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:

Ian Agol, 3-Manifold topology and hyperbolic geometry.
Alexander Givental, Symplectic and contact geometry, Singularity theory, Mathematical physics.
Michael Hutchings, Low Dimensional and Symplectic Topology and Geometry.
Robion Kirby, Low-dimensional topology.
Peter Teichner, Geometric topology, 4-manifolds, elliptic cohomology.
Constantin Teleman, Lie groups, Algebraic geometry, Topology, Quantum field theory.
John Wagoner, Differential topology, Algebraic K-theory, Dynamical systems.
Alan Weinstein, Poisson and symplectic geometry, groupoids, mathematical physics.

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.
Joe Wolf, Lie groups, Functional analysis, Riemannian geometry.

Visitors and postdoctoral fellows (2006-2007):

Christian Blohmann, Noncommutative geometry, symplectic geometry, representation theory and mathematical physics.
David Cimasoni, Geometric topology and knot theory.
Benoit Dherin, Mathematical physics, Poisson geometry, Deformations, Quantization.
Dagan Karp, Algebraic geometry, Gromov-Witten theory.
John Lott, Differential 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, Error analysis, Numerical computations, Computers, Convexity, Large matrices, Trajectory problems.
Nicolai Reshetikhin, Mathematical physics, Low-dimensional topology, Representation theory.
Marc A. Rieffel, Non-commutative harmonic analysis, operator algebras, quantum geometry.
Hung-Hsi Wu, 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)
Jeff Brown (Givental)
Santiago Canez (Weinstein)
Hanh Do (Weinstein)
Jana Comstock (Teichner)
Andrew Cotton-Clay (Hutchings)
Matthias Goerner (Teichner)
David Farris (Hutchings)
Fei Han (Teichner)
Matt Harvey (Kirby)
Qin Li (Teichner)
Aaron McMillan (Weinstein)
Arturo Prat-Waldron (Teichner)
Chris Pries (Teichner)
Andy Wand (Kirby)
Jiangang Yao (Kirby)

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

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

2006

Alfonso Gracia-Saz (Alan Weinstein) The Symbol of a Function of a Pseudodifferential Operator
Thomas R. Fleming (UCSD) (Peter Teichner) Generalized Link Homotopy Invariants
J. Elisenda Grigsby (Robion Kirby)&nbap;Knot Floer Homology in Cyclic Branched Covers
Henning Hohnhold (UCSD) (Peter Teichner) Supersymmetry in the Stolz-Teichner Project on Elliptic Cohomology
Rajan Mehta (Alan Weinstein) Supergroupoids, Double Structures, and Equivariant Cohomology

2005

Norah Esty (Charles Pugh) Orbit Structures for Groups of Homeomorphism on SI
Todor Milanov (Alexander Givental) Singularity Theory and Integrable Systems
Mark Warren Rinker (John Stallings) Height Functions, Maps to the Integers, and the Whitehead Graph of a Finite Presentation
Hsian-Hua Tseng (Alexander Givental) Quantum Riemann-Roch, Lefschetz and Serre Theorems for Orbifold Gromov-Witten Theory

2004

H. Tracy Hall (Robion Kirby) Counterexamples in Discrete Geometry
Tamas Kalman (Michael Hutchings and Robion Kirby) Contact Homology and One Parameter Families of Legendrian Knots
Robert Myers (Charles Pugh) Global Transverse Disks and Suspendibility Criteria
Nicholas Proudfoot (Allen Knutson) Hyperkahler Analogues of Kahler Quotients
Kevin Purbhoo (Allen Knutson) Vanishing and Non-Vanishing Criteria for Branching Schubert Calculus
Lawrence Roberts (Robion Kirby) Heegaard-Floer Homology and d-Based Links in Three Manifolds
Xiang Tang (Alan Weinstein) Quantization of Noncommutative Poisson Manifolds
Marco Zambon (Alan Weinstein) Submanifold Averaging in Riemannian, Symplectic and Contact Geometry
Chenchang Zhu (Alan Weinstein) Integrating Lie Algebroids via Stacks and Applications to Jacobi Manifolds

2003

Donald Barkauskas (John Stallings) Centralizers in Fundamental Groups of Graphs of Groups
Michael Burns (Vaughan Jones) Subfactors, Planar Algebras and Rotations
Tom Coates (Alexander Givental) Riemann-Roch Theorems in Gromov-Witten Theory
Megumi Harada (Allen Knutson) The Symplectic Geometry of the Gel'fand-Cetlin-Molev Basis for Representation of Sp(2n,C)
Andrew Miller (Charles Pugh) Upsiloids with Nonpositive Curvature
Alexandru Scorpan (Robion Kirby) Existence of Foliations on 4-Manifolds
Christopher Tuffley (Robion Kirby) Finite Subset Spaces of Graphs and Surfaces

2002

Tolga Etgu (Robion Kirby) Symplectic Forms on Product Four-Manifolds
Charles G. Holton (John Wagoner) The Rohlin Property for SFT C*-Automorphisms and Ergodic Properties of 3-Interval Exchanges
Olga Valerievna Radko (Alan Weinstein) Some Invariants of Poisson Manifolds
Sarah Anne Reznikoff (Vaughan Jones) Representations of the Temperley-Lieb Planar Algebra

2001

Henrique Bursztyn (Alan Weinstein) Morita Equivalence in Deformation Quantization
Benjamin Lent Davis (Alan Weinstein) On Poisson Spaces Associated to Finitely Generated Poisson R-Algebras
Karen Edwards (Andrew Casson) Stabilizations of Heegaard Splittings with Respect to Connect-Sums of 3-Manifolds
Saul Schleimer (Andrew Casson) Almost Normal Heegaard Splittings
Robert Schneiderman (Robion Kirby and Peter Teichner) 4-Dimensional Intersection Numbers of Knots and Links in 3-Manifolds

2000

Aaron Abrams (Andrew Casson) Configuration spaces and braid groups of graphs
Stephen Bigelow (Robion Kirby) Homological Representations of Braid Groups
Danny Calegari (Andrew Casson) Thesis title: Foliations and the Geometry of Three-Manifolds
Hsiang-Ping Huang (Vaughan Jones) Commutators Associated to a Subfactor and Its Relative Commutants
Sungjoon Ko (Robion Kirby) More about Tight Contact Structures on Lens Space
Johanna Neaderhouser (Morris Hirsch) Classifying One-Dimensional Attractors in Flows on Surfaces
Dylan Paul Thurston (Vaughan Jones) Wheeling: A Diagrammatic Analogue of the Duffo Isomorphism