New Advances in Arithmetic Geometry
Arithmetic geometry, a branch of mathematics that has been a source of much interest for mathematicians since ancient times, is concerned with the number theoretic properties of the solutions of polynomial equations in higher dimensions. One mathematician who shares in this time-honoured fascination is Stephen Kudla, currently the Canada Research Chair in Automorphic Forms and Arithmetic Geometry.
As a Canada Research Chair, Kudla is continuing his work in arithmetic mathematics. He maintains that most important recent advances in this area of mathematics have been achieved by establishing connections between integer solutions of polynomial equations and automorphic forms to which analytic techniques may be applied. He himself is focusing on a new approach to establish such connections, one that is based on the construction of generating series for arithmetic cycles. His technique provides a method for calculating important number theoretic quantities, including arithmetic volumes that were unattainable by earlier methods.
Kudla believes that arithmetic geometry should be of interest to all of us, not only because of its applications to cryptography and computer science, but also because it provides a deeper understanding of the structure of the world. His view is based on the idea that the patterns of mathematics are just as much a part of the fundamental structure of the universe as are the laws of particle or quantum physics. And just as astronomers peer through ever more powerful telescopes for faint traces of the origins of the physical universe, mathematicians like Kudla peer into the hidden mysteries of our world with chalk on blackboard.