I am a Morrey Visiting Assistant Professor / NSF postdoc at Berkeley
In fall 2017, I will move to an assistant professorship at Brown.
Research interests: Topology, geometry, geometric group theory, dynamics...
I study actions of infinite groups on manifolds and the moduli spaces of such actions: character varieties, spaces of flat bundles or foliations, and spaces of left-invariant linear or circular orders on groups.
As part of this, I've worked on the rich relationship between the algebraic and topological structure of diffeomorphism and homeomorphism groups, the large-scale geometry of such groups (e.g. subgroup distortion and dynamical consequences of this), and rigidity phenomenon for group actions, often arising from some geometric structure.
If you'd like to know more, here's a survey paper I wrote for the Handbook of group actions. It's an introduction to groups acting on the circle, through work of Etienne Ghys, Danny Calegari, Bill Goldman, Shigenori Matsumoto, and others, that has served as tools in and inspiration for (one facet of) my own research.
And for a very condensed read, here's the 1 page introduction to a description of my research from September 2016.
Office: 793 Evans hall.
Office hours: by appointment.
My CV is available.
News(4/2017) New paper with A. Clay and C. Rivas: On the number of circular orders on a group
(3/2017) Intorductory minicourse: Groups, Geometry and Rigidity
I'll be giving a series of introductory lectures at MIT on Geometric group theory and a proof of Mostow's strong rigidity theorem.
The course will be self-contained, with no prior background in geometric group theory assumed.
Lectures will be M/W/F 10-11am, from March 6th to March 24th (no lecture March 10) in room 2-255
Course notes and references will be updated here as we go along.
(11/2016) Updated version of Group orderings, dynamics, and rigidty available. This paper discusses the (surprising!) relationship between the space of circular orders on a group G and the space of actions of G on the circle. The updated version has some new results, including a dynamical characterization of isolated circular orders on free groups.
(10/2016) New preprint available: Strong distortion in transformation groups. with Frederic Le Roux.
We answer a question of Schreier from the Scottish Book, and show that many groups of diffeomorphisms have the strong distortion property, implying for instance that any isometric action on a metric space has bounded orbits.
Papers and preprints
- On the number of circular orders on a group. With Adam Clay and Cristobal Rivas.
- Realizing maps of braid groups by surface diffeomorphisms. A short note on some realization problems.
- Strong distortion in transformation groups. with Frederic Le Roux.
- Group orderings, dynamics, and rigidty. With Cristobal Rivas
- The large-scale geometry of homeomorphism groups. With Christian Rosendal.
To appear in Ergodic Theory and Dynamical Systems.
- PL(M) has no Polish group topology.
To appear in Fundamenta Mathetmaticae.
- Rigidity and flexibility of group actions on S^1.
To appear in the Handbook of group actions. L. Ji, A. Papadopoulos, and S.-T. Yau, eds
- Automatic continuity for homeomorphism groups and applications.
With an appendix on the structure of groups of germs of homeomorphism, written with Frederic Le Roux.
Geometry & Topology 20-5 (2016), 3033-3056.
- A short proof that the group of compactly supported diffeomorphisms on a manifold is perfect
following a strategy of Haller, Rybicki and Teichmann. In New York J. Math 22 (2016), 49-55.
- Left-orderable groups that don't act on the line.
Math. Zeit. 280 no 3 (2015) 905-918
- Spaces of surface group representations.
Inventiones Mathematicae. 201, Issue 2 (2015), 669-710. (link to published version)
- Diffeomorphism groups of balls and spheres.
New York J. Math. 19 (2013) 583-596.
- The simple loop conjecture is false for PSL(2,R).
Pacific Journal of Mathematics 269-2 (2014), 425-432.
- Homomorphisms between diffeomorphism groups.
Ergodic Theory and Dynamical Systems, 35 no. 01 (2015) 192-214.
- Bounded orbits and global fixed points for groups acting on the plane.
Algebraic and Geometric Topology 12 (2012) 421-433
- My dissertation, Components of representation spaces (2014) mostly overlaps with the content of the paper "Spaces of surface group representations" above, although I also very briefly discussed rigidity of universal circle actions of 3-manifold groups, and the thurston norm, at the end.
An exposition of some problems involving translation surfaces, from MSRI's dynamics on moduli spaces program.
- With Maxime Wolff: Rigidity and geometricity of surface groups acting on S^1.
- (abandoned project) Extending group actions from dM to M.     E-mail me if you'd like a copy.
Lectures on homeomorphism and diffeomorphism groups (in progress, notes from 2015 summer school, ~40 pages)
Related: many lecture notes from a seminar on Cohomology of diffeomorphism groups here .
Do-it-yourself Hyperbolic Geometry. A course I taught at Mathcamp.
Notes are a work in progress, feedback welcome!
- The mini-course I taught at "Beyond Uniform Hyperbolicity 2015" turned into the survey paper Rigidity and flexibility of group actions on S^1.
Slides and videos from recent talks:
- Boundedness and Distortion in transformation groups at MSRI (December 2016) slides, and a video
- Orderability and groups of homeomorphisms of the circle (Luminy, fall 2016) video
- Large scale geometry of homeomorphism groups (Young Geometric Group Theory, feb. 2016) slides
Groups acting on the circle (MSRI, January 2015) slides and video
(warning: the "notes" from the talk on the video page seem to contain some errors!)
- Three proofs of rigidity of surface group actions at MSRI ``Dynamics on moduli spaces" conference (2015) video
- Components of representation spaces (2013) slides (these are fairly minimalist!)
- Homomorphisms between diffeomorphism groups (2012) slides (2012)
- Many notes from my lectures in the 2014-15 ``cohomology of diffeomorphism groups" seminar can be found here
- Slides from a public lecture Geometry: a walk through mathematical spaces at the Sonoma State University M*A*T*H colloquium, fall 2015.
TeachingMnicourse at MIT, March 2017
- Intorductory 3-week minicourse: Groups, Geometry and Rigidity
Course notes and references for further reading will be updated here as we go along.
Math 141, Elementary Differential Topology.   Course website
- (Fall 2015) Math 113, Abstract Algebra.   Course website
- (Fall 2015) Math 130, The Classical Geometries.   Course website
(Fall 2014) Math 113 Abstract algebra;   sections 4 and 5.   Course website
- (2013-2014) Math 161-162-163 (Honors Calculus). Inquiry based learning. Co-instructor with E. Herman
- (2013) Math 196 Linear Algebra
- (2012) Math 195 Mathematical methods for the social sciences. (multivariable calculus)
- (2012) Math 113 Studies in Mathematics (geometry).   Course materials.
- (2011) Math 112 Studies in Mathematics (number theory).   Course materials.
- (2010-2011) Math 131-132-133 (single-variable calculus)
Other teaching and supervisionI mentored students in the University of Chicago REU in 2009 and 2011. You can find information and student papers here.
I was also involved for many years wtih the U Chicago Directed Reading Program.
And you may have also seen me at Mathcamp!
Currently (2016, spring) I'm supervising S. Ellia's undergraduate honors thesis. See her amazing website !
Seminars and activities
- I co-organize the Berkeley topology seminar with Hongbin Sun
- The Noetherian Ring, a group for women in math at UC Berkeley, has activities and occasional speakers.
- Berkeley-Stanford Foliations seminar from Spring 2016. Co-organized with Sander Kupers and Bena Tshishiku.
The Graduate student summer school: Diffeomorphism groups: algebra, topology, homology that I ran June 8-12, 2015 at UC Berkeley.
Lecture notes from my and from Bena Tshishiku's minicourses, and other information are available through the link.
2014-15 year-long Berkeley-Stanford joint seminar on cohomology of diffeomorphism groups, co-organized with Sam Nariman.
Upcoming travel and invited talks(for past schedule, see here )
- Jan.-Feb. Visiting Stanford University. I'm in office 384-E
Colloquium on Jan. 12
Informal geometry/topology seminar on Jan. 13
- Jan. 26-27 Visiting Caltech
Geometry and Topolgoy Seminar
- February 27 - March 3 Workshop: Groups of dynamical origin
- March 4-24 Visiting MIT.
- March 9-11 Spring Topology and Dynamics Conference
- March 14 Tufts Geometric Group Theory seminar
- March 27 U Chicago dynamics seminar
- April--May--June: Visiting Inst. Math. Jussieu
- Week of May 22 Georgia interntational topology conference
- May 8-12 Approximation, deformation, quasification at the Newton Institute.
- June 27-30 Summer conference on topology and applications
- August 7-11 GEAR retreat at Stanford