# 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 2021**,

**Math 276.**Knot Theory (Agol)

**,**

*Fall 2020***Math 277.**Poisson Geometry (Reshetikhin)

**,**

*Fall 2020***Math 277.**Ricci Flow (Bamler)

**,**

*Spring 2020***Math 277.**Categorical Structures in Symplectic Geometry (Wehrheim)

**,**

*Spring 2018***Math 277.**Complex/Kaehler Geometry (Sun)

**,**

*Fall 2017***Math 276.**Knots and Links (Agol)

**,**

*Spring 2017***Math 277.**Ricci curvature (Bamler)

**,**

*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 3.

## Senate Faculty

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

Mina Aganagic | Professor | String theory, mathematical physics |

Ian Agol | Professor | Low-dimensional topology |

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 Emeritus | Mathematical physics, Low-dimensional topology, Representation theory |

Song Sun | Professor | Differential geometry, Complex geometry, Geometric analysis |

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 | educating to counter oppression |

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 |
---|---|---|

Luis Giraldo Suarez | Visiting Scholar | Algebraic Geometry, Holomorphic dynamical systems, Symplectic and contact geometry. |

Yi-Jen Lee | Visiting Scholar | Gauge Theory and Symplectic Topology (e.g. Seiberg-Witten theory), Heegaard Floer Homology and Pseudo-Holomorphic Curves |

Jason D. Lotay | Chancellor's Professor | Differential geometry, geometric analysis |

Alexandru Scorpan | Visitor | 4-manifolds, mathematical publishing |

Norman Sheu | Unit 18 Lecturer | Differential geometry and mathematical physics |

Alex Waldron | Visiting Scholar | Geometric flows, gauge theory, differential geometry |

Rui Wang | Unit 18 Lecturer | Symplectic geometry and contact geometry, Mathematical physics |

## Postdocs

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

Eric Chen | NSF Postdoctoral Scholar | Geometric analysis, differential geometry |

Ivan Danilenko | Morrey Visiting Assistant Professor | Algebraic geometry, representation theory, mathematical physics. |

Colleen Delaney | Postdoctoral Scholar | Mathematical physics, quantum topology, applied category theory, quantum computation |

McFeely Jackson Goodman | NSF Postdoctoral Scholar & Unit 18 Lecturer | Differential geometry and topology. |

Peter Haine | NSF Postdoctoral Scholar and UC President's Postdoctoral Fellow | Homotopy theory, (derived) algebraic geometry, and related subjects |

Tobias Shin | NSF Postdoctoral Scholar | Homotopy theory of varieties in positive characteristic, and almost complex geometry |

Peng Zhou | Morrey Visiting Assistant Professor | Mathematical physics, mirror symmetry, semiclassical analysis |

## Faculty with Related Research Interests

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

Tony Feng | Assistant Professor | Number Theory, Arithmetic Geometry, Langlands Program, Algebraic Topology, Representation Theory |

Mark D. Haiman | Professor | Algebra, combinatorics, and algebraic geometry |

Jenny Harrison | Professor | Functional analysis, Geometric measure theory, Calculus of variations, Mathematical physics |

William M. Kahan | Professor Emeritus | Error analysis, Numerical computations, Computers, Convexity, Large matrices, Trajectory problems |

Arthur E. Ogus | Professor Emeritus | Algebraic geometry |

Andrei Okounkov | Professor | Representation theory (of combinatorial flavor) and its application to algebraic geometry, mathematical physics, probability, special functions, and other fields. |

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 | Associate Professor | Geometry |

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

## Graduate Students

## Recent Ph.D.s

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

Irit Huq-Kuruvilla | Twisted K-Theoretic Gromov-Witten Invariants and Euler Characteristics | Alexander Givental | 2023 |

Hongyi Liu | Compactness theorems on hyperkähler 4-manifolds | Song Sun | 2023 |

Eduardo Oregon Reyes | Some aspects of geometric actions of hyperbolic and relatively hyperbolic groups | Ian Agol | 2023 |

Chi Cheuk Tsang | Veering triangulations and pseudo-Anosov flows | Ian Agol | 2023 |

Luya Wang | New structures on embedded contact homology and applications to low-dimensional topology | Michael Hutchings | 2023 |

Nicolas Brody | Groups Acting on Products of Trees | Ian Agol | 2022 |

Christopher Kuo | Symplectic geometric methods in microlocal sheaf theory | Vivek Shende | 2022 |

Kyle Miller | Singularity Theory for Extended Cobordism Categories and an Application to Graph Theory | Ian Agol | 2022 |

Ritvik Ramkumar | The Geometry of Hilbert Schemes on Projective Space | David Eisenbud | 2022 |

Xiaohan Yan | Quantum K-Theory of Flag Varieties and Non-Abelian Localization | Alexander Givental | 2022 |

Julian Chaidez | Convexity In Contact Geometry And Reeb Dynamics | Michael Hutchings | 2021 |

Yi LAi | Ricci flows with non-compact initial conditions | Richard Bamler | 2021 |

Calvin McPhail-Snyder | SL_2(C)-holonomy invariants of links | Nicolai Reshetikhin | 2021 |

Eugene Rabinovich | Factorization Algebras for Bulk-Boundary Systems | Peter Teichner | 2021 |

Michael Smith | Curvature Bounds in Riemannian Geometry | Richard Bamler | 2021 |

German Stefanich | Higher Quasicoherent Sheaves | David Nadler | 2021 |

Christopher Eur | The Geometry of Divisors on Matroids | David Eisenbud | 2020 |

Yingdi Qin | Coisotropic Branes on Tori and Homological Mirror Symmetry | 2020 | |

Alexander Sherman | Spherical and Symmetric Supervarieties | 2020 | |

Ben Wormleighton | Numerics and stability for orbifolds with applications to symplectic embeddings | David Eisenbud | 2020 |

Yue Zhang | Guts, Dehn Fillings and Volumes of Hyperbolic Manifolds | Ian Agol | 2020 |

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 |