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4th European Women in Mathematics Summer School
HISTORY OF MATHEMATICS
Gendered working realities in Enlightenment mathematics
In the Enlightenment, different patterns for women to participate in the mathematical life can be described. The most visible, and the only one studied by traditional scholarship, is best exemplified by Émilie du Châtelet, an aristocrat who has an impressive scientific record, even if she published anonymously. The puella docta or exceptionally gifted and trained virgin offers another pattern, of which Maria Gaetana Agnesi’s life is a well studied example. In recent years, history of science has focused on spaces which are not primarily dedicated to scientific enquiry. Thus domestic spaces have been considered as sites of knowledge production. This inclusive approach has put to the fore a host of scientific workers – spouses, children, domesticity – who have contributed, by their most often invisible working force, to the scientific enterprise, and among them women, even if they were excluded from scientific institutions (academies and universities). In my talk in Leiden, I will present this approach and use it in order to investigate forms of mathematical work in the Enlightenment, where women’s activities were not absent, even if their achievements remained largely unacknowledged (Madame de l’Hôpital, Clairaut’s cook, etc).
"A woman may win the victory, though she may not wear the wreath": Women and mathematics in late 19th century Cambridge
In 1880 Charlotte Scott caused a sensation by being the first woman to gain success in the Cambridge Mathematical Tripos, being ranked equal to eighth wrangler (i.e. having marks equivalent to those of the male student who had come eighth in the order of merit). Ten years later Philippa Fawcett caused an even greater sensation by being ranked above the senior wrangler. Their remarkable results, which were widely published in the local and national press, added fuel to contemporary debates on the emancipation of women. Nevertheless, despite their success and the success of others who followed them, it was not until 1948 that women could be awarded degrees at Cambridge. In my talk I shall discuss the achievements of women who studied mathematics in late 19th century Cambridge, putting them into the broader context of the Cambridge mathematical culture of the period.
Annette B. Vogt
From Nina K. Bari to Paulette Libermann - Women in mathematics in the 20th century
The talk gives an overview on women mathematicians in the 20th century from a comparative perspective. The situation for women mathematicians in different countries will be described, from Russia/Soviet Union, other European countries, to the USA. Also the problem of different scientific cultures in these countries will be described. In the talk I will discuss some aspects of the development as well as some problems like political and economical ones, and I will describe some barriers in institutions and thought systems. I will also discuss the broader context of the mathematical culture between 1900 and the 1990s. In the talk there will be given some examples of women mathematicians and their achievements, like Emmy Noether (1882-1935) and Hilda Pollaczek-Geiringer (1893-1973) in Germany and later in exile, Nina K. Bari (1901-1961), Pelageja Ja. Kochina (1899-1999), Olga A. Ladyshenskaja (1922-2004) and Olga A. Oleinik (1925-2001) in the USSR, Cecilia Krieger (1894-1974) and Evelyn M. Nelson (1943-1987) in Canada, Julia Bowman Robinson (1919-1985) and Emma Trotskaia Lehmer (1906-2007) in the USA, and Paulette Libermann (1919-2007) in France.
Progress in Certified Mathematics
Progress in the foundations of mathematics has made it possible to formulate all thinkable mathematical concepts, algorithms and proofs in one language and in an impeccable way. This not in spite of, but partially based on the famous results of Gödel and Turing. In this way statements are about mathematical objects and algorithms, proofs show the correctness of statements and computations, and computations are dealing with objects and proofs. Interactive computer systems for a full integration of defining, computing and proving are based on this. The human defines concepts, constructs algorithms and provides proofs, while the machine checks that the definitions are well-formed and the proofs and computations are correct. Results formalized so far demonstrate the feasibility of computer mathematics. There are very good applications. The challenge is to make the systems more mathematician-friendly, by building libraries and tools. The eventual goal is to help humans to learn, develop, communicate, referee and apply mathematics.
Languages, logics and automata
The theory of formal languages has a strong tradition in theoretical computer science, and was developed for a large variety of discrete structures - such as words, trees, partial orders, pictures, graphs, etc. One of the most beautiful parts of these theory is the relationship between automata and logics. The starting point of this topic goes back to the early sixties, to the work of Buechi and Elgot about the equivalence between automata and monadic second order logic.
This research was motivated by decision problems for various logics and by Church's circuit synthesis problem. Monadic second order logic over various structures was shown to be decidable, and these results are based on the conversion of logics into automata. This conversion has also applications in the area of automated verification.
In this lecture I will introduce the notion of finite automaton over words (finite and infinite)
and present the relationships with first order logic and monadic second order logic. Then I will discuss the case of trees and its applications to the synthesis problem and to games.
Model theory and applications
The field of model theory is relatively young within mathematics. Its origins may be traced back 100 years or more, but most of the activity began in the latter half of the twentieth century, led by Tarski, A. Robinson, and Mal'cev. In model theory we study mathematical structures from the point of view of first-order logic. Information can be obtained about the structures by examining which sets are first-order definable. The most fundamental tool of model theory is the compactness theorem, which tells us that a set of first order sentences is satisfiable in a structure just in case each finite subset is satisfiable. In this tutorial we will describe the model theoretic approach to mathematics, making use of specific examples to illustrate how model theory provides information concerning bounds for certain functions, and establishes transfer theorems between structures. We will also indicate how the model theory of enriched fields, such as fields with an ordering, derivation, valuation, or automorphism, provides a setting in which to address certain questions in number theory or arithmetic geometry. We will attempt to coordinate the choice of examples in our lecture and in the problem sessions with the background of the participants.
Ana Cannas da Silva
Symplectic manifolds and folded symplectic manifolds
Symplectic manifolds are manifolds equipped with a closed nondegenerate 2-form. The combination of this algebraic condition (nondegeneracy) with this differential equation (closedness) generated a rich and independent field within differential geometry and differential topology having significant interactions with many areas of mathematics.
In the first part we will cover basics of symplectic manifolds, such as Darboux's theorem, Moser's trick and hamiltonian vector fields. In the second part we will discuss the existence of symplectic forms and of forms that locally look like the pullback of a symplectic form by a folding map.
In directed geometry and topology, there is an extra
structure, namely a local (time)direction. This is
given by prescribing which paths in the space are directed. This may be defined
via the underlying space: In Euclidean space, the directed paths are often
chosen to be the paths which increase in each coordinate. A directed graph is
Basic algorithmic problems in real algebraic geometry
The systematic study of real algebraic geometry started about 30 years ago, but classical results by Descartes, Sturm, Artin and Tarski remain very influential. One type of algorithmic problem is related to the geometric description of a set described by equations and inequalities: is it empty? Connected? What is its dimension? How to describe its projection? Its topological invariants? Another type of algorithmic problem is related to the algebraic certification of a geometric property: e.g. if a polynomial is everywhere positive, is it a sum of squares? These two types of algorithmic problems in real algebraic geometry are deeply related to mathematical logic, the first one to model theory and the second one to proof theory.
Who’s that girl? Why girls choose science – in their own words
The academia, industry and politics are calling for more scientists and engineers, and the media are writing about a recruitment crisis. Meanwhile, more young people than ever before are taking higher educations. Politicians and institutions of higher education are touting the unexploited potential of European women, also because girls do at least as well on international tests that measure skills and knowledge. Others justify the need for women in STEM (Science, Technology, Engineering and Mathematics) by contending that subjects need 'female values'. Both arguments take their point of departure in society's needs more than in the individual's needs. I will discuss the arguments based on analyses of seventeen girls’ own stories. The written stories indicate that the girls who choose STEM differ in many ways. This is not very surprising. However, the informants describe barriers in the form of myths and prejudices, experience of being alone and feeling different; that the girls who choose STEM do not fit into the stories told about scientists, mathematicians and engineers. Many of these girls are highly motivated for success, and they are resolute and determined despite some descriptions of serious objections from their surroundings since childhood. Another common characteristic of the stories is the part played by the parents. Fathers seems to play an especially important role, even in families where the mother or other close relatives have a scientific background. Why? The girls describe different experiences with science and mathematic during childhood, for better or for worse. Their interest in STEM is often described in positive and emotional terms, and mathematics are often mentioned as whetting their appetite for STEM, at an early age. The results are also considered in connection with analyses of data from the quantitative survey in the collaborative EU funded research project IRIS – adressing challenge that few young people, and women in particular, choose education and career in STEM.
This research is hopefully informative and provocative as it will raise awareness of why some girls choose to study subjects where females are in minority – and give some ideas why most girls do not choose to study STEM.