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.
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.
Senate Faculty
| Name | Title | Research Interests |
|---|---|---|
| Mina Aganagic | Associate Professor | String theory |
| Ian Agol | Professor | Low-dimensional geometry |
| Denis Auroux | Professor | Symplectic topology and mirror symmetry |
| Robert Bryant | Professor | Nonlinear partial differential equations and differential geometry, exterior differential systems, algebraic geometry, and Finsler geometry |
| Jacob Feldman | Professor Emeritus | Ergodic theory, Stochastic processes |
| Alexander Givental | Professor | Symplectic and contact geometry, Singularity theory, Mathematical physics |
| Robin C. Hartshorne | Professor Emeritus | Algebraic geometry, History of geometry |
| Morris W. Hirsch | Professor Emeritus | Dynamical systems, Neural networks |
| Wu-Yi Hsiang | Professor Emeritus | Transformation groups, Differential geometry |
| Michael Hutchings | Professor | Low Dimensional and Symplectic Topology and Geometry |
| Robion Kirby | Professor Emeritus | Topology of manifolds |
| John Lott | Professor | Differential geometry |
| David Nadler | Professor | Geometric representation theory, symplectic geometry |
| Charles C. Pugh | Professor Emeritus | Dynamical systems, normal hyperbolicity |
| Nicolai Reshetikhin | Professor | Mathematical physics, Low-dimensional topology, Representation theory |
| Isadore M. Singer | Professor Emeritus | Geometry, Partial differential equations, Physics |
| Peter Teichner | Professor | Geometric topology, 4-manifolds, elliptic cohomology |
| Constantin Teleman | Professor | Lie groups, Algebraic geometry, Topology, Quantum field theory |
| John B. Wagoner | Professor Emeritus | Differential topology, Algebraic K-theory, Dynamical systems |
| Alan D. Weinstein | Professor Emeritus, Professor of the Graduate School | Symplectic geometry, Mathematical physics |
| Joseph A. Wolf | Professor Emeritus, Professor of the Graduate School | Lie groups, Functional analysis, Riemannian geometry |
Visiting Faculty
| Name | Title | Research Interests |
|---|---|---|
| Dan Asimov | Visiting Scholar | Topology, geometry, scientific visualization |
| Alberto Ibort | Visiting Scholar | Semi-Riemannian and contact geometry, operator theory |
| Alexandru Scorpan | Visiting Scholar | 4-manifolds, mathematical publishing |
| Slobodan Simić | Visiting Associate Professor | Dynamical systems, Differential geometry |
| Masaru Tanaka | Visiting Scholar | Information geometry |
| Kevin Walker | Visiting Scholar | Low-dimensional topology, topological quantum field theories |
| Ilya Zakharevich | Visiting Lecturer | Integrable systems |
Postdocs
| Name | Title | Research Interests |
|---|---|---|
| Jonathan Dahl | RTG Postdoctoral Fellow | Metric Geometry & Optimal Transport |
| Sean Fitzpatrick | Lecturer | Symplectic, contact & poisson geometry, Quantization, and Equivariant index theory. |
| Matthew Gill | RTG Postdoctoral Fellow | Geometric analysis, Kähler-Ricci flow, Monge-Ampère equations |
| Madeleine Jotz | Postdoc | Poisson geometry, Lie groupoids, and differential geometry |
| David Li-Bland | NSF Postdoctoral Fellow | Differential geometry; mathematical physics; symplectic and Poisson geometry; Lie theory |
| Kate Poirier | RTG Postdoctoral Fellow | Algebraic topology, string topology |
| Rumen Zarev | Simons Visiting Assistant Professor | Heegaard Floer homology, low dimensional topology, contact and symplectic geometry |
Faculty with Related Research Interests
| Name | Title | Research Interests |
|---|---|---|
| Mark D. Haiman | Professor | Algebra, combinatorics, and algebraic geometry |
| Jenny Harrison | Professor | Geometric analysis |
| Vaughan F. R. Jones | Professor Emeritus | Von Neumann algebras |
| William M. Kahan | Professor Emeritus | Error analysis, Numerical computations, Computers, Convexity, Large matrices, Trajectory problems |
| Marc A. Rieffel | Professor | Non-commutative harmonic analysis, Operator algebras, Quantum geometry |
| Hung-Hsi Wu | Professor Emeritus | Real and complex geometry |
Graduate Students
| Name | Dissertation Supervisor |
|---|---|
| Dario Beraldo | Constantin Teleman |
| Daniel Berwick Evans | Peter Teichner |
| Harrison Chen | |
| Keon Choi | Michael Hutchings |
| Daniel Cristofaro-Gardiner | Michael Hutchings |
| Andrew J. Critch | Bernd Sturmfels |
| Chris Gerig | Michael Hutchings |
| Vinicius Gripp Barros Ramos | Michael Hutchings |
| Conrad Hengesbach | Robert Bryant |
| Sebastian Hurtado | |
| Kevin H. Lin | Constantin Teleman |
| Aaron Mazel-Gee | Peter Teichner |
| Shawn McDougal | Peter Teichner |
| Benjamin B. McMillan | |
| Eric C. Peterson | Constantin Teleman |
| Franco Vargas Pallete | |
| Qiao Zhou |
Recent Ph.D.s
| Name | Dissertation Title | Dissertation Supervisor | Year |
|---|---|---|---|
| Benoit Jubin | The Tangent Functor Monad and Foliations | Alan D. Weinstein | 2012 |
| Yi Liu | Nonzero Degree Maps BetweenThree Dimensional Manifolds | Ian Agol | 2012 |
| Daniel Pomerleano | Curved String Topology and TangentialFukuya Categories | Constantin Teleman | 2012 |
| Santiago Valencia Canez | Double Groupoids,Orbifolds, and Symplectic Category | Alan Weinstein | 2011 |
| Ka Lun Choi | Constructing a Broken LefschetzFibration with a Span or Twist-spun Knot Fiber | Robion Kirby | 2011 |
| Jeffrey Doker | Geometry of Generalized Permutohedra | Federico Ardila at SFSU, Matthias Beck at SFSU, Lior Pachter, Bernd Sturmfels, Lauren Williams | 2011 |
| David Michael Farris | The embedded contact homology of nontrivial circle bundles over Riemann surfaces | Michael Hutchings | 2011 |
| Matthias Goerner | Visualizing Regular Tesselations: Principal Congruence Links and Equivariant Morphisms from Surace to3-Manifolds | Peter Teichner | 2011 |
| Qin Li | Pontrjagin forms on certain string homogeneous spaces | Peter Teichner | 2011 |
| Aaron Fraenkel McMillan | On Embedding Singular Poisson Spaces | Alan Weinstein | 2011 |
| Dmitri Pavlov | A decomposition theorem for noncommutative L_p-spaces and a new symmetric monoidal bicategory of von Neumann algebras | Peter Teichner | 2011 |
| Alan Tarr | Smooth Field Theories and HomotopyField Theories | Peter Teichner | 2011 |
| Valentin Tonita | Twisted Gromov-Witten invariants and applications to quantum K-theory | Alexander Givental | 2011 |
| Andy Wand | Factorizations of Diffeomorphisms of Compact Surfaces with Boundary | Robion Kirby | 2010 |
| Qingtao Chen | Some Mathematical Aspects of Quantum Field Theory | Nicolai Reshetikhin | 2009 |
| Andrew Cotton-Clay | Symplectic Floer homology of area-preserving surface diffeomorphisms and sharp fixed point bounds | Michael Hutchings | 2009 |
| Arturo Felipe Prat-Waldron | Pffaffian Line Bundles over Loop space, Spin Structures and the Index Theorem | Peter Teichner | 2009 |
| Christopher J Schommer-Pries | The Classification of Two-Dimensional Extended Topological Field Theories | Peter Teichner | 2009 |
| Noah Snyder | Quantum groups, tensor categories and knot invariants | Nicolai Reshetikhin | 2009 |
| Fei Han | Sypersymmetric QFTS. Super Loop Spaces and Bismut-Chern Character | Peter Teichner | 2008 |
| Jiangang Yao | Codimension one embedding of manifolds & expanding maps and solenoid attractors | Robion Kirby | 2008 |
