Szmielew began undertaking research, advised by Tarski, while she was still studying for her first degree and the two became friends. In August 1938 Tarski went to Zakopane for his summer holidays and he invited Szmielew and her husband to join him for a week in August when they all went on hikes in the Tatra mountains. Szmielew produced her first important result in 1938 concerning the axiom of choice for finite sets. She wrote this up but, before it could be published, Germany invaded Poland on Friday 1 September 1939. Advancing at a rapid rate, despite brave resistance by Polish forces, Warsaw fell on 27 September and the country was forced to surrender two days later. Szmielew's studies at the University of Warsaw came to an abrupt end. Tarski, by good fortune, had travelled to Harvard University in the United States to attend a meeting and was there when war broke out. He was given permission to remain in the United States. Adolf Lindenbaum was killed by the Nazis in 1941. Jan Łukasiewicz suffered great hardships during World War II and fled from Poland with his wife. The occupying German forces closed the University of Warsaw and the main campus was turned into a military barracks.
During World War II Szmielew worked as a surveyor. However, she continued to undertake research on her own and, during these years she proved the decidability of the theory of abelian groups. The closing of most educational institutions in Poland by the Germans did not stop the Poles organising underground educational establishments and, despite the obvious danger of taking part in these illegal activities, Szmielew engaged in underground teaching. In January 1945 Lodz was liberated by the Soviet Army and, in May 1945, the University of Lodz was founded built on several educational institutions which had operated in Lodz before the Germans invaded in 1939 and on the underground university that had operated during the war. Szmielew was appointed to the new University of Lodz and she also taught at the Lodz Institute of Technology. Her first paper, On choices from finite sets, written in 1938 but unpublished because of the war, finally appeared in Fundamenta Mathematicae in 1946. She had been unable to complete her undergraduate degree at Warsaw because of the war so now she worked to obtain a first degree and she was awarded an M.A. by the University of Warsaw in 1947. After the award of the M.A. she was appointed as a senior assistant to the Chair of Mathematics at the University of Warsaw. She attended the Tenth International Congress of Philosophy in Amsterdam in August 1948 and reported on her work on the decidability of the theory of abelian groups. She published the 3-page paper Decision problem in group theory in the Proceedings of the Congress. It was reviewed by Richard Milton Martin who wrote:-
This paper is concerned with the positive solution of the decision problem in the elementary (first order) theory of Abelian groups. (The solution of this problem for the elementary theory of arbitrary groups is known to be negative.) ... it is shown that there exists a method which enables us to decide in each particular case whether a given sentence (of the theory under consideration) is satisfied by every Abelian group. Another form of this theorem is also arrived at by a different method.Szmielew worked at the University of Warsaw for two years before moving to the United States to complete her doctorate advised by Tarski.
We mentioned above than Tarski had been lucky enough to be in the United States when Germany invaded Poland in 1939, but his wife and two children, a son Jan and a daughter Ina, were both in Poland. They were unable to leave during the war despite Tarski's strenuous efforts to get them out but after the war, in 1946, with Tarski in a permanent post at the University of California at Berkeley they were able to join him. To illustrate the danger his family had been in, we note that Tarski's father, mother, brother and sister-in-law all died at the hands of the Nazis during the war. Once his family had joined him, Tarski set about arranging an invitation for Szmielew to come to the University of California at Berkeley. Given the results she had obtained during the war, she was confident that she could complete her doctorate within a year. The authors of  related the events that followed:-
[Szmielew] arrived in the United States in 1949 as the first Pole to come to Berkeley under his sponsorship; given the Cold-War political situation, this must have taken some powerful intervention in both countries. ... Liberated, mysteriously beautiful, and independent, Szmielew set about doing what she wished with little concern for the reaction of others. Unconstrained by convention, she came to Berkeley, leaving her husband in Poland, and accepted Tarski's invitation to live in his house with his family as a guest for the year. On the face of it, this was quite reasonable. Wanda had a grant with limited funds, she had limited time, and she did not have a car; so it made eminently good economic sense for her to be right there. Furthermore, she was going to be working closely with Tarski, whose habit it was to work into the early hours of the morning. Jan's bedroom, adjacent to Tarski's study, was made hers and Jan [Tarski's son] moved to a room in the basement. He welcomed the change at first - at fifteen he was happy to have a secluded place to himself- but he was not happy when it became clear that the rapport between Alfred and Wanda went far beyond a shared interest in mathematical logic. Since Tarski was in the habit of doing as he wished at home without having to justify his actions, it is difficult to know whether he had any misgivings about creating this menage. He had had mistresses before and would have others in the future; it was, after all, part of the culture in Europe - especially among intellectuals, artists, and freethinkers. No one made a fuss, particularly if there was some discretion. The problem in this instance was the blatant lack of it. Why did Maria [Tarski's wife] put up with this? The simple answer is: She did not think she had much choice. In the United States she had no income of her own and, at first, no language competency; she was completely dependent upon her husband. ... in 1950, in the Cragmont Avenue house, she tried her best to live with a humiliating situation and keep her emotions under control. Periodically, though there would be a crisis, a boiling over, usually precipitated by Szmielew's presumption of privilege and the expectation that Maria would take care of domestic tasks for her, as if Alfred's wife were the housekeeper rather than the lady of the house. At those moments, Ina (who was then twelve) recalled Maria would blow up and there would be loud bitter arguments and recriminations, "shouting and crying; everyone in the family hearing everything." Eventually the heat would subside, and all would simmer down until the next crisis. But both Ina and Jan retained an everlasting resentment and dislike for Wanda Szmielew because of her insensitivity toward Maria as well as toward them.Szmielew's 250-page doctoral dissertation, Arithmetical properties of abelian groups, earned her a Ph.D. in 1950. This was an outstanding piece of work which was published as the 70-page paper Elementary properties of abelian groups in Fundamenta Mathematicae in 1954. She was awarded the Prize of the Minister of Higher Education in 1956 for this impressive paper. It was :-
... an essential contribution to the foundations of algebra and inspired the future research of Abraham Robinson and the pupils of Anatoly Malcev, as well as some other scholars.Dana Scott, reviewing Szmielew's paper in , writes:-
So many results on undecidability have been proved in the last few years that it is refreshing to see a theorem on the positive side again. In this careful study it is shown that there is a (primitive) recursive decision method for the first-order theory of Abelian groups. The exact algorithm is not given, but the details of the process of eliminating quantifiers and reducing sentences to a normal form are contained in the proofs and can be effectively extracted from the paper. In any case, one would hardly wish to carry out the decision method, and knowledge of its existence is satisfying enough. Instead, the metamathematical implications of the result are more to the point, and the emphasis of this paper lies in that direction. The main accomplishment is an exhaustive description of all finitely axiomatizable extensions of the theory of Abelian groups in terms of certain basic sentences expressing reasonably standard group-theoretical properties. Further all complete extensions of the theory are found together with an (infinite) set of algebraic invariants for explicitly telling whether two Abelian groups share the same first-order properties. ... The mode of presentation uses in an essential way Tarski's notions of elementary (arithmetical) functions and classes. A very good summary of the pertinent facts about these concepts is supplied. ... the author deserves credit for having carried the plan through successfully.After the award of her doctorate, Szmielew returned to the University of Warsaw where she was promoted to assistant professor. Further promotions followed around the time her daughter Aleksandra Szmielew was born on 18 December 1954. She worked closely with Karol Borsuk who had played a major role in reconstructing the University of Warsaw after World War II. He had been made a professor in 1946. Szmielew and Borsuk published the book The foundations of geometry (Polish) in 1955. Herbert Busemann writes in a review:-
This book deviates from other works with the same title both in content and in the extent to which details often omitted in other books are carried out. It begins with an introduction on the history of the subject and on the very elements of point-set topology. The first four chapters are concerned with absolute geometry, in particular with the axioms of incidence and order. ... An English edition of this book might well be an answer to the problem of how to acquaint young students with exact reasoning, presenting at the same time a coherent course.Indeed an English edition under the title Foundations of geometry, Euclidean and Bolyai-Lobachevskian geometry, projective geometry was published in 1960. Donald Coxeter writes that the book :-
... was re-written for the English edition with many improvements, e.g. Hilbert's axioms have been replaced by simpler ones along the lines of Eliakim Moore and Oswald Veblen.In 1956 Tarski visited Szmielew, Borsuk and Mostowski in Warsaw :-
By then, Szmielew had a daughter, Aleksandra, who was two years old. Tarski would form an attachment to her as she grew older, and he always asked about her in his letters to Wanda. Tarski had professional business to attend to as well; he was determined to do all he could to bring Polish logicians to the United States and especially to Berkeley for conferences and for longer stays. It was at this time that he set the wheels in motion for Szmielew and Borsuk to attend the Axiomatic Method Symposium planned for 1957 ...Szmielew attended the Berkeley Symposium, held 26 December 1957 to 4 January 1958, and gave the lecture Some metamathematical problems concerning elementary hyperbolic geometry. On the same theme, she announced important results (without proofs) in her 1959 paper Absolute calculus of segments and its metamathematical implications. Dana Scott writes:-
Absolute geometry consists of those theorems common to both the Euclidean and hyperbolic theories. However, without distinguishing between the two cases the author builds up the theory of proportion (except, of course, for the results depending on the existence of the fourth proportional which fails in hyperbolic geometry) in a new way which thereby leads to the absolute calculus (or algebra) of (equivalence classes of free) segments.Further papers on the foundations of geometry followed. Maria Semeniuk-Polkowska, who studied at the University of Warsaw in the 1960s, wrote:-
For a short time I wavered between choosing a seminar on the foundations of geometry, which Professor Szmielew encouraged me to take, and that on foundations of mathematics by Professor Rasiowa. I admired greatly both their beautiful lectures, and I was under their good influence.The authors of  write that:-
Wanda was bold, determined, self-assured, and independent, as well as beautiful, and Tarski admired all these attributes. One gets the impression that once she was launched on her own, Tarski was the one trying to please her and not the other way around. He courted her favour and was most solicitous of her until her death. While she was ill he helped her buy a car and sent her money for a vacation in Italy ...The illness mentioned in this quote was cancer and Szmielew was already suffering badly from the disease when, in 1974, she decided to write a book giving her approach to the foundations of geometry which, by that time, had taken a rather different line to the views of Tarski. Her colleague Maria Moszynska wrote:-
[Szmielew] was no longer able to carry on her university duties. During the next two years she worked intensively. The monograph soon began to live its own life, to develop beyond the initial framework. But by May 1976 it became clear that Wanda Szmielew had neither enough time nor strength to complete the work. [After Wanda's death] I tried to change only what was absolutely necessary, so as to preserve as far as possible the original text.The book was published in Polish in 1981 and an English translation under the title From affine geometry to Euclidean geometry. An axiomatic approach was published in 1983. Leslaw W Szczerba writes in a review:-
The main theme of the book is the interrelations between geometric systems and corresponding algebraic theories. Mutual interpretability of several geometric theories and their algebraic counterparts is established. All geometric theories are treated as one-sorted elementary theories: theories about points. The only exceptions are parts of Sections 7 and 8 of Chapter 2, where Archimedean and continuity axioms are discussed, and therefore a non-elementary language is admitted. ... The book has been published posthumously, and this is apparent in Part two and especially in Part three. There is an abundance of open problems concerning the material, as well as a lot of material relevant to the book, known even from earlier publications of the author, which has not been included.Victor Pambuccian, who is writing a book about the axiomatic foundation of geometry which will survey all contributions made from the 1882 beginning of the modern axiomatisation of geometry to the present, also reviewed Szmielew's book:-
This book leaves the reviewer with a feeling such as one experiences watching the throwing of the javelin at the Olympic Games. Every movement seems simple, natural, the way it has to be. One almost forgets that behind this apparent ease and grace there are years of hard work, of sweat and tears ...Szmielew's daughter Aleksandra, asked about her mother's character, said she was :-
... authoritarian, towards me, towards everyone, and she was never worried much about what other people thought of her.The authors of  write:-
She expended a great deal of care and ingeniousness on teaching. It was typical her to strive for beauty and perfection in everything. She gave her best both in her lectures and in directing the work of others. She was extremely sensitive to other people. She tried to instil into her pupils her own passion for creative work, her perseverance and lore of method and precision, and also her rare gift for organizing work. For her pupils she was always the best friend and protector.
Article by: J J O'Connor and E F Robertson