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 and Topology, 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, and a one semester course Math 214 in the foundations of differential topology.

Math 214. Differentiable Manifolds
Math 215A. Algebraic Topology
Math 215B. Algebraic Topology

Two or more topics courses are given yearly:

Math 270. Hot Topics
Math 276. Topics in Topology
Math 277. Topics in Differential Geometry

Recent topics include:

Spring 2016, Math 276. Factorization Algebras (Teichner)
Spring 2016, Math 277. Applied Holomorphic Curve Theory (Hutchings)
Fall 2015, Math 270. Gauge Theory (Wehrheim)
Spring 2015, Math 276. Gauge Field Theory (Teichner)
Spring 2015, Math 277. Ricci Flow (Lott)
Spring 2014, Math 276. Functorial Field Theories and Ring Spectra (Teichner)
Spring 2014, Math 276. Regularization of Moduli Spaces (Wehrheim)
Fall 2013, Math 278. Pseudoholomorphic Curves (Wehrheim)
Spring 2013, Math 277. The Quantum Riemann-Roch-Hirzebruch Theorem (Givental)
Fall 2012, Math 277. Contact Homology (Hutchings)
Spring 2012, Math 277. Kaehler Geometry (Lott)
Spring 2011, Math 270. The Blob Complex (Teichner)
Spring 2011, Math 276. The Index Theorem (Teichner)
Fall 2010, Math 276. Seiberg-Witten-Floer Theory (Hutchings)
Fall 2010, Math 277. Optimal Transport (Lott)
Spring 2010, Math 270. Perturbative Quantization (Teichner)
Spring 2010, Math 276. Three-Manifold Topology (Agol)
Fall 2009, Math 277. Mirror Symmetry (Auroux)
Fall 2009, Math 277. Exterior Differential Systems (Bryant)
Fall 2009, Math 277. Ricci Flow (Lott)
Spring 2009, Math 270. Topological Field Theory (Teichner)
Fall 2008, Math 276. Rational Homotopy Theory (Teleman)

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.

There are various seminars related to symplectic geometry. On the first Monday of 7 months per year, there is a meeting of 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.

The Differential Geometry seminar is held weekly throughout the year, normally Mondays at 5.

Senate Faculty

Name Title Research Interests
Mina Aganagic Professor String theory
Ian Agol Professor Low-dimensional topology
Denis Auroux Professor Symplectic topology and mirror symmetry
Richard H. Bamler Assistant Professor Geometric analysis, differential geometry, topology
Robert Bryant Professor Emeritus Nonlinear partial differential equations and differential geometry, exterior differential systems, algebraic geometry, and Finsler geometry
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, geometric analysis
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
Katrin Wehrheim Associate Professor Low-dimensional and symplectic topology
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
Zhiqi Chen Visiting Scholar Differential geometry
Joseph Maher Visiting Scholar Geometric topology, random walks on groups
Alexandru Scorpan Visiting Scholar 4-manifolds, mathematical publishing

Postdocs

Name Title Research Interests
James Conway Visiting Assistant Professor Contact and symplectic geometry
Ved Datar Postdoc Differential geometry, Kaehler-Einstein metrics, Kaehler-Ricci flow, Gromov-Hausdorff convergence
Marina Iliopoulou Morrey Visiting Assistant Professor Harmonic analysis, geometric measure theory, incidence geometry
David Li-Bland NSF Postdoctoral Fellow Differential geometry; mathematical physics; symplectic and Poisson geometry; Lie theory
Kathryn Mann Morrey Visiting Assistant Professor Geometry, topology, dynamics
Khoa L Nguyen Postdoc Symplectic topology
Michele Schiavina SNSF Postdoc Mathematical Physics, Field Theory, Quantum Information, Quantum Gravity
Hongbin Sun Morrey Visiting Assistant Professor Low dimensional topology and hyperbolic geometry
Michael Viscardi Postdoc Geometric representation theory; mirror symmetry

Faculty with Related Research Interests

Name Title Research Interests
Mark D. Haiman Professor Algebra, combinatorics, and algebraic geometry
Jenny Harrison Professor Functional analysis, Geometric measure theory, Calculus of variations, Mathematical physics
Vaughan F. R. Jones Professor Emeritus Von Neumann algebras
William M. Kahan Professor Emeritus Error analysis, Numerical computations, Computers, Convexity, Large matrices, Trajectory problems
Arthur E. Ogus Professor Algebraic geometry
Marc A. Rieffel Professor Non-commutative harmonic analysis, Operator algebras, Quantum geometry
James A. Sethian Professor Applied mathematics, Computational physics, Partial differential equations
Vivek Shende Assistant Professor Geometry
Hung-Hsi Wu Professor Emeritus Real and complex geometry

Graduate Students

Name Dissertation Supervisor
Alexander Appleton
Nicolas Brody
Catherine Cannizzo Denis Auroux
Julian Chaidez
Harrison Chen David Nadler
Yael Degany Katrin Wehrheim
Kevin Donoghue Ian Agol
Christopher Eur
Kuan-Ying Fang Denis Auroux
Benjamin Filippenko Katrin Wehrheim
Benjamin Gammage David Nadler
Chris Gerig Michael Hutchings
Andrew Hanlon Denis Auroux
Jeff Hicks Denis Auroux
Alvin Kerber Ian Agol
Daniel Lowengrub
Calvin McPhail-Snyder
Kyle Miller
Mihai Munteanu
Christopher Policastro Katrin Wehrheim
Yingdi Qin
Eugene Rabinovich
Ritvik Ramkumar
Wolfgang Schmaltz Katrin Wehrheim
Michael Smith
Tao Su Richard E.Borcherds, Vivek Shende
Thunwa 'Nics' Theerakarn John Lott
Ryan George Thorngren Vivek Shende
Franco Vargas Pallete Ian Agol
Morgan Weiler Michael Hutchings
Patrick Wilson John Lott
Ben Wormleighton David Nadler
Qiaochu Yuan David Nadler
Qiao Zhou (Elaine) David Nadler
Zhengyi Zhou Katrin Wehrheim

Recent Ph.D.s

Name Dissertation Title Dissertation Supervisor Year
Chang-Yeon Cho Topological types of Algebraic stacks 2016
Aaron Mazel-Gee Goerss--Hopkins obstruction theory via model ?-categories Peter Teichner 2016
Benjamin B. McMillan Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations Robert Bryant 2016
Christian Hilaire John Lott 2015
Xin Jin David Nadler 2015
Heather Lee Denis Auroux 2015
Eric C. Peterson Constantin Teleman 2015
Zachary Aaron Sylvan Denis Auroux 2015
Sebastian Hurtado Homomorphisms between groups of diffeomorphisms Ian Agol 2014
Pablo Solis Antimicrobial Applications of Ambient-Air Plasmas Constantin Teleman 2014
Renato Vianna On Exotic Lagrangian Tori in CP2 Denis Auroux 2014
Nathaniel Watson Peter Teichner 2014
Dario Beraldo Loop Group Actions on Categories and Whitaker Invariants Constantin Teleman 2013
Daniel Berwick Evans Supersymmetric sigma models, partition functions, and the Chern-Gauss-Bonnet Theorem Peter Teichner 2013
Keon Choi The embedded contact homology of toric contact manifolds Michael Hutchings 2013
Daniel Cristofaro-Gardiner Some Results Involving Embedded Contact Homology Michael Hutchings 2013
Vinicius Gripp Barros Ramos The Asymptotics of ECH Capacities and Absolute Gradings on Floer Homologies Michael Hutchings 2013
Daniel Halpern-Leistner Geometric Invariant Theory and Derived Categories of Coherent Sheaves Constantin Teleman 2013
Shawn McDougal Representing Sato-Levine Invariants by Whitney Tower Intersections Peter Teichner 2013
Benoit Jubin The Tangent Functor Monad and Foliations Alan D. Weinstein 2012
Haijian Kevin Lin Two Studies of Topological Quantum Field Theory in Two Dimensions Constantin Teleman 2012
Yi Liu Nonzero Degree Maps Between Three Dimensional Manifolds Ian Agol 2012
Daniel Pomerleano Curved String Topology and Tangential Fukuya Categories Constantin Teleman 2012
George W. Melvin Constantin Teleman