Our Geometry and Topology group specializes in connections between low-dimensional topology, hyperbolic geometry, arithmetic groups, and geometric group theory.
Andrew Clickard, Graduate Student
Webpage. Advisor: David Futer. Research interests: low-dimensional topology, 3-manifold geometry, quantum topology.
David Futer, Professor
Webpage. Research interests: low-dimensional topology and geometry, geometric group theory.
Ross Griebenow, Graduate Student
Advisor: Samuel J. Taylor. Research interests: geometric topology, dynamics, computational topology.
Rob Oakley, Graduate Student
Webpage. Advisor: David Futer. Research interests: low-dimensional topology and geometric group theory.
Igor Rivin, Professor
Research interests: hyperbolic geometry, convexity, combinatorial geometry, algebraic groups, probability theory, graph theory, dynamics, finance, and computational crystallography.
Adam Smith, Graduate Student
Advisor: Samuel J. Taylor. Research interests: geometric group theory.
Matthew Stover, Professor
Webpage. Research interests: algebraic geometry, low-dimensional topology, hyperbolic geometry, dynamics, and number theory.
Samuel J. Taylor, Associate Professor
Webpage. Research interests: geometric topology, geometric group theory, and dynamics.
The Geometry and Topology Seminar takes place on Wednesdays at 2:30PM in Room 617 of Wachman Hall. Current contacts for the seminar are David Futer, Matthew Stover and Samuel Taylor.
The PATCH (Philadelphia Area Topology, Contact and Hyperbolic) Seminar is organized jointly with Bryn Mawr, Haverford, and Swarthmore Colleges, as well as the University of Pennsylvania. The seminar takes place approximately three times per semester, rotating among the campuses.
Selected Courses
Math 8061-62: Differential Geometry and Algebraic Topology
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.
Math 9021: Riemannian Geometry
This semester-long course will survey the rapidly expanding field of geometric group theory, focusing on the role played by negative curvature. We will begin with classical combinatorial techniques used to construct and study infinite discrete groups. After introducing basic concepts in coarse geometry, we will turn our attention to Gromov's notion of hyperbolic groups. In addition to studying geometric, algebraic, and algorithmic properties of these groups, we will keep an eye towards several generalizations of hyperbolicity that have recently played a large role in understanding many geometrically significant groups. As examples, we will also touch on the study of mapping class groups, outer automorphism groups of free groups, and cubical groups.
Math 9023-24: Knot Theory and Low Dimensional Topology
This semester-long course will survey the rapidly expanding field of geometric group theory, focusing on the role played by negative curvature. We will begin with classical combinatorial techniques used to construct and study infinite discrete groups. After introducing basic concepts in coarse geometry, we will turn our attention to Gromov's notion of hyperbolic groups. In addition to studying geometric, algebraic, and algorithmic properties of these groups, we will keep an eye towards several generalizations of hyperbolicity that have recently played a large role in understanding many geometrically significant groups. As examples, we will also touch on the study of mapping class groups, outer automorphism groups of free groups, and cubical groups.
Math 9061-62: Lie Groups
This semester-long course will survey the rapidly expanding field of geometric group theory, focusing on the role played by negative curvature. We will begin with classical combinatorial techniques used to construct and study infinite discrete groups. After introducing basic concepts in coarse geometry, we will turn our attention to Gromov's notion of hyperbolic groups. In addition to studying geometric, algebraic, and algorithmic properties of these groups, we will keep an eye towards several generalizations of hyperbolicity that have recently played a large role in understanding many geometrically significant groups. As examples, we will also touch on the study of mapping class groups, outer automorphism groups of free groups, and cubical groups.
Math 9073: Geometric Group Theory
This semester-long course will survey the rapidly expanding field of geometric group theory, focusing on the role played by negative curvature. We will begin with classical combinatorial techniques used to construct and study infinite discrete groups. After introducing basic concepts in coarse geometry, we will turn our attention to Gromov's notion of hyperbolic groups. In addition to studying geometric, algebraic, and algorithmic properties of these groups, we will keep an eye towards several generalizations of hyperbolicity that have recently played a large role in understanding many geometrically significant groups. As examples, we will also touch on the study of mapping class groups, outer automorphism groups of free groups, and cubical groups.