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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).

Faculty

Michael Artin Algebraic Geometry, Non-Commutative Algebra

Roman Bezrukavnikov Representation Theory, Algebraic Geometry

François Greer Algebraic Geometry

Davesh Maulik Algebraic Geometry

Andrei Neguț Algebraic Geometry, Representation Theory

Bjorn Poonen Number Theory, Algebraic Geometry

Andrew Sutherland Computational Number Theory and Arithmetic Geometry

Zhiwei Yun Geometric Representation Theory, Number Theory

Wei Zhang Number Theory, Automorphic Forms, Algebraic Geometry

Instructors & Postdocs

Felix Gotti

Aaron Landesman Classical algebraic geometry, moduli spaces, arithmetic statistics, monodromy, mapping class groups

Thomas Rüd Number theory, representation theory of p-adic groups, algebraic geometry

David Yang Algebraic Geometry, Representation Theory, Geometric Langlands

Researchers & Visitors

Shiva Chidambaram Algebraic number theory, abelian varieties, Galois representations, arithmetic geometry

Edgar Costa Computational Number Theory, Arithmetic Geometry

David Roe Computational number theory, Arithmetic geometry, local Langlands correspondence

Samuel Schiavone Computational number theory, arithmetic geometry

Graduate Students*

Niven Achenjang

Zihong Chen

Ryan Chen

Anlong Chua

Vasily Krylov Geometric Representation Theory

Calder Morton-Ferguson Geometric representation theory

Ivan Motorin Cluster Algebras, Resolution of Singularities, Representation Theory, Integrable Systems

Hao Peng

Oron Propp Geometric representation theory

Vijay Srinivasan

*Only a partial list of graduate students