THE ONLY FEMALE PROFESSOR OF MATHEMATICS SSU EASILY SOLVES "ILL-POSED" PROBLEMS
Galina Khromova started visiting Saratov University as a three-year-old child ... Together with her student father. And her entire future family (her husband, her son, and her grandson) also "grew up" in the university. And also exclusively mathematical.
Mathematics is a mysterious country with unlimited opportunities. Our interlocutor also matches it. If the majority of her colleagues well-posed tasks all their lives, Galina Khromova takes on precisely the “ill-posed” ones. This is the topic of her research in which she has achieved the greatest success. At SSU, she is the first and so far the only woman among its graduates who owns the title of Doctor of Physics and Mathematics in Mathematics. Since the foundation of the university in 1909!
On September 16, Professor of the Department of Mathematical Physics and Numerical Analysis Galina Khromova celebrates her 85th anniversary. Neither we nor she can believe it.
– A point-blank question: are mathematicians born or made?
– What does it mean to be born a mathematician? It means having some kind of special ability. But even if you are born a mathematician, you may not become one. You need appropriate circumstances. For example, I was not born a mathematician. I was just an excellent student all my life, graduated from school with a gold medal. But I became famous for my essays, I also really liked the foreign language, in general, all the humanities subjects. After school, I was even going to enter the Faculty of Philology at Moscow State University. And my relatives promised that if I did not go to Moscow (so as not to live in poverty there, as they believed), they would buy me a piano. And I had a dream to study music, and it overruled. As I remember now, I was sitting, studying piece To Eliza, Beethoven's piano, and I was absolutely happy!
– But mathematics is also an abstract science to some extent. When did you realise that it was yours?
– The first Russian female mathematician Sofia Kovalevskaya believed that mathematics and poetry are very close to each other. The poet must see what others do not see. And the same gift of imagination is required from a mathematician. But I realised that later. And at first, having stayed in Saratov, I decided to enter the Faculty of Physics, the new Department of Radio Electronics, thinking that I would apply for that mostly demanded course. And only at the last minute I applied the documents to the Faculty of Mathematics. Besides, my father graduated from it. At school, I built such self-confidence that I could do anything! So at the university I could not go wrong.
In our family, education has always been in the first place, my main task was to study. My father lived in Syzran. I entered Saratov University when I was already born. In order not to live in a dormitory, the parents exchanged a comfortable three-room flat in Syzran for a room without amenities, but on the other hand, it took only five-minutes to walk to the university. Money did not mean much to us. And then the war began, as a result we lived in it for many years, I even got married living there.
My father often took me with him to the university, since there was no one to leave me with. So I feel in Building like at home, there were physics and mathematics classes took place (then it was divided into two Faculties – of Physics and Mathematics). While my father was taking the exams, I talked with other students, I know all the windows, all the corridors here. A historical photograph has survived, depicting the entire course of my father, who graduated in 1941, in the university courtyard. Many students went to the front and never returned. And then I was only six years old, and I was standing in the front row.
And when I got married, this fact also played a huge role in my mathematical destiny. My future husband and I studied in the same group and got married when we were 4-year students. To say the least, I can say that August Khromov is every inch a mathematician! He is devoted to mathematics, he lives by it. He is a recognized leader of the Saratov scientific mathematical school. And I have lived next to him, seen his success, his interest, and his admiration for great mathematicians. Thanks to this, I have also taken to mathematics, this wonderful science!
Today my husband and I are the oldest professors and scientists at our faculty. Surprisingly, being older than 80 my husband demonstrates outstanding research results. Of course, he laughs about this: it's good, he says, that we are still alive. I am fully aware that he is head and shoulders above me (not only physically!). But he has his own way in science, I have my own.
When I defended my terminal degree, it turned out that I was the first female SSU graduate who became Doctor of Physical and Mathematical Sciences in Mathematics. In 1998, I once had to congratulate an SSU graduate, academician Pyotr Ulyanov on his birthday, and one room on the second floor of the Faculty of Mechanics and Mathematics in Building IX is named after him. And so at the celebration at Moscow State University I presented him a gift, and he later gave me a photo from that event with the inscription “the first female doctor in mathematics from the first male academician in mathematics of the USSR Academy of Sciences and the Russian Academy of Sciences – SSU graduates”.
– Please, tell me about your professors: who taught you and what? Is there such a thing as "Saratov mathematical school"?
– When, after graduation, I worked at the SSU Computation Centre, we carried out contractual work with the Institute of Geophysics. I was the leader of the group. And so they gave me a task, and, as always, I was sure that I would solve it. But I coldn’t do it! There were no methods to solve it then. And while I was tormented with her, I learned that the first works of Russian mathematicians appeared, who substantiated the theory of ill-posed problems. These were outstanding Russian researchers – academicians Andrei Tikhonov, Mikhail Lavrentyev, and Valentin Ivanov. And then the university received an invitation to a conference on ill-posed problems. And so I listened to those lectures in Kiev - and it began ... In the end, I got engrossed by the theory, but it turned out that there were no textbooks or manuals, so I began disassembling everything on my own, I organised a group in the Computation centre and, thus, I opened my way to science.
In 1973, supervised by our prominent mathematician, Doctor of Physics and Mathematics, SSU Professor SSU Nikolai Kuptsov I defended my Ph.D. thesis. My opponent was just one of the founders of the ill-posed problems theory, Corresponding Member of the USSR Academy of Sciences, Lenin Prize Winner Valentin Ivanov, and the review of the leading organisation was signed by Gurii Marchuk, who soon became President of the Academy of Sciences. It was such a famous company!
If in my Ph.D. I applied Ivanov's method to problems that had not yet been solved, then already in my doctoral dissertation I presented my own method and defended it in 1998 at the Institute of Mathematics and Mechanics of the Ural Branch of the Academy of Sciences. So today I give lectures to master degree students at the university using my own materials – I have developed several special courses on the ill-posed problems theory and methods.
As for the Saratov mathematical school, our mathematicians are highly estimated. Many eminent Moscow scientists have graduated from our university. At that time, they even told jokes that a branch of SSU was located in Moscow.
– Is it possible to explain in lay terms what is the theory and methods of ill-posed problems?
– A well-posed problem always has a mathematical solution, it is unique and stable. In a classical university, we solve the problems like that. But researchers also deal with other problems, the initial data of which are obtained as a result of some experimental activity and, therefore, are known only approximately. This is mathematics on the verge of theory and practice. For example, I suggest that students solve a school problem: an airplane is flying, its path is known, and you need to determine the speed. So, to solve it, I tell them, a whole dissertation has been written. Because the given function is "not smooth", as in the textbook, since in reality the device tracks the plane's path and draws a "saw" instead of a smooth line. And there are a lot of such ill-posed problems related to astrophysics and biophysics. They can arise during geophysical, geological, astronomical observations and in solving problems of optimal control and planning. Now we know how to solve them. And I am developing these methods. This is a new theory that is applied to practical problems.
– In this case, can I ask you an ill-posed question? Will you answer anyway? The role of mathematics in global digitalisation must increase. But as its result we may create artificial intelligence (AI). Will it replace all mathematicians at once in the future or not?
– It’s a very difficult question even for specialists dealing with this problem. I can only quote a well-known aphorism on this matter: "The real danger is not that computers will start thinking like people, but that people will start thinking like computers."
– It is known that Rector of Moscow State University, academician Viktor Sadovnichy, likes to tell jokes about mathematicians. Do you have any of your own, especially a favorite one?
– Somehow jokes slip my mind. I'd rather again quote the statement of Sofya Kovalevskaya, which I like a lot: mathematics is a magical land, mysterious and beautiful, the entrance to which is available only to the elite. In this respect, I was lucky!
