# Algebra & Algebraic Geometry

Polynomial equations and systems of equations occur in all branches of mathematics, science and engineering. Understanding the surprisingly complex solutions (algebraic varieties) to these systems has been a mathematical enterprise for many centuries and remains one of the deepest and most central areas of contemporary mathematics.

The research interests of our group include the classification of algebraic varieties, especially the birational classification and the theory of moduli, which involves considerations of how algebraic varieties vary as one varies the coefficients of the defining equations. The Minimal Model Program offers one promising route toward classification. Another active research area involves Hodge theory, which relates the topology of an algebraic variety with harmonic functions. The Hodge Conjecture is one of the seven Clay Millennium Problems with a million dollar reward. Gromov-Witten theory, the study of the derived category, Calabi-Yau manifolds, and mirror symmetry are active areas partially inspired by their connections with theoretical high energy particle physics, especially string theory. Noncommutative algebraic geometry, a generalization which has ties to representation theory, has become an important and active field of study by several members of our department. The advent of high-speed computers has inspired new research into algorithmic methods of solving polynomial equations, with many interesting practical applications (e.g., to economics, genetics and robotics).