Obituary of Tadeusz Ważewski

Czesław Olech, Jacek Szarski and Zofia Szmydt wrote an obituary of Tadeusz Ważewski which was published as: Tadeusz Ważewski, Annales de la Société Polonaise de Mathématique 29 (1974), 1-13. The obituary was written in French. We give below an English version of the obituary.

Tadeusz Ważewski (1896-1972)

On 5 September 1972, Tadeusz Ważewski, a famous mathematician, professor at the Jagellonian University in Kraków, one of the founders of the Mathematical Institute of the Polish Academy of Sciences, member of the Polish Academy of Sciences, founder of the Kraków School of Differential Equations, died in Zaryte.

Tadeusz Ważewski was born on 24 September 1896 in Wygnanka. Having finished high school in Tarnów in 1914, he began his studies at the Jagiellonian University. He first studied physics and then took up mathematics under the supervision of Stanisław Zaremba. His interest was attracted by set theory and topology, which were developing at that time, but were not cultivated in Kraków. In 1920, his work on a singular continuum was published [1]. In the period 1921-1923 Ważewski continued his studies in Paris. In 1924, he received the degree of Doctor of Mathematical Sciences from the University of Paris, defending his thesis on so-called dendrites [3], i.e. locally connected continua containing no simple, closed curves. The members of the review committee were: E Borel, A Denjoy and P Montel.

In 1927 Ważewski passed his professorship examination at the Jagiellonian University, presenting the thesis on rectifiable continua [4]. His subsequent works were already concerned with mathematical analysis. Among his early works in this field there are two that are particularly worthy of mention. In one, [8] in these notes, Ważewski introduced the notion of asymptotic Jacobians and in the other [12] he used it to prove the theorem on the change of variables in the multiple integral without the assumption that the transformation is invertible.

In 1933 Ważewski was appointed as an extraordinary professor at the Jagiellonian University. During this period he began to take an interest in differential equations, the field of mathematics in which the greatness of his scientific discoveries was to be revealed. This change in his field of research was partly due to the influence of Stanisław Zaremba, but it was mainly due to Ważewski's interest in natural phenomena and the need he felt to put these phenomena into mathematical form. Those areas of mathematics that are related to other branches of mathematics and to other sciences were considered by Ważewski to be the most important. In one of his lectures he said: "Nature has a mathematical character. Without mathematics, it would be impossible to give the laws of nature a quantitative form. It is precisely this quantitative form that alone can be of service to technology."

Ważewski was an extremely astute researcher. He knew how to penetrate deeply into the nature of the problem, to highlight its essential meaning and to discover analogies between seemingly distinct problems. Thanks to these qualities Ważewski became an eminent specialist in the field of differential equations, a field which he enriched until the end of his days with ideas of extraordinary originality and in which he began new directions of research.

Among his works on differential equations published before 1939 there are three particularly interesting ones, in which Ważewski applied, for the first time, in a systematic way the theory of ordinary differential inequalities to that of first-order partial differential equations. The first note [17] concerns a general criterion, of the Kamke type, of the uniqueness of the solution of the Cauchy problem for nonlinear first-order partial differential equations; the second note [30] contains the effective assessment of the ___domain of the existence of such a solution under the weak regularity hypotheses. The third note [33], concerning the existence of the solution of the Cauchy problem for a diagonal system of first-order partial differential equations, contains the first result on this subject in the mathematical literature. Before 1939 he also published the work [16] giving a negative result concerning the global existence of the first integral for the ordinary differential equation. Ważewski notably constructed the differential equation

y' = f(x, y)

, whose right-hand side is of the class

C^{∞}

in a ___domain

\Omega

, for which every first integral of class

C^{1}

\Omega

is necessarily constant. After the war this result was the starting point for numerous publications by Ważewski's students.

In October 1939, Tadeusz Ważewski was among the group of professors from Kraków's higher education institutions who were arrested by the German fascists and then transported to the Oranienburg concentration camp, where many of them lost their lives. Kazimierz Stołyhwo, a renowned Polish anthropologist who was also a prisoner in the Oranienburg camp, spoke about Ważewski: "I love the city of Kraków very much, because Tadeusz Ważewski lives there, and I do not regret being a prisoner in Oranienburg, because there I met Ważewski." It should be mentioned that due to the interventions of many scholars, Stołyhwo could have been released much earlier, but he refused, saying that he could not leave the camp as long as his colleagues remained there.

After his return to Kraków, Ważewski taught at the underground university, continuing his scientific research intensively despite the harsh conditions of the fascist occupation. His works on the assessment of the ___domain of existence of implicit functions [40] and his fundamental dissertation on ordinary differential inequalities [57] belong to this period. The latter work determined one of the directions of research conducted after the war in the mathematical centre in Kraków.

In 1945 Tadeusz Ważewski was appointed full professor at the Jagiellonian University. During the first years after the war his attention was drawn to the problems of the qualitative path of solutions of ordinary differential equations. During this period Ważewski created a new topological method for examining the qualitative appearance of solutions [48]. This method, involving the notion of a retract, which had been introduced by Karol Borsuk, is based on a simple but profound idea and has proven to be very fruitful in applications. We will illustrate it with an elementary example.

Consider a system of two differential equations

(1)

\Large\frac{dx}{dt}\normalsize = f(t, x)

where

x = (x_{1}, x_{2}), f = (f_{1}, f_{2})

, whose solutions are uniquely determined by the initial conditions in an open set containing the parallelepiped

(2)

-∞ < t < ∞, |x_{1}| ≤ 1, |x_{2}| ≤ 1

Let us assume that

x_{1}f_{1} > 0

for

|x_{1|} = 1

while

x_{2}f_{2} < 0

for

|x_{2}| = 1

. These assumptions mean that by the opposite faces of the parallelepiped (2):

A

on which

x_{1} = 1

and

B

on which

x_{1} = -1

, the integrals of the system (1) leave the set (2), while by the other two faces they enter it. Ważewski's theorem says that under these conditions on each segment joining the faces

A

and

B

there exists a point

p

such that the integral from

p

remains in the set (2) for

t \rightarrow +∞

. The reasons why this is so are quite simple. In the example considered, the integrals of the system (1) coming from the interior of the set (2) can only leave it by the faces

A

and

B

. Let us choose on these faces two points

a \in A

and

b \in B

. If the integral coming from a point

p

of the set (2) leaves it for the first time by the face

A

, then let us make point

a

correspond to

p

; if it leaves it for the first time by the face

B

, then let us make point

b

correspond to

p

. In this way we have defined in a subset

E

of (2) a continuous mapping. The intersection of

E

with the segment joining the points

a

and

b

is different from this segment; indeed, otherwise our map would transform the segment on its ends in a continuous way, which is impossible. Consequently, there exists a point

p

of this segment such that the integral coming from

p

does not leave the set (2) for

t \rightarrow +∞

.

The retract method, due to T Ważewski, proved to be very effective in the study of asymptotic problems of the theory of ordinary differential equations. It became the starting point for the investigations of the author himself, his students and specialists abroad. To this day, new generalisations of the method find applications in many new problems, also in the theory of partial differential equations. Ważewski's topological method has entered the monographs and textbooks of the world mathematical literature. At the invitation of the Organising Committee Tadeusz Ważewski gave a lecture on his method at the International Congress of Mathematicians, held in Amsterdam in 1954 [80].

The retract method is one of the greatest achievements of Polish mathematics after the Second World War. According to the opinion of Solomon Lefschetz, a famous American mathematician, expressed in 1961, T Ważewski's retract method is one of the greatest discoveries in the theory of ordinary differential equations after 1945.

It was a characteristic feature of the scientific activity of Tadeusz Ważewski, the author of more than 100 memoirs, that the problems he dealt with belonged to the most topical ones.

Since 1960 Ważewski has published a whole series of works [100]-[106]. [108]-[112] on control systems. These are the differential equations of the form

(3)

\Large\frac{dx}{dt}\normalsize = f(t, x, u)

where u is the control vector. Equation (3) becomes the ordinary differential equation, if we replace the vector u by a function u(t), called the control function. It follows that the initial Cauchy problem for equation (3) admits several solutions and that in general there are as many as admissible control functions. Control systems intervene in optimal control problems, in which the aim is to find in the class of admissible control functions the one which is optimal from the point of view of a criterion given in advance.

Ważewski made the following observation which later proved to be very fruitful. If the point

(t, x)

is fixed, we denote by

F(t, x)

the set which is the image of the ___domain of the control parameter

u

, by means of the transformation

y = f(t, x, u)

then each solution of the system (3) is a solution of the following problem: find an absolutely continuous function

x(t)

satisfying for almost every

t

in a suitable interval the condition

\Large\frac{dx}{dt}\normalsize \in F(t, x(t))

This condition is called a differential equation with a multivalued second member or an equation with orientators. The notion of the orientator was introduced by Ważewski. Thanks to this observation Ważewski linked the theory of control systems with that of contingent equations which was developed before the war by S K Zaremba in Poland and by A Marchaud in France. This led to many important and interesting results obtained by Ważewski himself and his students.

Speaking about Ważewski's scientific achievements, it should be mentioned that his interest in physics, dating back to the beginning of his studies, found expression in his work on the vibrating string [75], [76], [79].

Ważewski not only knew how to grasp essential problems, but through deep analysis he managed to find the proper method of solution, which often proved unexpectedly simple. The concepts and methods he introduced were always extraordinarily fruitful and served as a starting point for investigations and generalisations. Besides the above-mentioned method of retraction and the theory of equations for orienteers, other examples could be given. It should be emphasised that Ważewski also focused his attention on didactic and methodical questions, as well as on the elegance of mathematical proofs. The unification of the proofs of about sixty cases of de l'Hôpital's rule is a striking example of this [49], [52]. Here it should be noted that de l'Hôpital's rule and its generalisations led Ważewski to the theorem on finite increments for a function of the real variable and to values in a Banach space [56]. This theorem served as a starting point for the theory of differential inequalities in Banach spaces and for the ingenious constructions of solutions to differential equations in these spaces [71], [72], [114], [115].

The training of young mathematicians was, besides his personal scientific activity, Tadeusz Ważewski's passion. It was already among the first-year students that Ważewski knew how to find, without any errors, future researchers. Ważewski was a master at finding simple and interesting problems to which he drew the attention of the students by making them undertake investigations on their own. His lectures were extremely lively and fascinated the audience. Interesting and topical problems posed by Ważewski as well as his talent and infectious enthusiasm for mathematics attracted the most gifted students. Ważewski did not ignore any effort of his students, and analysed and appreciated every good idea. In case of any error he indicated other possibilities of treating and solving the problem. Simple and striking ideas found his lively, often enthusiastic reception. This lively reaction of Ważewski to even very simple results was an additional encouragement for his students.

Ważewski's seminars were characterised by a deep and insightful analysis of the issues discussed and the care taken to elucidate all the difficulties. At Ważewski's seminars, if the results of other authors were presented, the aim was always to reveal the essential points and the guiding idea of the theorems, to find their genesis. If the lecturer did not succeed in doing this himself, it was done with the assistance of the seminar participants. In this way, a new and more general conception of the problem was often arrived at, a new and more natural way of solving it was found, and new results were discovered. Ważewski's seminars were a real scientific school.

In his university courses as well as in his scientific lectures, Ważewski insisted on presenting the main idea of the proof, by highlighting the genesis of the problem or the definition introduced. He indicated not only the consequences of a definition but also hinted at the situation where the conditions of a definition are lacking. It is necessary not only - he said - to show students what happens if a property holds, but also what happens if it is lacking. It was this way of presenting their own results and those of other authors that Ważewski demanded from his students. The fact that the result obtained was correct never satisfied him. He demanded a proof which was as simple as possible, the highlighting of the main idea and the indication of the essential hypotheses with the help of properly constructed counterexamples. Ważewski attached great importance to the way in which notes were written and he also taught this to his students. He prepared his students for scientific work as well as for pedagogical work. He watched over the first exercises and lessons conducted by his students, giving them meritorious and didactic advice. Ważewski was a tireless teacher. He devoted a lot of time and feeling to working with students. He cared about their material conditions, provided them with scholarships at home and abroad, and presented them for prizes. He was a very careful patron of his students. He facilitated their entry into the mathematical world. Their participation in congresses and conferences engaged him personally, their successes pleased him.

The establishment of the Kraków School of Differential Equations was the crowning achievement of the scientific and pedagogical activity of Tadeusz Ważewski, a master and scholar. It is an unparalleled achievement in the history of Polish mathematics after 1945. The influence of this school on the development of research in certain branches of the qualitative theory of ordinary differential equations is quite pronounced and distinct.

Several professors, three of whom are already members of the Polish Academy of Sciences, graduated from Ważewski's school. Ważewski's students form a large part of the mathematical centre in Kraków.

The Kraków School of Differential Equations was formed in the years when the country was rising from the ruins of the war, when Polish mathematics was ravaged by the fascist occupation. Ważewski contributed to the reconstruction of Polish mathematics, to its recovery of its high position in the world.

Ważewski maintained scientific contacts with foreign countries, taking part in international conferences and congresses and giving lectures at the invitation of foreign universities. In 1959, for example, he visited the universities of Florence, Rome, Pisa, Genoa, Lille and Rennes, as well as the Collège de France in Paris. Foreign visitors came to get acquainted with Ważewski's school.

To appreciate Ważewski's great merits, we must go back to the period between the two world wars, when the magnificent development of the two Polish mathematical schools began, which has lasted to this day: the Warsaw School of Topology and Set Theory and the Leopold School of Functional Analysis. At that time, only a few mathematicians were working on differential equations in Poland. Among them were such renowned mathematicians as Stanisław Zaremba, Juliusz Schauder and Alfred Rosenblatt, but they worked in isolation. It was therefore not easy to focus the attention of young mathematicians on problems in the theory of differential equations, which were often very difficult problems, and to form a group that would deal with them. This work is due to Tadeusz Ważewski, who knew how to demand and inspire. It was thanks to his initiative and the Mathematical Institute of the Polish Academy of Sciences that research in the field of differential equations began to develop with great momentum in Poland after 1945.

Soon after the war Ważewski was elected a corresponding member of the Polish Academy of Sciences and Letters. From the moment of the establishment of the Polish Academy of Sciences he was appointed a corresponding member, becoming a full member in 1957. In 1959 he was elected president of the Polish Mathematical Society, which in 1967 awarded him the title of honorary member. In the same year he received the title of doctor honoris causa at the Jagiellonian University. He was an editor of the journal Annales Polonici Mathematici. He was awarded the State Science Prize twice. For his merits the State Council awarded him high distinctions: the Knight's Cross and the Officer's Cross of the Order Polonia Restituta and twice the Standard of Labour, first class. In 1949 he received the Scientific Prize of the Kraków Region and then the Gold Medal of the City of Kraków, for which he had so much merit.

Ważewski was a man of great character. Sensitive and shy, he was at the same time tough and determined to fulfil the most difficult duties he had taken upon himself. He knew how to choose the truth that could be told and the truth that should be told. Kindness was allied in his nature with wisdom. He asked little for himself while giving generously to others. In his demeanour he was prodigiously charming. He was also witty, his mind springing from the precision of his critical thinking. Very sensitive to the great works of science and art, he concentrated his energy and insight on mathematics, which was the passion of his life.

He will always remain in the hearts of those who were fortunate enough to work with him and draw from his boundless wisdom and kindness.

Last Updated November 2024