Webpage. Advisor: David Futer. Research interests: low-dimensional topology, 3-manifold geometry, quantum topology.

Webpage. Research interests: low-dimensional topology and geometry, geometric group theory.

Advisor: Samuel J. Taylor. Research interests: geometric topology, dynamics, computational topology.

Advisor: Matthew Stover. Research interests: dynamics and rigidity.

Webpage. Advisor: David Futer. Research interests: low-dimensional topology and geometric group theory.

Research interests: hyperbolic geometry, convexity, combinatorial geometry, algebraic groups, probability theory, graph theory, dynamics, finance, and computational crystallography.

Webpage. Advisor: Samuel J. Taylor. Research interests: TEXT

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

Webpage. Research interests: geometric topology, geometric group theory, and dynamics.

This is the foundational year-long course that gives an introduction to the terminology and methods of differential geometry and algebraic topology.  The first semester (Math 8061) is devoted to geometry and topology of manifolds and smooth maps. In this part of the course, we introduce (co)tangent bundle, vector (and tensor) fields and exterior forms. We discuss in detail local behavior of smooth maps. We learn Sard's theorem and Whitney's embedding theorem. This part of the course also includes oriented intersection theory and integration on manifolds. The second semester (Math 8062) is devoted to the fundamental tools of algebraic topology: CW complexes, fundamental group, van Kampen's theorem, covering spaces, homology, cohomology, and Poincare duality.

The main goal of this one-semester course is to provide a solid introduction to the two central concepts of Riemannian geometry, namely, geodesics and curvature and their relationship. After taking this course, students will have an intimate acquaintance with the tools and concepts that are needed for pursuing research in Riemannian geometry or applying its ideas to other fields of mathematics such as analysis, topology, and algebraic geometry. The topics covered include Riemannian metrics, Riemannian connections, geodesics, curvature (sectional, Ricci, and scalar curvatures), the Jacobi equation, the second fundamental form, and global results such as the Gauss-Bonnet Theorem, the theorems of Hopf-Rinow and Hadamard, variations of energy, the theorems of Bonnet-Myers and of Synge-Weinstein, and the Rauch comparison theorem.

This is the first semester of a year-long course surveying the modern theory of knots and providing an introduction to some fundamental results and techniques of low-dimensional topology. The course will start at the very beginning of knot theory; it will then proceed to several classical knot invariants (Alexander, Jones, HOMFLY polynomials). The first semester will also touch on braid groups and mapping class groups, and use these groups to show that every (closed, orientable) 3-manifold can be constructed via knots.

This course develops Lie theory from the ground up. Starting with basic definitions of Lie group-manifolds and Lie algebras, the course develops structure theory, analytic and algebraic aspects, and representation theory. Interactions with other fields, e.g., differential equations and geometry are also discussed.