Research interests: algebraic and analytic number theory.

Webpage. Research interests: geometric Galois actions, operads and questions related to finite type invariants of knots.

Webpage. Research interests: number theory and machine learning.

Webpage. Research interests: modular forms, Galois representations, elliptic curves.

Webpage. Research interests: noncommutative algebra, representation theory and invariant theory.

Webpage. Research interests: algebraic geometry, low-dimensional topology, hyperbolic geometry, dynamics, and number theory.

This is the foundational year-long course that gives an introduction to the terminology and methods of modern abstract algebra. The course sequence should preferably be taken during the first year of graduate studies, since all other courses on algebraic topics build on it. The main topics are: groups, rings, fields, Galois theory, modules, and (multi-)linear algebra.

This is a year-long course on the fundamental concepts of commutative algebra and classical as well as modern algebraic geometry. Topics for the first semester include: ideals of commutative rings, modules, Noetherian and Artinian rings, Noether normalization, Hilbert's Nullstellensatz, rings of fractions, primary decomposition, discrete valuation rings and the rudiments of dimension theory. Topics for the second semester include: affine and projective varieties, morphisms of algebraic varieties, birational equivalence, and basic intersection theory. In the second semester, students will also learn about schemes, morphisms of schemes, coherent sheaves, and divisors.

This is a course on the principal methods and results of algebraic representation theory. It starts with an introduction to the fundamental notions, tools and general results of representation theory in the setting of associative algebras. This is followed by a thorough coverage of the classical representation theory of finite groups over an algebraically closed field of characteristic zero. If time permits, then the semester concludes with a brief introductory discussion of the representation theory of the general linear group. Other possible additional topics depend on the instructor.

The course is devoted to fundamental notions of homological algebra: chain complexes, abelian categories, derived functors, and spectral sequences. A portion of this course is also devoted to rudiments of category theory. Students will learn how to apply constructions of homological algebra and category theory to questions from abstract algebra, topology and deformation theory.