Overview
Algebra and Number Theory
Gauss famously referred to the theory of numbers as the “Queen of Mathematics”. It is certainly one of the oldest and most beautiful branches of mathematics, and almost unique with respect to the wide array of tools it uses from subjects such as analysis, algebra and topology. This makes number theory both an extraordinarily rich and challenging field for research.
Rochester has a large, active research group working across the many subfields of number theory including algebraic number theory, analytic number theory, p-adic arithmetic, p-adic cohomology, function field arithmetic, modular forms, the theory of L-functions, Diophantine geometry, and arithmetic dynamics.
Analysis
Analysis studies functions, equations, and mathematical structures using methods of approximation, continuity, and limiting processes. It is a broad and fundamental area of mathematics, with connections to geometry, probability, physics, and many other fields.
Combinatorics
The combinatorics group is interested in intersections of combinatorics with other areas of mathematics such as analysis, algebra, topology, geometry and number theory.
Geometry
The geometry group works in a number of different areas. Geometry has been one of the most popular fields of mathematics in recent decades. It combines techniques from apparently unconnected areas, such as the use of analytical tools for answering topological questions, and it has strong relations with quantum field theory and string theory.
Recent results in geometry include the proof of the Poincare conjecture, by G. Perelman in 2002-2003 following the analytic program of R. Hamilton, and the differentiable sphere theorem, proved by S. Brendle and R. Schoen in 2008-2009.
Probability, Ergodic Theory, and Mathematical Physics
Probability theory studies random phenomena in many areas of mathematics including statistical mechanics, combinatorics, dynamical systems and geometry.Topology
The department has experts in algebraic topology, particularly in homotopy theory. The 20th century has been called the century of topology. General topology now forms one of the foundations of modern analysis and geometry.
The development of algebraic topology led to the development of many of the key concepts of algebra: homological algebra, category theory, Lie groups/algebras and K-theory among them. The use of cobordism invariants and intersection theory developed in differential topology form the foundations for invariants such as the Seiberg-Witten invariants and Gromov-Witten invariants so important in current day geometry.
Today topology continues to flourish as a pure endeavor. Symplectic topology and Voevodsky’s recent introduction of the techniques of homotopy theory into algebraic geometry are among some examples. In addition, there are growing connections with theoretical physics, dynamical systems, computer science and DNA biology.