Lyudmila Vsevolodovna spent time in several different cities as the family moved around. They were in Helsinki in 1905 and St Petersburg in 1907. By 1909 they were in Riga where Vsevolod Mikhailovich lectured at the Polytechnic Institute. In 1915 the German army invaded Latvia and the staff at the Riga Polytechnic Institute were evacuated to Moscow. The Keldysh family had no relations to help them in Moscow and they suffered considerable hardship. For several years they lived in Losinoostrovskaya, just outside the city, but since the parents loved classical music they often attended concerts in the city. After these concerts, they had no alternative but to make their way home on foot, a journey of several miles. Lyudmila recalled the difficult times in Losinoostrovskaya, remembering one day when her mother fed the whole family just fried onions, having no other food. She completed her school education in Ivanovo-Voznesensk, a city about 250 km northeast of Moscow. The family had moved there towards the end of 1918 when Vsevolod Mikhailovich began teaching at the Polytechnic Institute - this Institute was joined to the Riga Polytechnic Institute. However, he often returned to Moscow to work on engineering projects.
Nikolai Nikolaevich Luzin gave a lecture in Ivanovo-Voznesensk which Lyudmila Keldysh and other high school pupils attended. This lecture firmly established in Keldysh's mind that she wanted to study mathematics and, if possible, become a professional mathematician like Luzin. He had been appointed as Professor of Pure Mathematics at Moscow State University in 1917 and, by the time he gave the lecture, had built an impressive research group with students P S Aleksandrov, M Ya Suslin, D E Menshov and A Ya Khinchin. Keldysh enrolled in the Physics and Mathematics Department at Moscow State University and graduated in 1925. She had joined Luzin's research seminar in 1923, the same year as Petr Sergeevich Novikov joined the research group. The authors of  describe Keldysh in the family's Moscow apartment (where they moved in 1923):-
Young people often gathered at the father's family home. They were mainly friends of the oldest daughter, Lyudmila, students and postdoctoral fellows, mathematicians, among whom were P S Novikov (her future husband), P S Aleksandrov, A N Kolmogorov, I V Arnol'd (who would become a known mathematician and the father of a future member of the Academy of Sciences) and others. There were also musicians, friends of the son Yurii. There were games, laughter, and serious conversations. The strong personality of Lyudmila Keldysh influenced the people around her. Her brother Mstislav, a future president of the Academy, followed her advice when he chose mathematics against the desire of his father that he should become a construction engineer, like his father.Her first important mathematical result did not appear in one of her own publications :-
Her name became known after her initial result, which was first published in 1930 in Luzin's lecture notes [Leçons sur les ensembles analytiques et leurs applications Ⓣ(1930)]. This was a construction of an arithmetic example (using continued fractions) of a set belonging to the fourth Borel class. It was the first significant advance in the study of the Borel classification since the first non-trivial example of a third-class set was given by Baire in 1905.From 1930 to 1934 Lyudmila Keldysh taught at the Moscow Aviation Institute. Her first son Leonid Veniaminovich Keldysh, from a first marriage, was born in April 1931 - he became a famous physicist and member of the USSR Academy of Sciences. She continued working in Luzin's research group and also began working at the V A Steklov Mathematical Institute of the USSR Academy of Sciences as soon as it moved from St Petersburg to Moscow around 1934. It was in 1934 that her first papers were published: On the Homeomorphism of Canonical Elements of the 3rd Class; On Simple Functions of Class a; and On the Structure of B Measurable Functions of Class a. She married P S Novikov, her second marriage, in 1935 but these were difficult years. Joseph Stalin had begun his attacks on the peasants in 1928 but in 1934 he launched an even more terrifying purge of party members who he considered his rivals. Several members of Keldysh's family became victims - an uncle on her mother's side and his son were murdered. Her mother was arrested and her father was held for questioning in 1935 but both were released. However, for a long time after this the family lived in fear, terrified if their doorbell rang in the evening. Over the next few years Keldysh and Novikov had two sons, Andrei Petrovich and Sergei Petrovich (born 1938). Both became famous mathematicians but Andrei Petrovich, who was an expert in algebraic number theory, died at a young age. Sergei Petrovich Novikov won a Fields Medal and, naturally, has a biography in this archive.
Keldysh continued to undertake research on the structure of Borel sets and in 1941 she defended her doctoral thesis Structure of B-Sets (Russian). This 76-page thesis was published in 1945, the delay being due to German invasion of Russia during World War II. Although World War II started in September 1939 with the German invasion of Poland, this had little effect in Moscow. However, on 22 June 1941 the German armies invaded their former allies pushing rapidly east into Soviet lands. At first their main advance was aimed towards Moscow and evacuations took place. We give a fairly long quote from  where Keldysh's struggle over the next few years is vividly described:-
On the critical day of 16 October 1941, she had to leave Moscow with her three sons. The oldest, Leonid ..., was ten years old and the youngest, Sergei, three. It was the last possibility to get on the train going eastward from Moscow (although nobody could tell to what town). Only with the help of Kolmogorov, who presented them as members of his own family, did they succeed at the last moment in finding room on the train carrying away the academic institutes. First they appeared in Gorkii, then several days later they arrived in Kazan on the Volga river. Here the evacuated institutes of the Academy of Sciences were settled. However, Keldysh and her children did not have the status of evacuees but only of refugees. They were without belongings, without warm clothes for the winter, without the guarantee of lodging, practically without any means to exist. They eventually found a roof over their heads: they were put up in the large gym of Kazan University together with several hundred similar refugees. A month later Novikov arrived in Kazan; he was ill and in grave condition. They were given a room in the hall of the student dormitory. Surgery was recommended for Novikov; the doctors were not sure about its outcome. Keldysh spent the next several months of the cold and hungry winter of 1941 and spring of 1942 between the room in the dormitory, where the children were awaiting her, and the hospital, where she nursed her husband. The distance was approximately 10 km and there was no transportation in this district of Kazan. The daily ration per person was then 300 grams of bread and a teaspoon of granulated sugar. The money in the family was only enough to buy a sack of coarse-ground rye flour at the market once a month. This flour, cooked in water and called zatirka, became the main meal of the family for the next two years. After they returned to Moscow, life got better little by little.After returning to Moscow, two daughters were born. The elder daughter was Nina, the younger being Elena who went on to study humanities and taught French and Portuguese at the Moscow State Institute of International Relations. The end of World War II marks a time of change of topic for Keldysh who moved from her research in set theory to look at more topological topics. First she published the paper Sur la structure des ensembles mesurables B Ⓣ (1944) which summarised her very considerable contributions to the structure of sets of higher Borel class. A Russian monograph published in 1945 also gave a detailed description of the results she had obtained over several years concerning the classification of Borel sets in the space of irrational numbers lying in (0, 1). Then she published the paper Open mappings of A-sets (Russian) in 1945 which in some sense bridges the gap between her set theoretical work and her topological research. Over the next few years she published papers such as Continuous Mappings of a Zero-Dimensional Compact Set (1947) and Continuous Mappings of a Segment onto an n-Dimensional Cube (1949). This research, looking at Peano curves, was motivated by a conjecture by Pavel Sergeevich Aleksandrov, the head of the Moscow topology seminar.
Sergei Adian writes in  about Keldysh at this time:-
At home Lyudmila Vsevolodovna was a real homemaker. She successfully combined domestic concerns with her fruitful scientific work. There was a rare mutual understanding between them in all questions, scientific or worldly. [She and her husband Petr Sergeevich Novikov] had a very simple lifestyle in the family, no excesses, simple furniture, only interesting pictures on the walls, and warm human relations. Petr Sergeevich and Lyudmila Vsevolodovna were enthusiastic about literature, both Russian and foreign. After lunch or after supper, a relaxed discussion of new publications usually sprang up, often with the participation of guests.In the late 1950s Keldysh organised her famous seminar on geometric topology, concentrating in particular on topological embeddings, at the V A Steklov Mathematical Institute. In 1964 she was appointed as a professor at Moscow State University. She published the book Topological imbeddings into Euclidean space (Russian) in 1966 which she wrote to help her research students gain the necessary background to enable them to read current papers on the topics discussed in her seminar. In the same year she published the paper Topological embeddings in Euclidean space writing in the Introduction:-
Here we give an account of the work on geometric topology which has been carried out in our seminar at the Steklov Institute (Moscow). It can be divided into three parts: structure of the homeomorphism group of Rn and locally flat embeddings of manifolds in Rn; topological embeddings of manifolds, polyhedra and compacta in Rn; monotone mappings of manifolds.Indeed her seminar proved very important in the development of mathematics in Russia and it continued until 1974. One of her students, A V Chernavskii, who was the author or joint author of nine of the articles referenced below, wrote about the L V Keldysh seminar when he joined it in the early 1970s:-
... we began to organize our workshops systematically in various parts of the Moscow countryside, combining topology, football or skiing depending on the season, anecdotes, serious conversations about life, and so on.In 1974 one of Keldysh's students was expelled from the Institute and, as a protest, Keldysh left the Steklov Mathematical Institute. By this time, however, Petr Sergeevich Novikov was seriously ill and the strain of the two years leading up to his death in January 1975 undermined Keldysh's formerly robust health. She died a year later and was buried with her husband in the Novodevich cemetery. It is fitting that the stone over their grave displays the same simplicity as they showed in their lives. It reads:
Let us note her interests outside mathematics  by giving a quote from the article that honoured her 70th birthday:-
She is a person of the most various interests. She makes a point of not missing the exhibits of artists who interest her. She can climb up a rickety wooden staircase to the top of a Gothic cathedral. She loves to spend her holidays at tourist centres, she loves old architecture, and she loves nature.Among the honours that Keldysh received we mention the Red Banner of Labour, the medal "Maternal Glory", and the Prize of the Presidium of the USSR in 1958. In August 2004 the conference Geometric Topology, Discrete Geometry and Set Theory was held in Moscow to mark the centenary of her birth.
Article by: J J O'Connor and E F Robertson