Life and Work of Sarah Rees 2 nd New York Women in Mathematics Network Conference

Presenter: Weiyan Guo

New York City College of Technology(CUNY) Mentor: Dr. Delaram Kahrobaei

May 2 nd 2008

Sarah Rees 1957‐‐

Sarah Rees was born in Cambridge(UK), both her parents were mathematicians. She is one of four girls in the family. Her family moved to Exeter when she was a baby and grew up and went to school there. Three of the four of them studied math at university (the fourth did natural sciences) and two went on to become academic mathematicians.

She studied in Cambridge University as an undergraduate and for a postgraduate taught course(Known as path III) and then went on to postgraduate Studies in Oxford. After that she travelled the world for a while as a postdoc. She did a year in Brussels, with Francis Buekenhout, who was very kind and gave her lots of helpful advice. He made sure she met people who would help her further, and it was at a conference that year that she met Mark Ronan and Steve Smith, who invited her to the University of Illinois in Chicago for two quarters. she decided, influenced she is sure by Francis Buekenhout, that she should stay a little longer in the US (she thought she should stay 2 years), So she applied for an instructorship at Ohio State; she stayed there for a year, under the patronage of Ron Solomon, and then went back to a temporary lectureship in Birmingham, England.

But at that stage she became rather disenchanted, finding the university situation in the UK rather depressed, the job market very bare, and feeling she was in a mathematical blind alley research wise. She looked around for something else to do, thought of going to Africa, then of moving into computer science. At that stage she met David Epstein, who suggested that she should in fact move to Warwick as his RA, and that there she could learn computer science through his automatic groups project. So off she went (after 3 months at the University of Lusaka, Zambia on an exchange program they ran with Birmingham, fulfilling her plan to go to Africa), and worked with David Epstein and Derek Holts for 3 years. She didn't move into computer science, but she learned a lot, and somehow reoriented herself. And at the end of 3 years she got a permanent position in Newcastle, to which she went via Universitaet Essen (where she stayed for 3 months in the Institut fuer Experimentelle Mathematik).

She’s been in Newcastle ever since, apart from a year long sabbatical in Germany (1996,7) and a shorter sabbatical in Nebraska and Paderborn in 2004, and various trips to Australia, Germany and the US in particular. She'd been wanting a child for a long time, and she finally managed that in 2000.She's 7 now. They travel a lot together, because she is a single parent.

Her research interests have always been at the junction of group theory, geometry and combinatory. For her D.Phil thesis and for three years subsequently she worked on Tits buildings and related structures; these provide a geometrical picture of simple groups. More recently, she has worked on geometrical, combinatorial and computational aspects of group theory. Her work is motivated by questions which arise within central strands of group theory. She has worked largely with groups defined by finite presentations (which are generally infinite), but also with matrix groups specified by a set of generating matrices. For finitely presented groups she has looked at problems of decidability and computability.

Of her most recent projects, one examines connections between group theory and formal language theory, the other studies quantum computation from a group theoretic perspective. For both finitely presented and matrix groups she has also worked on the theoretical development and analysis of high performance algorithms, and further she has developed practical implementations of these, which have been freely and widely distributed within the mathematical community. Her work in the development of algorithms has been a mix of theory and implementation, and has naturally led her to be involved in the development of larger scale computer algebra systems, such as GAP and Baumslag's New York based MAGNUS system in addition to her own, widely distributed, standalone software.

Word Problem

G = <X | R > (finite presentation) Word Problem (WP) :∃ 9? Algorithm decide w = G 1?

The word problem is said to be soluble if the set W(G,X) can be recognized by a halting Turing machine, that is, if there is a terminating algorithm which, given a word w over X, returns the answer ‘yes’ if w represents the identity element and ‘no’ otherwise. Identity w means that repeated application of the equations

in R together with the rules of free cancellation transforms w into the empty word є.

For example: xx ­1 or x ­1 x, for x є X

In general, there is no clear reason why every route to reduce any particular word w to the trivial word should not pass through a much longer word; if there is no upper bound on the length of intermediate words which must be encountered on any reduction of a word to the identity, then the design of an algorithm to do this is clearly difficult, if not impossible.

When the word problem of a group is soluble, it is natural to ask how hard the problem is. It is standard to use bounds on time, or on space, or at least to use measures which give information about those bounds. It is also common to analyze the type of computational machine which is needed to perform the calculation (which could range from the most general Turing machine to a simple finite state automation).

Max Dehn’s work

The word problem was first studied by Max Dehn, together with the conjugacy and isomorphism problem. Dehn’s motivation for the solution of the word problem was geometric; he wanted to solve the problem for the fundamental group of a surface, in which case a word represents the identity precisely when the corresponding loop contracts.

Any such group has a Dehn presentation over any generation set X, with the following properties. First, for each of the equations u=v in the finite defining set R, u the left hand side, is longer than v, the right hand side.

Second, whenever w is a non­trivial word over X which represents the identity, the left hand side of at least one equation in R is a subword of w.

If a non­trival input word w contains no left hand side as a subword then the answer ‘no’ can be returned. Otherwise w contains a left hand side u as a subword, substitution of u by the corresponding right hand side v gives a shorter representative of the same group element, and iteration solves the problem.

EXAMPLES OF GROUPS WITH SOLVABLE WORD PROBLEM

¢ Automatic groups, including: Finite groups Polycyclic groups Negatively curved groups Euclidean groups Coxeter groups Braid groups Geometrically finite groups

¢ Finitely generated recursively absolutely presented groups, including: Finitely presented simple groups.

¢ Finitely presented residually finite groups ¢ One relator groups, including Fundamental groups of closed orientable two­dimensional

manifolds. ¢ Combable groups

PRESENTATIONS A presentation of a group is a description of a set and a

subset of the free group generated by , written <S|R>. Some common examples: Group Presentation

the free group on S < S | Ø >

C n , the cyclic group of order n < a | a 2 > D ∞ , the infinite dihedral group <r, f | f 2 , (rf) 2 > Z × Z <x, y | xy = yx> Z m × Z n <x, y | x m =1, y n = 1, xy = yx> the free abelian group on S < S | R> where R is the set of all commutators

of elements of the free group on S

A Turing machine A Turing machine A Turing machine is a kind of state machine. At any time the machine is in any one of a finite number of states. Instructions for a Turing machine consist in specified conditions under which the machine will transition between one state and another. A Turing machine has an infinite one­ dimensional tape divided into cells. Traditionally we think of the tape as being horizontal with the cells arranged in a left­ right orientation. The tape has one end, at the left say, and stretches infinitely far to the right. Each cell is able to contain one symbol, either ‘0’ or ‘1’.

Automaton Automaton

An automaton is a quintuple (∑, X, E, A, Ω), where ∑ and X are sets, A and Ω are subsets of ∑, and E is set of triples of the form (σ, U, τ) is in E, then we think of e as a directed edge going from σ to τ and having label U (under the figure).

U σ τ

REFERENCES

1) Sarah Rees, How hard is the word problem? Proceedings of Conference for European Women in Mathematics, Varna, Bulgaria, September 2002.

2) Charles C. Sims, Computation with finitely presented groups , Cambridge University Press, 1994.

3) Sarah Rees Homepage: http:www.mas.ncl.ac.uk/~nser

4) Communication with Sarah Rees