Angus Macintyre

Angus John Macintyre FRS,^[1] FRSE (born 1941) is a British mathematician and logician known for his work in Model theory, logic, and their applications in algebra, algebraic geometry, and number theory. He is Emeritus Professor of Mathematics, at Queen Mary University of London.^[3]

Education

After undergraduate study at the University of Cambridge, he completed his PhD at Stanford University under the supervision of Dana Scott in 1968.^[2]

Career and research

From 1973 to 1985, he was Professor of Mathematics at Yale University, where he taught combined graduate and undergraduate courses in recursive function theory and philosophical foundations of mathematics.^[4] From 1985 to 1999, he was Professor of Mathematical Logic at the University of Oxford. In 1999, Macintyre moved to the University of Edinburgh, where he was professor of mathematics until 2002, when he moved to Queen Mary College, University of London. Macintyre was the first Scientific Director of the International Centre for Mathematical Sciences (ICMS) in Edinburgh.

Macintyre is known for many important results. These include classification of aleph-one categorical theories of groups and fields in 1971, which was influential in geometric stability theory. In 1976, he proved a result on quantifier elimination for p-adic fields from which a theory of semi-algebraic and subanalytic geometry for p-adic fields follows (in analogy with that for the real field) as shown by Jan Denef and Lou van den Dries. The quantifier elimination theorem was used by Jan Denef in 1984 to prove a conjecture of Jean-Pierre Serre on rationality of various p-adic Poincaré series. Macintyre worked with Zoé Chatzidakis and Lou van den Dries on definable sets over finite fields (generalising the Lang-Weil estimates of Serge Lang and André Weil to definable sets). He initiated the model theory of difference fields and of Frobenius automorphisms. His work on first-order aspects of intersection theory relates to Alexander Grothendieck's standard conjectures on algebraic cycles.

Macintyre has had many works on the model theory of real and complex exponentiation. With Alex Wilkie he proved the decidability of real exponential fields (solving a problem of Alfred Tarski) modulo Schanuel's conjecture from transcendental number theory. With Lou van den Dries he initiated and studied the model theory of logarithmic-exponential series and Hardy fields and the model theory for the theory of algebraic integers related to Rumely's local-global principle in number theory. Macintyre together with David Marker and Lou van den Dries proved several important results on the model theory of the real field equipped with restricted analytic functions, which has had many applications to exponentiation and O-minimality. Macintyre has proved results on Boris Zilber's theory of the complex exponentiation, and Zilber's pseudo-exponential fields.

Macintyre and Jamshid Derakhshan have developed a model theory for the adele ring of a number field where they prove results on quantifier elimination and measurability of definable sets. They use and extend important foundational work of Solomon Feferman and Robert Vaught on the first-order theory of products of algebraic structures. The adele ring was introduced by Claude Chevalley.(The word "adele" is short for "additive idele"^[2] and it was invented by André Weil. The previous name was the valuation vectors.) The initial purpose for introducing adeles was simplifying and clarifying class field theory. It quickly found applications outside that area after John Tate's thesis on Hecke zeta functions (of number fields) and work of André Weil on Tsuneo Tamagawa numbers of algebraic groups and varieties, and later work of Robert Langlands (related to Langlands conjectures) and the theory of automorphic L-functions and representations.

Macintyre and Marek Karpinski have proved several results on VC-dimension, which has had applications to theoretical computer science and neural networks.

Awards and honours

He was elected a Fellow of the Royal Society in 1993.^[1] In 2003, he was awarded the Pólya Prize by the London Mathematical Society. From 2009 to 2011, he was President of the London Mathematical Society (LMS).

References