# Research Areas

Faculty in algebraic geometry study a diverse set of topics including the cohomology and geometry of the moduli space of curves, the foundations of Gromov-Witten theory, the geometry of algebraic cycles, and problems of enumerative geometry

Analysis and PDE are a major strength of Stanford’s Department of Mathematics, with strong connections to geometry and applied mathematics (since PDE describe fundamental aspects...

Applied mathematics at the Stanford Department of Mathematics focuses, very broadly, on the areas of scientific computing, stochastic modeling, and applied analysis. Some of the more specific...

Combinatorics concerns the study of discrete objects. It has applications to diverse areas of mathematics and science, and has played a particularly important role in...

Currently research in financial mathematics at Stanford is in two broad areas. One is on mathematical problems arising from the analysis of financial data; it involves...

Modern geometry takes many different guises, ranging from geometric topology and symplectic geometry to geometric analysis (which has a significant overlap with PDE and geometric measure...

Contemporary number theory is developing rapidly through its interactions with many other areas of mathematics. Insights from ergodic theory have led to dramatic progress in...

The probability group at Stanford is engaged in numerous research activities, including problems from statistical mechanics, analysis of Markov chains, mathematical finance, problems at the...

Representation theory is fundamental in the study of objects with symmetry. It arises in contexts as diverse as card shuffling and quantum mechanics. An early...

Symplectic topology is at the crossroads of several mathematical disciplines such as low-dimensional topology, algebraic geometry, representation theory, Hamiltonian dynamics, integrable systems, mirror symmetry, and string theory. It...

- Topology studies properties of spaces that are invariant under deformations. A special role is played by manifolds, whose properties closely resemble those of the physical universe. Stanford...