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

Polynomial equations and systems of equations appear 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 be a fundamental focus in modern mathematics.

Our research group specializes in 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 key approach to this classification. Another vibrant research direction explores Hodge theory, connecting algebraic variety topology with harmonic functions. The Hodge Conjecture, one of the seven prestigious Clay Millennium Problems, carries a million-dollar prize. Active research areas like Gromov-Witten theory, derived category studies, Calabi-Yau manifolds, and mirror symmetry draw inspiration from connections to theoretical particle physics, particularly string theory. Noncommutative algebraic geometry, an extension linked to representation theory, has emerged as a significant focus for several department members. Advances in computing power have also spurred new investigations into algorithmic solutions for polynomial equations, with diverse real-world applications in fields such as economics, genetics, and robotics.