Geometry/Topology Research - UC Berkeley Department of Mathematics

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.

Faculty

Tenured and tenure-track:

Mina Aganagic, Geometry, gauge theory and string theory.
Ian Agol, 3-Manifold topology and hyperbolic geometry.
Denis Auroux, Symplectic topology and mirror symmetry.
Robert Bryant, Geometry of differential equations.
Alexander Givental, Symplectic and contact geometry, Singularity theory, Mathematical physics.
Michael Hutchings, Low Dimensional and Symplectic Topology and Geometry.
Robion Kirby, Low-dimensional topology.
John Lott, Differential geometry.
Nicolai Reshetikhin, Mathematical physics, Low-dimensional topology, Representation theory.
Peter Teichner, Geometric topology, 4-manifolds, elliptic cohomology.
Constantin Teleman, Lie groups, Algebraic geometry, Topology, Quantum field theory.

Emeriti

Lester Dubins, Probability, gambling theory, geometry.
Jacob Feldman, Ergodic theory, stochastic processes.
Robin Hartshorne, Algebraic geometry.
Morris Hirsch (Not in residence), Dynamical systems, neural networks.
Wu-Yi Hsiang, Transformation groups, differential geometry.
Shoshichi Kobayashi, Riemannian and complex manifolds, Infinite Lie groups.
Charles Pugh (Not in residence), Global theory of differential equations.
John Wagoner, Differential topology, Algebraic K-theory, Dynamical systems.
Alan Weinstein, Poisson and symplectic geometry, groupoids, mathematical physics.
Joe Wolf, Lie groups, Functional analysis, Riemannian geometry.

Current Visitors and Postdoctoral Fellows:

Chris Douglas, Algebraic and geometric topology.
Paul Woon Yin Lee, Optimal transportation, Differential geometry.
Konrad Waldorf, Differential geometry of gerbes and quantum field theories.
Christian Zickert, Low dimensional topology, Hyperbolic geometry.

Faculty listed under other sections who are also interested in geometry and topology:

Mark Haiman, Algebra, combinatorics, and algebraic geometry.
Jenny Harrison, Geometric analysis, differential topology, mathematical physics.
Vaughan Jones, Von Neumann algebras.
William Kahan, (Emeritus), Error analysis, Numerical computations, Computers, Convexity, Large matrices, Trajectory problems.
Marc A. Rieffel, Non-commutative harmonic analysis, operator algebras, quantum geometry.
Hung-Hsi Wu, (Emeritus), Mathematics Education.

Courses

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

Graduate courses

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

Seminars

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.

Current Thesis Students: Name (Adviser)

Chris Atkinson (Agol)
Daniel Berwick-Evans (Teichner)
Santiago Canez (Weinstein)
Ka Choi (Kirby)
Hanh Do (Weinstein)
Matthias Goerner (Teichner)
David Farris (Hutchings)
Benoit Jubin (Weinstein)
Qin Li (Teichner)
Yi Liu (Agol)
Aaron McMillan (Weinstein)
Dmitri Pavlov (Teichner)
Sobhan Seyfaddini (Weinstein)
Alan Tarr (Teichner)
Andy Wand (Kirby)
Nate Watson (Teichner)

Recent Ph.D.s: Name (Adviser) Thesis title

2009

Jeffrey Steven Brown (Givental) Gromov-Witten Invariants of Toric Fibrations
Qingtao Chen (Reshetikhin) Some Mathematical Aspects of Quantun Field Theory
Andrew Walker Cotton-Clay (Hutchings) Symplectic Floer Homology of Area-preserving Surface Diffeomorphisms and Sharp Fixed Point Bounds
Arturo Prat-Waldron (Teichner) Pfaffian Line Bundles Over Loop Spaces, Spin Structures and the Index Theorem
Christopher John Schommer-Pries (Teichner) The Classification of Two-Dimensional Extended Topological Field Theories
Noah Joseph Snyder (Reshetikhin) Quantum Groups, Tensor Categories and Knot Invariants

2008

Fei Han (Teichner) Supersymmetric QFTs, Super Loop Spaces and Bismut-Chern Character
Jiangang Yao (Kirby) Codimension One Embedding of Manifolds & Expanding Maps and Solenoid Attractors

2007

Yanfeng Chen (Kirby) Categorification of Representations of Quantum Groups and Invariants of Tangle Cobordisms
Taiyo Inoue (Kirby) Organizing Volumes of Right-Angled Hyperbolic Polyhedra
Eli Bohmer Lebow (Hutchings) Embedded Contact Homology of 2-Torus Bundles Over the Circle
David Spivak (Teichner) Quasi-Smooth Derived Manifolds
Jiajun Wang (Kirby) Cosmetic Surgeries, Nice Heegaard Diagrams and Floer Homology