安德列·尼古拉耶维奇·吉洪诺夫Andrey Tikhonov

像大多数俄罗斯数学家一样，吉洪诺夫 Nikolaevich 安德列·尼古拉耶维奇·吉洪诺夫的名字从罗马字母转写有多种方式。除吉洪诺夫 Nikolaevich 吉洪诺夫之外，最常见的写法是吉洪诺夫 Nikolayevich Tychonoff。

吉洪诺夫作为走读生上了中学，并于1922年进入莫斯科大学，这一年他完成了中学教育。他在莫斯科大学数学物理系数学部学习，取得了显著进步，1925年还在本科课程中途时就发表了他的第一篇论文。

这第一部著作与帕维尔·亚历山德罗夫和帕维尔·萨穆伊洛维奇·乌雷松关于topological空间可度量化条件的结果有关。然而他并未止步于此，而是继续在拓扑学中进行研究。到1926年，他已经发现了今天以他命名的拓扑构造，即定义在拓扑空间乘积上的吉洪诺夫拓扑。帕维尔·亚历山德罗夫在[4]中回忆起他当时未能认识到吉洪诺夫这些想法的重要性时，回忆道：-

……他非常清楚地记得，当吉洪诺夫提出那个定义时，他是多么怀疑。一个通过如此巨大的邻域引入的拓扑——这些邻域与整个空间仅由有限个坐标相区别——怎么可能捕捉到拓扑乘积的任何本质特征呢？

吉洪诺夫确实给出了正确的定义，而这个对即使像帕维尔·亚历山德罗夫这样伟大的拓扑学家来说都反直觉的想法，使吉洪诺夫得以继续证明诸如任意一组紧拓扑空间的乘积是紧的这样重要的拓扑结果。

很少有数学家在开始研究生涯之前就赢得世界声誉，但这基本上就是吉洪诺夫的情况。他关于乘积的吉洪诺夫拓扑的结果是在1927年毕业之前取得的。凭借这一令人瞩目的记录，他于1927年成为莫斯科大学的研究生。人们可能会认为，一个显然对拓扑学思想有如此直觉把握的人，会非常乐意在这一领域施展才华。然而，吉洪诺夫在数学的其他领域同样有才华。他的工作范围在[3]中被总结如下：-

我们要感谢吉洪诺夫在现代数学广泛主题中取得的深刻而基本的结果。他在拓扑学和泛函分析、常微分方程和偏微分方程理论、地球物理学和电动力学的数学问题、计算数学和数学物理方面的一流成就都广为人知。吉洪诺夫的科学工作的特点是，在所谓纯数学的非常抽象的领域中取得了辉煌成就，同时深入研究了与实际需求直接相关的数学学科。

事实上，吉洪诺夫的工作从拓扑学导向了泛函分析，他在1935年提出了关于局部凸拓扑空间凸紧子集上连续映射的著名不动点定理。这些结果在拓扑学和泛函分析中都很重要，并被吉洪诺夫应用于解决数学物理中的问题。

他于1936年以Functional equations of Volterra type and their applications to mathematical physics为题通过了教授资格论文（Habilitation）学位论文答辩。该论文应用了埃米尔·皮卡近似求解微分方程方法的推广，并给出了在热传导中的应用，特别是服从约瑟夫·斯特凡和路德维希·玻尔兹曼所给定律的冷却。成功通过学位论文答辩后，吉洪诺夫于1936年被任命为莫斯科大学教授，三年后当选为苏联科学院通讯院士。

吉洪诺夫处理数学物理问题的方法在[14]中有描述：-

吉洪诺夫研究的一个特点是将自然科学中的一个具体主题与对基本数学问题的探讨结合起来。在讨论自然界中的某个一般问题时，他总是知道如何挑选出一个典型的、具体的物理问题，并给它一个清晰的数学表述。然而，他的数学研究从不局限于解决某个给定的具体问题，而是作为提出一个一般数学问题的起点，这个一般数学问题是第一个问题的广泛推广。

吉洪诺夫对数学物理中若干一般问题的极其深刻的研究，源于他对地球物理学和电动力学的兴趣。因此，他对地壳的研究导致了对抛物型方程适定奥古斯丁·路易·柯西问题的研究，以及构造求解维多·沃尔泰拉型一般泛函方程的方法。...

吉洪诺夫在数学物理方面的工作一直持续到20世纪40年代，他因这项工作于1953年获得国家奖。然而，1948年，当他考虑最高阶导数项中含小参数的方程组解的行为时，他开始研究一类新问题。在引入该主题的一系列基础性论文之后，这项工作由他的学生继续推进。

吉洪诺夫做出基础性贡献的另一个领域是计算数学（[11]和[12]）：-

在他的指导下，许多用于解决电动力学、地球物理学、等离子体物理学、气体动力学……以及自然科学其他分支中各种问题的算法被发展出来并付诸实践。……计算数学中最杰出的成就之一是同构差分格式理论，这是吉洪诺夫与Samarskii合作发展的……

20世纪60年代，吉洪诺夫开始发表一系列关于不适定问题的重要论文。他定义了一类可正则化的不适定问题，并引入了正则化算子的概念，用于这些问题的求解。吉洪诺夫将自己的计算技能与求解此类问题相结合，给出了用于计算他在求解这些问题时所使用的算子的算法的计算机实现。1966年，吉洪诺夫因在不适定问题方面的工作而被授予列宁奖。同年，他当选为苏联科学院的正式成员。

吉洪诺夫在数学领域的广泛兴趣使他在莫斯科大学担任了多个不同的讲席，特别是数学物理系的讲席和工程数学系的计算数学讲席。他还成为莫斯科大学计算与控制论系主任。吉洪诺夫被任命为苏联科学院应用数学研究所副所长，并担任该职位多年。

Like most Russian mathematicians there are different ways to transliterate Andrei Nikolaevich Tikhonov's name from the Roman alphabet. The most common way, other than Andrei Nikolaevich Tikhonov, is to write it as Andrey Nikolayevich Tychonoff.

Andrei Nikolaevich Tikhonov attended secondary school as a day pupil and entered the Moscow University in 1922, the year in which he completed his school education. His studied in the Mathematics Department of the Faculty of Mathematics and Physics at Moscow University and made remarkable progress, having his first paper published in 1925 while he was still in the middle of his undergraduate course.

This first work was related to results of Aleksandrov and Urysohn on conditions for a topological space to be metrisable. However he did not stop there and continued his investigations in topology. By 1926 he had discovered the topological construction which is today named after him, the Tikhonov topology defined on the product of topological spaces. Aleksandrov, recalling in [4] how he failed to appreciate the significance of Tikhonov's ideas at the time he proposed them, remembered:-

... very well with what mistrust he met Tikhonov's proposed definition. How was it possible that a topology introduced by means of such enormous neighbourhoods, which are only distinguished from the whole space by a finite number of the coordinates, could catch any of the essntial characteristics of a topological product?

Tikhonov certainly had given the right definition and this idea, which was counterintuitive to even as great a topologist as Aleksandrov, allowed Tikhonov to go on and prove such important topological results as the product of any set of compact topological spaces is compact.

Few mathematicians have gained a worldwide reputation before they even start their research careers but this was essentially how it was for Tikhonov. His results on the Tikhonov topology of products were achieved before he graduated in 1927. With this impressive record he became a research student at Moscow University in 1927. It might be thought that someone who had clearly such an intuitive grasp of topological ideas would be only too pleased to use his talents in that area. Tikhonov, however, had equal talents for other areas of mathematics. The range of his work is summarised in [3]:-

We owe to Tikhonov deep and fundamental results in a wide range of topics in modern mathematics. His first-class achievements in topology and functional analysis, in the theory of ordinary and partial differential equations, in the mathematical problems of geophysics and electrodynamics, in computational mathematics and in mathematical physics are all widely known. Tikhonov's scientific work is characterised by magnificent achievements in very abstract fields of so-called pure mathematics, combined with deep investigations into the mathematical disciplines directly connected with practical requirements.

In fact Tikhonov's work led from topology to functional analysis with his famous fixed point theorem for continuous maps from convex compact subsets of locally convex topological spaces in 1935. These results are of importance in both topology and functional analysis and were applied by Tikhonov to solve problems in mathematical physics.

He defended his habilitation thesis in 1936 on Functional equations of Volterra type and their applications to mathematical physics. The thesis applied an extension of Émile Picard's method of approximating the solution of a differential equation and gave applications to heat conduction, in particular cooling which obeys the law given by Josef Stefan and Boltzmann. After successfully defending his thesis, Tikhonov was appointed as a professor at Moscow University in 1936 and then, three years later, he was elected as a Corresponding Member of the USSR Academy of Sciences.

Tikhonov's approach to problems in mathematical physics is described in [14]:-

A characteristic of Tikhonov's research is to combine a concrete theme in natural science with investigations into a fundamental mathematical problem. In discussing some general problem in nature he always knows how to pick out a typical concrete physical problem and to give it a clear mathematical formulation. However, his mathematical investigations are never confined to the solution of a given concrete problem, but serve as the starting point for stating a general mathematical problem that is a broad generalisation of the first problem.

The extremely deep investigations of Tikhonov into a number of general problems in mathematical physics grew out of his interest in geophysics and electrodynamics. Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type. ...

Tikhonov's work on mathematical physics continued throughout the 1940s and he was awarded the State Prize for this work in 1953. However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest derivative. After a series of fundamental papers introducing the topic, the work was carried on by his students.

Another area in which Tikhonov made fundamental contributions was that of computational mathematics ([11] and [12]):-

Under his guidance many algorithms for the solution of various problems of electrodynamics, geophysics, plasma physics, gas dynamics, ... and other branches of the natural sciences were evolved and put into practice. ... One of the most outstanding achievemnets in computational mathematics is the theory of homogeneous difference schemes, which Tikhonov developed in collaboration with Samarskii....

In the 1960s Tikhonov began to produce an important series of papers on ill-posed problems. He defined a class of regularisable ill-posed problems and introduced the concept of a regularising operator which was used in the solution of these problems. Combining his computing skills with solving problems of this type, Tikhonov gave computer implementations of algorithms to compute the operators which he used in the solution of these problems. Tikhonov was awarded the Lenin Prize for his work on ill-posed problems in 1966. In the same year he was elected to full membership of the USSR Academy of Sciences.

Tikhonov's wide interests throughout mathematics led him to hold a number of different chairs at Moscow University, in particular a chair in the Mathematical Physics Faculty and a chair of Computational Mathematics in the Engineering Mathematics Faculty. He also became dean of the Faculty of Computing and Cybernetics at Moscow University. Tikhonov was appointed as Deputy Director of the Institute of Applied Mathematics of the USSR Academy of Sciences, a position he held for many years.