意识到他不能讲授高等数学，所以他做了关于“数学与装饰学的共同根源”和“数学思想发展中的某些时刻”的讲座。他获得了那里的永久职位，月薪25美元。他坚持要求40美元一个月，并得到了同意。登于1945年加入教职，并一直留在那里直到去世。登是唯一一位在该学院任教的数学家，该学院于1956年关闭。弗洛伦斯·南丁格尔·大卫 Peifer写道[14]：-

登教授数学、哲学、希腊语和意大利语。在数学方面，他的课程包括数学史和射影几何。他因热爱自然和艺术而被铭记。他的讲座经常包括关于哲学、艺术及其与数学联系的题外话。他以苏格拉底式风格讲课时最为出色。他喜欢在与学生徒步穿越树林时进行这些讲座。他被铭记为一位关怀和敬业的老师。

登杰出的研究记录与他最终职位的低微形成了鲜明对比。他是一位直觉几何学家，受到大卫·希尔伯特对该学科公理化方法的启发。他在拓扑学方面的工作使他进入群的研究，特别是从拓扑学考虑自然产生的群表示。在他1911年的论文Über unendliche diskontinuierliche Gruppen Ⓣ（论无限不连续群）中，登提出了关于群表示的重要问题，即字问题和同构问题。字问题提出了一个基本问题：是否存在一种算法来确定由表示给出的群中的一个字是否平凡。此后已证明，一般情况下不存在这样的算法。对这类问题的研究在组合群论中仍然具有重大意义。登还撰写了关于静力学、射影平面的著作，以及如上所述，关于数学史的著作。

第二次世界大战结束后，German Mathematical Society（DMV）于1948年重建，Erich Kamke（1890-1961）写信给所有被驱逐的人，请他们重新加入。登拒绝了，我们给出他1948年8月写的回复，其中他清楚地陈述了他的理由（例如见[2]）：-

我没有任何怨恨。你可能知道，我再次与德国的几位数学家保持密切联系，当然主要是与那些我特别亲近的人。但我不能重新加入德国数学学会。我已经失去了信心，认为这样的协会将来会与1935年有所不同。我担心它会再次不抵制来自外部的不公正措施。DMV没有守护非常重要的价值。我的负面印象是由以下事实造成的：它没有在1935年自行解散，甚至没有相当数量的数学家离开该协会。我不担心DMV会再次驱逐犹太人，但也许下一次会是所谓的共产主义者、无政府主义者或“有色人种”。与德国的联系，特别是与德国数学家的联系，对我来说非常珍贵。

Paul L Chessin描述了一些关于登的有趣个人回忆：-

1945年，[登]接替了当时威斯康星州麦迪逊数学系主任Rudolph Langer，后者休假前往德克萨斯大学。登教授来教授几门研究生课程。我参加了他关于非线性偏微分方程的课程。

我们很高兴有一位讲德语的教师，因为我们即将参加德语语言考试（博士学位要求）。但他拒绝了，因为那个学期他要参加美国公民身份考试！

登会在Rathskellar（十大大学联盟内唯一的校园啤酒场所）高谈阔论。我们在啤酒馆里学到了更多——比如他的个人生活。他是家里的败家子。只要他留在大学里，他就得到支持。因此他获得了许多博士学位。

他会手持啤酒杯，用希腊语朗诵一些古典作品段落。到了给最终成绩的时候，他只是点名，并逐一询问教科书各章的标题。显然我们可以提前看一眼以确保正确答案。我相信他给了大约18个A和3-4个B。

在整个课堂过程中，他会打断并表达一些关于通过公民身份考试的担忧。

登在1951-1952学年结束时退休，并被授予荣休教授称号。他继续在黑山学院工作，为教职员工和学生提供建议。在监督从学院校园移走一些山茱萸树后不久，他生病并死于栓塞。他被安葬在黑山学院校园内他喜爱的树林中。

Max Dehn wrote one of the first systematic expositions of topology (1907) and later formulated important problems on group presentations, namely the word problem and the isomorphism problem.

Let us begin by giving some details of Max Dehn's family. His father, Maximillian Moses Dehn (15 March 1841 - 16 April 1897), was a medical doctor. Maximillian was one of a large family having nine siblings: Arnold, Henne Hannah, Gustav, Rudolf Joseph, Otto Carl Isaak, Hannah Bertha, Martin, Elizabeth Charlotte, and Moses Wilhelm. Maximillian married Berta Raf Hirsch (15 April 1845 - 2 February 1926) in Hamburg, Germany, on 2 May 1867. They had nine children: Rudolph Bernhard, Elizabeth Hanne Adeline, Henriette Marianne, Max Wilhelm (the subject of this biography), Berta Meta, Karl Arnold, George Ernst, Charlotte, and Marie Auguste. The family was racially Jewish but did not think of themselves as Jews, rather they were Germans. Max was brought up in Hamburg where he attended the Wilhelm Gymnasium. This school, founded in 1881, was named after emperor Wilhelm I and when Dehn attended the school it was situated on Moorweidenstrasse. This school prepared students for university and, after graduating, Dehn entered the University of Freiburg.

At Freiburg, Dehn studied mathematics but, as was common for students at this time, he did not spend his whole time at one university. He also studied at Göttingen where, under David Hilbert's supervision, he obtained his doctorate in 1900 for a thesis entitled Die Legendreschen Sätze über die Winkelsumme im Dreieck Ⓣ. In this thesis he proved the Saccheri-Legendre theorem which states that in absolute geometry the sum of the angles in a triangle is at most 180°. By absolute geometry, we mean geometry satisfying the axioms of Euclidean geometry except for the parallel postulate. In August 1900, Hilbert gave his famous address at the International Congresses of Mathematicians held in Paris. In it he presented ten problems which he believed were important for the development of mathematics. In fact, the published version of his talk contains 23 such problems. In 1900 Dehn solved the 3rd of these problems namely:-

Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

Dehn showed that the answer to this question is "no", constructing a counterexample using what is today called the 'Dehn invariant'. This was the first of Hilbert's problems to be solved. He submitted his paper solving the 3rd Hilbert Problem to the University of Münster as his Habilitation thesis before the end of 1900 and, in the following year, he was appointed as a Privatdocent at Münster. He held this position until 1911. In 1907, jointly with Poul Heegaard, Dehn wrote one of the first systematic expositions of topology for the Enzyklopädie der mathematischen Wissenschaften [2]:-

It offers the first systematic representation of a research direction of great depth and beauty - going back to Euler, Gauss, Listing, Riemann, Möbius, Betti and Poincaré - which was then called 'analysis situs' and today probably be called geometric topology.

Dehn next attempted to solve the Poincaré conjecture but, of course, he failed. However, his attempts led him to publish the paper Über die Topologie des dreidimensionalen Raumes Ⓣ (1910) in which he discussed 3-dimensional manifolds with trivial homology, constructed from the complement of the trefoil knot and various other knots. In order to study such manifolds, he constructed the Cayley graph of their fundamental groups. In 1911 Dehn was appointed as an extraordinary professor at Kiel, and from 1913 until 1921 he was a full professor at the University of Breslau. He married Antonie Landau on 23 August 1912; they had three children Helmut (1914-2007), Maria (1915-2013), Eva Agathe (1919-2008) all born in Breslau.

In 1914 Dehn published the paper Die beiden Kleeblattschlingen Ⓣ. Joan Birman describes this paper in [5]:-

Written in 1914, not long after the discovery of the fundamental group of a topological space, it tackles a simple and beautiful problem: to confirm a property of the simplest knot which is suggested by five minutes of experimentation: that the right and left trefoil knots are not isotopic. The fundamental group, very new in 1914, was used by Dehn in a sophisticated fashion, via the investigation of its outer automorphism group. Having used it to prove the trefoil is not amphicheiral, Dehn's paper concludes with a brief investigation of the outer automorphism group of the fundamental group of the figure eight knot, which is amphicheiral. He identifies (as was shown later) its generators. ... before 1984 we really didn't have any simple tests for non-amphicheirality, so that Dehn's work (which at first glance looks like the use of a cannon to kill a sparrow) remained central to the subject for nearly 70 years.

Of course, 1914 marks the beginning of World War I and Dehn's career was interrupted while he served in the army from 1915 to 1918. After military service, he returned to his position as professor at Breslau, but in 1921 he was appointed to the chair of Pure and Applied Mathematics at the University of Frankfurt, succeeding Ludwig Bieberbach. There he organised a famous seminar on the history of mathematics. André Weil writes [4]:-

A humanistic mathematician who saw mathematics as one chapter - certainly not the least important - in the history of human thought, Dehn could not fail to make an original contribution to the historical study of mathematics, and to involve his colleagues and students in the project. This contribution, or rather this creation, was the historical seminar of the Frankfurt mathematics institute. Nothing could have seemed simpler or less pretentious. A text would be chosen and read in the original, with an effort to follow closely not only the superficial lines but also the thrust of the underlying ideas. ... It was only later that I attended it, on subsequent visits to Frankfurt, a place I made a point of visiting as often as I could. I am not sure whether it was already in the summer semester of 1926 that, during a seminar session devoted to Cavalieri, Dehn showed how this text had to be read from the viewpoint of the author, taking into account both what was commonly accepted in his lifetime and the new ideas that Cavalieri was trying the best of his ability to implement. Everyone participated in the discussion, contributing what he could to the group effort.

In 1932 Dehn wrote the essay Das Mathematische im Menschen Ⓣ which he published in Scientia, an Italian journal produced in Bologna by Federigo Enriques [2]:-

In this essay, Dehn attempted, as he wrote "to link the science of mathematics with the whole of human activity". According to Dehn, only a few preconditions are necessary to bring about mathematics. Mathematics, he claimed, arises of its own accord whenever mathematical emotion and logical conclusions come together - and all human beings have both capacities. Dehn thus positioned "mathematical ability" at a basic anthropological level. The belief that mathematics is deeply rooted in human nature, in much the same way that music is based on innate musical abilities, gave him the confidence that non-professional mathematicians could also be made to appreciate this science.

See the first few paragraphs of Dehn on Mathematical abilities at THIS LINK.

On 30 January 1933 Hitler came to power in Germany and on 1 April there was the so-called "boycott day" when Jewish shops were boycotted and Jewish lecturers were not allowed to enter the university. On 7 April 1933 the Civil Service Law was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired but there were exemptions for those who had fought for Germany in World War I. Dehn qualified under this exemption clause so continued to head the mathematics seminar at University of Frankfurt. However, decisions at the Nuremberg party congress in the autumn of 1935 made it clear that non-Aryans would no longer be able to keep their posts even if they had served in World War I. Dehn was forced to resign but remained in Frankfurt. In 1936 he sent his children out of Germany. His son Helmut went to the United States and his daughters Maria and Eva were sent to England to continue their education. Dehn lectured in a number of European countries over the next couple of years and spent from January to April 1938 in England with his daughters. He continued to publish articles. In 1936, he published Raum, Zeit, Zahl bei Aristoteles vom mathematischen Standpunkt aus Ⓣ in Scientia.

See the first few paragraphs of Max Dehn on Space, Time and Number in Aristotle at THIS LINK

In 1938 he published Die Gruppe der Abbildungsklassen Ⓣ which contains the important ideas known today as a 'Dehn twist'. Dehn and his wife were in Frankfurt on Kristallnacht (so called because of the broken glass in the streets on the following morning), the 9-10 November 1938. Many Jews were murdered, hundreds were seriously injured, and thousands were subjected to horrifying experiences. Thousands of Jewish businesses were burnt down together with over 150 synagogues. The Gestapo arrested 30,000 well-off Jews and a condition of their release was that they emigrate. On 11 November 1938 Dehn was arrested but, with so many Jews being held, the prisons were full and he was released later on the same day. Immediately Dehn and his wife, fearing re-arrest, fled to the home of Willi Hartner, friends in Bad Homburg north of Frankfurt. Hartner wrote [11]:-

Unforgettable for those who saw him at the time was his calmness, his philosophical composure. For the conversations centred not on the events of the day, but on the relationship of mathematics to art, on problems of archaeology, and finally on the concept of humanity of Confucius.

After a few weeks the Gestapo became less active in their persecution of Jews and Dehn and his wife risked fleeing to Hamburg where they lived for a while at the home of one of Dehn's elder sisters before escaping, first to Denmark in January 1939 and from there to Norway. Dehn was soon given a temporary position at he Technische Hochschule in Trondheim as a replacement for Viggo Brun (1885-1978) who was on leave. However, on 1 March 1940 Germany invaded Norway and Trondheim fell to the Germans on 9 April. Dehn fled the city but, despite the great risks, was back in the city by June and began to plan his move to the United States.

In October 1940 the Dehns emigrated to the USA, travelling from Norway to Stockholm in Sweden, then on to Moscow in Russia from where they took the trans-Siberian train to Vladivostok. From there they took a ship to Kobe in Japan and then another ship to San Francisco where they arrived on 1 January 1941. Once in the USA Dehn taught at several universities and colleges, for instance at the University of Idaho in Pocatello, the Illinois Institute of Technology and St John's College in Annapolis, Maryland. Paul L Chessin has described some interesting personal recollections of Dehn while at Madison, Wisconsin:-

In 1945, [Dehn] replaced Rudolph Langer, then head of the mathematics department in Madison, Wisconsin, who went on sabbatical to the University of Texas. Prof Dehn came to teach several graduate courses. I attended his course in 'Non-linear Partial Differential Equations'. We were delighted to have a German-speaking instructor since we were to take our German language examinations (required for the doctorate). He, in turn, refused since it was that term that he was to take his examination for US citizenship! Max would hold forth in the Rathskellar (the only campus beer establishment within the Big Ten Universities). We learned more in the beer stube - such as his personal life. He was the black sheep in his family. So long as he remained in his university, he was supported. Thus he received many doctorates. He would declaim in Greek, some passages from the classics, beer stein in hand. At the time for setting out final grades, he merely called the attendance and one by one asked essentially for the titles of the chapters in the textbook. Clearly we could glance ahead to be assured of the correct answer. I believe he gave something like 18 A's and 3-4 B's. Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination.

However Dehn was unable to find a full-time position and Saunders Mac Lane recently wrote:-

... most mathematicians fleeing Europe were helped to some sort of position in the United States ... I recall two cases of failures: Max Dehn, noted for work in topology, got only a weak position.

The weak position, referred to by Mac Lane, was at Black Mountain College. This college had no accredited degrees and taught mainly creative arts. There was no trained mathematician on the staff when Dehn was invited to give two lectures there in 1944. He realised that he could not lecture on advanced mathematics so he gave his lectures on 'Common roots of mathematics and ornamentics' and 'Some moments in the development of mathematical ideas'. He was offered a permanent post there at $25 a month. He held out for $40 a month which was agreed. Dehn joined the Faculty in 1945 and remained there until his death. Dehn was the only mathematician ever to teach at the College which closed in 1956. David Peifer writes [14]:-

Dehn taught Mathematics, Philosophy, Greek, and Italian. In Mathematics his courses included History of Mathematics and Projective Geometry. He is remembered for his love of nature and the arts. His lectures frequently included tangents on philosophy, the arts, and their connections to mathematics. He was best when lecturing in a socratic style. He was fond of giving these lectures while hiking with his students through the woods. He is remembered as a caring and devoted teacher.

Dehn's outstanding research record is in stark contrast with the low level of his final post. He was an intuitive geometer, stimulated by Hilbert's axiomatic approach to the subject. His work in topology had led him into the study of groups, particularly group presentations which arise naturally from topological considerations. In his 1911 paper Über unendliche diskontinuierliche Gruppen Ⓣ, Dehn formulated important problems on group presentations, namely the word problem and the isomorphism problem. The word problem asks the fundamental question of whether there is an algorithm to determine whether a word in a group given by a presentation is trivial. It has since been shown that no such algorithm exists in general. Research on questions of this type are still of major importance in combinatorial group theory. Dehn also wrote on statics, projective planes and, as we have noted above, on the history of mathematics.

After World War II ended, the German Mathematical Society (DMV) was re-established in 1948 and Erich Kamke (1890-1961) wrote to all those who had been expelled asking them to rejoin. Dehn refused and we give his reply, written in August 1948, in which he states clearly his reasons (see for example [2]):-

I bear no grudge of any kind. As you may be aware, I am once again in close contact with several mathematicians in Germany, of course primarily with those whom I was particularly close to. But I cannot rejoin the German Mathematical Society. I have lost the confidence that such an association would act differently in the future than it did in 1935. I fear it would, once again, not resist an unjust measure coming from outside. The DMV did not have the custody of very important values. My negative impression is caused by the fact that it did not dissolve itself in 1935 and that not even a considerable number of mathematicians left the association. I am not afraid that the DMV will once again expel Jews, but perhaps next time it will be so-called communists, anarchists or "coloured people". Contact with Germany, especially with German mathematicians, is very dear to my heart.

Paul L Chessin has described some interesting personal recollections of Dehn:-

In 1945, [Dehn] replaced Rudolph Langer, then head of the mathematics department in Madison, Wisconsin, who went on sabbatical to the University of Texas. Prof Dehn came to teach several graduate courses. I attended his course in Non-linear Partial Differential Equations.

We were delighted to have a German-speaking instructor since we were to take our German language examinations (required for the doctorate). He, in turn, refused since it was that term that he was to take his examination for US citizenship !

Max would hold forth in the Rathskellar (the only campus beer establishment within the Big Ten Universities). We learned more in the beer stube - such as his personal life. He was the black sheep in his family. So long as he remained in his university, he was supported. Thus he received many doctorates.

He would declaim in Greek, some passages from the classics, beer stein in hand. At the time for setting out final grades, he merely called the attendance and one by one asked essentially for the titles of the chapters in the textbook. Clearly we could glance ahead to be assured of the correct answer. I believe he gave something like 18 A's and 3-4 B's.

Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination.

Dehn retired at the end of the academic year 1951-1952 and was made Professor Emeritus. He continued to work at Black Mountain College, giving advice to staff and students. Shortly after supervising the removal of a number of dogwood trees from the college campus, he became ill and died of an embolism. He is buried in the woods that he loved on the Black Mountain College campus.