# 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)

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*Spring 2016***Math 277.**Applied Holomorphic Curve Theory (Hutchings)

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*Fall 2015***Math 270.**Gauge Theory (Wehrheim)

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*Spring 2015***Math 276.**Gauge Field Theory (Teichner)

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*Spring 2015***Math 277.**Ricci Flow (Lott)

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*Spring 2014***Math 276.**Functorial Field Theories and Ring Spectra (Teichner)

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*Spring 2014***Math 276.**Regularization of Moduli Spaces (Wehrheim)

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*Fall 2013***Math 278.**Pseudoholomorphic Curves (Wehrheim)

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*Spring 2013***Math 277.**The Quantum Riemann-Roch-Hirzebruch Theorem (Givental)

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*Fall 2012***Math 277.**Contact Homology (Hutchings)

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*Spring 2012***Math 277.**Kaehler Geometry (Lott)

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*Spring 2011***Math 270.**The Blob Complex (Teichner)

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*Spring 2011***Math 276.**The Index Theorem (Teichner)

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*Fall 2010***Math 276.**Seiberg-Witten-Floer Theory (Hutchings)

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*Fall 2010***Math 277.**Optimal Transport (Lott)

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*Spring 2010***Math 270.**Perturbative Quantization (Teichner)

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*Spring 2010***Math 276.**Three-Manifold Topology (Agol)

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*Fall 2009***Math 277.**Mirror Symmetry (Auroux)

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*Fall 2009***Math 277.**Exterior Differential Systems (Bryant)

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*Fall 2009***Math 277.**Ricci Flow (Lott)

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*Spring 2009***Math 270.**Topological Field Theory (Teichner)

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*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 | Associate 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 |

Song Sun | Associate Professor | Differential geometry |

Peter Teichner | Professor Emeritus | 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 |
---|---|---|

James Conway | Lecturer | Contact and symplectic geometry |

Joshua Evan Greene | Simons Fellow and Visiting Fellow | Geometric topology, combinatorics |

François Labourie | Chancellor's Professor | Differential geometry, discrete subgroups, hyperbolic dynamics |

JeongHyeong Park | Visiting Scholar | Differential geometry, harmonic spaces, spectral geometry, geometric structures on manifolds |

Alexandru Scorpan | Visiting Scholar | 4-manifolds, mathematical publishing |

Jean Taylor | Visiting Scholar | Geometric measure theory, models of crystal growth |

Rui Wang | Lecturer | Symplectic geometry, moduli spaces |

## Postdocs

Name | Title | Research Interests |
---|---|---|

David Corwin | RTG Post-doctoral Scholar | Arithmetic geometry, arithmetic topology |

Gabriel Dorfsman-Hopkins | RTG Post-doctoral Scholar | Arithmetic geometry, p-adic geometry, mathematical illustration, 3D printing |

Yu-Wei Fan | Morrey Visiting Assistant Professor | Algebraic geometry, mirror symmetry |

Nicholas Miller | Morrey Visiting Assistant Professor | Hyperbolic geometry, low-dimensional topology |

Antoine Song | Clay Postdoctoral Fellow | Differential geometry, minimal surfaces |

Martin Speirs | Postdoc | Topological Hochschild homology, algebraic K-theory |

## 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 Emeritus | 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 |

Melanie Matchett Wood | Professor | Number theory |

Hung-Hsi Wu | Professor Emeritus | Real and complex geometry |

## Graduate Students

Name | Dissertation Supervisor |
---|---|

Nicolas Brody | |

Aaron Brookner | |

Julian Chaidez | |

Ethan Dlugie | |

Christopher Eur | David Eisenbud |

Christopher Kuo | |

Yi LAi | |

Hongyi Liu | |

Calvin McPhail-Snyder | |

Kyle Miller | Ian Agol |

Xianglong Ni | |

Eduardo Oregon Reyes | |

Yingdi Qin | |

Eugene Rabinovich | |

Ritvik Ramkumar | David Eisenbud |

Alexander Sherman | |

Michael Smith | |

German Stefanich | David Nadler |

Dun Tang | Alexander B. Givental |

Chi Cheuk Tsang | |

Luya Wang | |

Ben Wormleighton | David Eisenbud |

Xiaohan Yan | |

Junsheng Zhang | |

Ziwen Zhao |

## Recent Ph.D.s

Name | Dissertation Title | Dissertation Supervisor | Year |
---|---|---|---|

Alexander Appleton | Singularities in U (2)-invariant 4d Ricci flow | Richard Bamler, Jon Wilkening | 2019 |

Catherine Cannizzo | Homological mirror symmetry for the genus 2 curve in an abelian variety and its generalized Strominger-Yau-Zaslow mirror | Denis Auroux | 2019 |

Alexander Carney | The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields | Xinyi Yuan | 2019 |

Kevin Donoghue | A Spin TQFT Related to the Ising Categories | Ian Agol | 2019 |

Benjamin Filippenko | Polyfolds and Persistence | Katrin Wehrheim | 2019 |

Benjamin Gammage | Microlocal sheaves and mirror symmetry | David Nadler | 2019 |

Andrew Hanlon | Monodromy of Fukaya-Seidel categories mirror to toric varieties | Denis Auroux | 2019 |

Jeff Hicks | Tropical Lagrangians and Homological Mirror Symmetry | Denis Auroux | 2019 |

Mihai Munteanu | Nontrivial tori in spaces of symplectic embeddings | Michael Hutchings | 2019 |

Morgan Weiler | Mean action of periodic orbits of area-preserving annulus diffeomorphisms | Michael Hutchings | 2019 |

Harrison Chen | A Localization Theorem for Derived Loop Spaces and Periodic Cyclic Homology | David Nadler | 2018 |

Chris Gerig | Seiberg-Witten and Gromov invariants for self-dual harmonic 2-forms | Michael Hutchings | 2018 |

Daniel Lowengrub | Applications of the Intersection Theory of Singular Varieties | Vivek Shende | 2018 |

Wolfgang Schmaltz | Gromov-Witten Axioms for Symplectic Manifolds via Polyfold Theory | Katrin Wehrheim | 2018 |

Tao Su | A Hodge-theoretic study of augmentation varieties associated to Legendrian knots/tangles | Vivek Shende, Richard E.Borcherds | 2018 |

Thunwa 'Nics' Theerakarn | Locally Volume Collapsed 4-Manifolds with Respect to a Lower Sectional Curvature Bound | John Lott | 2018 |

Ryan George Thorngren | Combinatorial Topology and Applications to Quantum Field Theory | Vivek Shende | 2018 |

Franco Vargas Pallete | On Renormalized Volume | Ian Agol | 2018 |

Patrick Wilson | Asymptotically Conical Metrics and Expanding Ricci Solitons | John Lott | 2018 |

Zhengyi Zhou | Morse-Bott and Equivariant Theories Using Polyfolds | Katrin Wehrheim | 2018 |

Kuan-Ying Fang | Geometric Constructions of Mapping Cones in the Fukaya Category | Denis Auroux | 2017 |

Alvin Kerber | Quasi-Fuchsian surface subgroups of infinite covolume Kleinian groups | Ian Agol | 2017 |

Qiao Zhou (Elaine) | Applications of Toric Geometry to Geometric Representation Theory | David Nadler | 2017 |

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 | |

George W. Melvin | Constantin Teleman |