Research in Geometry/Topology
Geometry and topology at Berkeley center around the study of manifolds, with the incorporation of methods from algebra and analysis.
The principal areas of research in geometry involve symplectic, Riemannian, and complex manifolds, with applications to and from combinatorics, classical and quantum physics, ordinary and partial differential equations, and representation theory.
Research in topology per se is currently concentrated to a large extent on the study of manifolds in low dimensions. Topics of interest include knot theory, 3- and 4-dimensional manifolds, and manifolds with other structures such as symplectic 4-manifolds, contact 3-manifolds, hyperbolic 3-manifolds. Research problems are often motivated by parts of theoretical physics, and are related to geometric group theory, topological quantum field theories, gauge theory and Seiberg-Witten theory, and higher dimensional topology.
A number of members of the Geometry/Topology group belong to the Research Training Group in Geometry, Topology and Operator Algebras, which runs activities and supports grad students and postdocs in its areas of interest.
Undergraduate upper division courses
The undergraduate courses, Math 140, 141, 142 are devoted to different topics in geometry and topology:
Math 140. Metric Differential Geometry
Math 141. Elementary Differential Topology
Math 142. Elementary Algebraic Topology
There is a 3 semester sequence of graduate courses in geometry, Math 240, 241, 242, two of which are taught each year.
Math 240. Riemannian Geometry
Math 241. Complex Geometry
Math 242. Symplectic Geometry
There is a 2 semester sequence in algebraic topology, 215A,B, taught every year, a one semester course Math 214 in the foundations of differential topology, and an advanced course in differential topology, Math 265.
Math 214. Differentiable Manifolds
Math 215A. Algebraic Topology
Math 215B. Algebraic Topology
Math 265. Differential Topology
Two or more topics courses are given yearly:
Math 276. Topics in Topology
Math 277. Topics in Differential Geometry
Recent topics include:
Fall 2005, Math 276. Elliptic cohomology via Quantum Field Theory (Teichner)
Spring 2005, Math 276. Heegaard Floer homology (Ozsvath)
Fall 2004, Math 276. Elliptic cohomology via Formal Groups (Teichner)
Spring 2004, Math 276. 4-Manifolds, (Kirby)
Spring 2004, Math 276. (Viro)
Spring 2004, Math 277. (Liu)
Spring 2003, Math 277. (Givental)
Fall 2002, Math 276. (Hutchings)
Spring 2002, Math 277. (Bao)
Fall 2001, Math 276. Heegaard Floer homology (Kirby)
Fall 2000, Math 277. Momentum Mappings (Weinstein)
Fall 2000, Math 277. (Reshetikhin)
Peter Teichner has been running a "Hot Topics" course/seminar which meets for two hours once a week on a topic of wide interest. The last three covered:
Fall 2006, Math 290. Derived Algebraic Geometry and Topology
Spring 2006, Math 290. Derived Algebraic Geometry and Topology
Spring 2005, Math 290. Non-Axiomatic Quantum Field Theory
The Topology seminar is held weekly throughout the year, normally
Wednesdays at 4pm. The speakers are normally visitors, but sometimes are
resident faculty or graduate students. Three times a year the Bay Area
Topology Seminar meets at Stanford (fall), Berkeley (winter) and Davis
(spring), with two lectures in the afternoon and dinner afterward.
The seminar in Symplectic Geometry (very broadly interpreted) meets on Mondays from 2 to 3 or 3:30. On the first Monday of 7 months per year, it becomes the Northern California Symplectic Geometry Seminar (Berkeley-Davis-Santa Cruz-Stanford), with two talks and a dinner, the venue alternating between Berkeley and Stanford. In the first (October) meeting of each academic year, one of the talks is the Andreas Floer Memorial Lecture, given by a distinguished invited speaker.
In the fall semester, 2007, Ian Agol will run a weekly seminar focused on topics in Kleinian groups, Teichmuller theory, and geometric group theory. It will complement the semester programs at MSRI on Geometric Group Theory and on Teichmuller Theory and Kleinian Groups, and will prepare students for the conferences at MSRI which will be occurring in November on these topics.