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).

Department Members in This Field

Faculty

• Michael Artin Algebraic Geometry, Non-Commutative Algebra
• Roman Bezrukavnikov Representation Theory, Algebraic Geometry
• Steven Kleiman Algebraic Geometry, Commutative Algebra
• Davesh Maulik Algebraic Geometry
• Andrei Neguț Algebraic Geometry, Representation Theory
• Bjorn Poonen Arithmetic Geometry, Algebraic Number Theory, Rational Points on Varieties, Undecidability
• Leonid Rybnikov Representation Theory, Integrable Systems, Quantization, Kashiwara Crystals
• Andrew Sutherland Computational Number Theory and Arithmetic Geometry
• Zhiwei Yun Representation Theory, Number Theory, Algebraic Geometry
• Wei Zhang Number Theory, Automorphic Forms, Algebraic Geometry

Instructors & Postdocs

• Manik Dhar Combinatorics, Theoretical Computer Science
• Felix Gotti Commutative Algebra and Semigroup Theory (in connection to Number Theory and Combinatorics)
• Artem Kalmykov Geometric representation theory, quantum groups
• Aaron Landesman Classical algebraic geometry, moduli spaces, arithmetic statistics, monodromy, mapping class groups
• Siyan Daniel Li-Huerta Arithmetic Geometry, Langlands Program
• Miguel Moreira Algebraic Geometry, Moduli Spaces, Enumerative Geometry
• Thomas Rüd Number theory, representation theory of p-adic groups, algebraic geometry
• Minh-Tâm Trinh
• David Yang Algebraic Geometry, Representation Theory, Geometric Langlands
• Robin Zhang Number Theory, Automorphic Forms, Diophantine Geometry

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
• Junliang Shen

Graduate Students*

• Niven Achenjang
• Ryan Chen
• Zihong Chen
• Anlong Chua Geometric representation theory
• Ilya Dumanski Geometric and combinatorial methods in representation theory
• Jose Guzman Algebraic Geometry, Enumerative Geometry
• Vasily Krylov Geometric Representation Theory
• Anne Larsen
• Calder Morton-Ferguson Geometric representation theory
• Ivan Motorin Cluster Algebras, Resolution of Singularities, Representation Theory, Integrable Systems
• Hao Peng
• Vijay Srinivasan

*Only a partial list of graduate students