PhD in Pure Mathematics - Algebra and Algebraic Geometry in Cambridge United States | Massachusetts Institute of Technology

Polynomial equations and systems emerge across all fields of mathematics, science, and engineering. The intricate solutions (known as algebraic varieties) to these systems have captivated mathematicians for centuries and continue to represent one of the most profound and fundamental areas of modern mathematics.

Our research group focuses on classifying algebraic varieties, particularly through birational classification and moduli theory, examining how these varieties change as equation coefficients shift. The Minimal Model Program presents a potential pathway for classification. Another key research direction explores Hodge theory, connecting algebraic variety topology with harmonic functions. The Hodge Conjecture, among the seven Clay Millennium Problems, carries a million-dollar prize. Active research areas include Gromov-Witten theory, derived category studies, Calabi-Yau manifolds, and mirror symmetry—fields partly motivated by links to theoretical particle physics, particularly string theory. Noncommutative algebraic geometry, an extension with ties to representation theory, has grown into a significant focus for several department members. Advances in computing have spurred new investigations into algorithmic solutions for polynomial equations, yielding practical applications in economics, genetics, and robotics.