It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

/pdf/quantum-field-theory-and-the-jones-polynomial-5897tp4ud9.pdf

Quantum field theory and the Jones polynomial

Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

https://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf

The Arithmetic of Elliptic Curves

Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.

https://www.springer.com/productFlyer_978-0-387-98463-6.pdf?SGWID=0-0-1297-1590487-0

One-Parameter Semigroups for Linear Evolution Equations

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II

We give a positive answer to the Berry-Robbins problem for any compact Lie group G, i.e. we show the existence of a smooth W-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold G/T . This map is constructed via solutions to Nahm’s equations and it is compatible with the S O(3) action, where S O(3) acts on G/T via a regular homomorphism from SU(2) to G. We then generalize this picture to include an arbitrary homomorphism from SU(2) to G. This leads to an interesting geometrical picture which appears to be related to the Springer representation of the Weyl group and the work of Kazhdan and Lusztig on representations of Hecke algebras.

Nahm’s equations, configuration spaces and flag manifolds

It must be many years since a pure mathematician was asked to address this society on such an occasion. Certainly none appears in the records going back to 1940. This is of course no reflection on the impartiality of the Council of the Royal Society, but simply an acknowledgement of the wide gap that separates pure mathematicians from other scientists and of the serious difficulties of communicating across that gap. Fortunately we have our intermediaries - the applied mathematicians - who extract from the body of mathematical knowledge the most useful parts and bring them to bear on recognizable scientific problems in a wide variety of fields, all the way from the traditional areas of physical science right through to the biological and social sciences. Many distinguished applied mathe­maticians have indeed spoken to the society on problems such as aerodynamic noise or stellar evolution, which are heavily dependent on mathematical analysis, but which can be explained in physical terms readily understood by a wide audience. On such occasions the mathematical techniques involved will quite properly have been relegated to decent obscurity. As a result scientists at large probably have only the vaguest ideas about mathematical research per se . They will understand mathematical work related to their particular field, frequently, I may add, better than the mathematicians themselves, but they must find it hard to visualize mathematics in the abstract. It is therefore perhaps worthwhile for a pure mathematician to attempt to explain how we view our subject and what motivates our research in the absence of any particular scientific interpretation or application. If I might summarize the situation diagrammatically, consider mathematics as some kind of giant computer with a large number of terminals on its periphery, representing fields of application. A practising scientist is like the terminal user. He is primarily interested in the output and will know something about what the computer can do for him, but he is not involved in what goes on inside the heart of the computer. In the early days of computers, users and designers were frequently the same people, but with their rapid growth and sophistication this is now the exception rather than the rule. Similarly it is the increasing sophistication of mathematics which has led to the large gap between ‘users’ and ‘designers’.

Bakerian Lecture, 1975: Global Geometry

A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.

https://www.intlpress.com/site/pub/files/_fulltext/journals/sdg/2003/0008/0001/SDG-2003-0008-0001-a001.pdf

Projective planes, Severi varieties and spheres

Topology and Geometry for Physicists. By Charles Nash and Siddhartha Sen. Academic Press: 1983. Pp. 310. £25, $39.50.

Mathematics for physics

We give a positive answer to the Berry-Robbins problem for any compact Lie group G, i.e. we show the existence of a smooth W-equivariant map from the space of regular triples in a Cartan subalgebra to the flag manifold G/T. This map is constructed via solutions to Nahm's equations and it is compatible with the SO(3) action, where SO(3) acts on G/T via a regular homomorphism from SU(2) to G. We then generalize this picture to include an arbitrary homomorphism from SU(2) to G. This leads to an interesting geometrical picture which appears to be related to the Springer representation of the Weyl group and the work of Kazhdan and Lusztig on representations of Hecke algebras.

Michael Atiyah

Papers

Nahm’s equations, configuration spaces and flag manifolds

Bakerian Lecture, 1975: Global Geometry

Projective planes, Severi varieties and spheres

Mathematics for physics

Nahm's equations, configuration spaces and flag manifolds