|Born||1941 (age 72–73)|
|Institutions||Queen Mary University of London
University of Oxford
|Alma mater||Stanford University|
|Thesis||Classifying Pairs of Real-Closed Fields (1968)|
|Doctoral advisor||Dana Scott|
|Notable awards||Polya Prize|
Angus John Macintyre FRS, FRSE (born 1941) is a British mathematician and logician and Professor of Mathematics at the Queen Mary, University of London. He is known for many fundamental, important, and widely influential contributions to Model theory, logic and their applications in algebra, algebraic geometry, and number theory. Macintyre has been a leading figure in model theory and its applications since the early 70's.
After undergraduate studies in Cambridge University, he completed his PhD at Stanford University under the supervision of Dana Scott in 1968 with a thesis entitled "Classifying Pairs of Real-Closed Fields". He was Professor of Mathematics at Yale University from 1973 until 1985, then Professor of Mathematical Logic at the University of Oxford and Professorial Fellow of Merton College, Oxford until 1999, then Professor at the University of Edinburgh until 2002, and then at Queen Mary.
Macintyre was the first Scientific Director of the International Center for Mathematical Sciences (ICMS) in Edinburgh. He was elected a Fellow of the Royal Society in 1993, and in 2003 he was awarded the Pólya Prize by the London Mathematical Society. He was elected a fellow of the Royal Society of Edinburgh. From 2009 to 2011, he served as the President of the London Mathematical Society. He has supervised a great number of doctoral students who have become leaders in the field.
Macintyre's celebrated papers from 1971 on aleph-one categorical theories of groups and fields created new paradigms for research in the field and have been highly influential in the development of model theory since the 1970's. They were used in the development of geometric stability theory (of Boris Zilber, Gregory Cherlin, Ehud Hrushovski and others) and in the development of model theory of groups, rings, and fields.
Macintyre' works from 1974 on model theory of algebraically closed groups and decision problems in algebra and the word problem for groups, rings and skew fields relate to works by Higman and B. Neuamann, and were pioneering in the connections of logic and algebra and have remained the only works in the topic since then.
His works on model theory and sheaves of structures from the early 70's introduced new ideas and established the first works on model theory and topos theory and functorialities in algebraic geometry. They remain an important area of logic of geometry for future investigations related to a "functorial" viewpoint of model theory.
One of Macintyre's most famous results is a theorem proved in 1976 stating that the field of p-adic numbers has quantifier elimination in a language which is called the Macintyre language. This result created theories of p-adic model theory and p-adic semi-algebraic geometry and has had many remarkable applications to arithmetic geometry and number theory where it naturally applies. It was used by Jan Denef to prove a conjecture of Jean-Pierre Serre on the rationality of various p-adic Poincare series. Together with Macintyre's work from 1992 on uniform rationality of p-adic integrals, it was used in the important work of Jan Denef and Francois Loeser on motivic integration.
Macintyre collaborated with Zoe Chatzidakis and Lou van den Dries on the model theory of finite and pseudo-finite fields. Their famous result on the number of points of definable sets in finite fields generalizes the Lang-Weil estimates (of Serge Lang and André Weil) from the case of a variety to a definable set. This work has had various important applications to model theory of finite and asymptotic structures, model theory of simple theories, and to analytic number theory including questions on Kloosterman and Weil sums over finite fields.
Macintyre developed model theory for the Frobenius automorphisms of algebraic geometry and proved the theory of Frobenius is decidable (this was independently proved by Ehud Hrushovski). This was the content of his 1998 Alfred Tarski Lectures (other languages) at the University of California at Berkeley.
He has had many important works on the logic of exponentiation. Among these is his work with Alex Wilkie on decidability of real exponential fields which solves a problem of Alfred Tarski modulo Schanuel's conjecture. He has related works on model theory of elliptic functions and 1-motives and Grothendieck's period conjecture. Macintyre developed a model theory for Weil cohomology theories and intersection theory. This provides a new connection between logic and algebraic geometry.
Macintyre worked with Thomas Scanlon and Luc Belair on the model theory for Frobenius on Witt vectors, and with van den Dries on various topics, including the model theory of Rumely's local-global principle in number theory and of logarithmic-exponential series and Hardy field. These have had several applications in model theory for difference and differential fields.
Macintyre's work with David Marker and Lou van den Driess on the model theory of restricted analytic fields with exponentiation gave important results as well as a new approach to O-minimality theorems for the exponential function.
He worked with M. Karpinski on VC dimension and computing volumes of definable sets. This has had applications in computer science and neural networks.
Macintyre and Jamshid Derakhshan have developed a model theory for the adele ring over a number field, a ring introduced by Chevalley and Weil and of importance in number theory and geometry. Macintyre has worked with Paola D'aquino and Giuseppina Terzo on Zilber's pseudo-exponential fields. Another famous work is the collaboration of Macintyre with Cherlin and van den Dries on the model theory of pseudo-algebraically closed fields. They gave a logic for Galois groups which has been the only work in the topic so far. This gave new connections of model theory with Galois theory and has had important applications in field arithmetic.
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