I am a Morrey Visiting Assistant Professor / NSF postdoc at Berkeley
Research interests: Topology, geometry, geometric group theory, dynamics...
More specifically...
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 leftinvariant 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 largescale 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
(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.
(6/2016) New preprint: The largescale geometry of homeomorphism groups. With Christian Rosendal.
We show the group of isotopically trivial homeomorphisms has a welldefined quasiisometry type and study its geometry.
(5/2016) New paper! Group orderings, dynamics, and rigidty with Cristobal Rivas
And here's an abstract (although you could also just read the introduction to the paper)
(5/2016) I'm honored to receive the Berkely mathematics distinguished teaching award. Thanks, MUSA!
(3/2016) Update on our Foliations seminar at Stanford. We're about to start a new topic: spaces of codimension 1 foliations on 3manifolds. See the link for details.
Papers and preprints
 Strong distortion in transformation groups. with Frederic Le Roux.
 Group orderings, dynamics, and rigidty. With Cristobal Rivas
 The largescale 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 205 (2016), 30333056.  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), 4955.  Leftorderable groups that don't act on the line.
Math. Zeit. 280 no 3 (2015) 905918  Spaces of surface group representations.
Inventiones Mathematicae. 201, Issue 2 (2015), 669710. (link to published version)  Diffeomorphism groups of balls and spheres.
New York J. Math. 19 (2013) 583596.  The simple loop conjecture is false for PSL(2,R).
Pacific Journal of Mathematics 2692 (2014), 425432.  Homomorphisms between diffeomorphism groups.
Ergodic Theory and Dynamical Systems, 35 no. 01 (2015) 192214.  Bounded orbits and global fixed points for groups acting on the plane.
Algebraic and Geometric Topology 12 (2012) 421433  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 3manifold groups, and the thurston norm, at the end.
In Progress:
 With Maxime Wolff: Rigidity and geometricity of surface groups acting on S^1.
 (abandoned project) Extending group actions from dM to M. Email me if you'd like a copy.
Lecture series:

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 . 
Doityourself Hyperbolic Geometry. A course I taught at Mathcamp.
Notes are a work in progress, feedback welcome!  The minicourse 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 201415 ``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.
Teaching
Courses at UC Berkeley:note: HW assignments and test solutions have been removed from many of the websites below. Email me if you are an instructor looking for resources)

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
 (20132014) Math 161162163 (Honors Calculus). Inquiry based learning. Coinstructor 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.
 (20102011) Math 131132133 (singlevariable calculus)
Other teaching and supervision
I 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 coorganize the Berkeley topology seminar with Hongbin Sun
 The Noetherian Ring, a group for women in math at UC Berkeley, has activities and occasional speakers.
 BerkeleyStanford Foliations seminar from Spring 2016. Coorganized with Sander Kupers and Bena Tshishiku.

The Graduate student summer school: Diffeomorphism groups: algebra, topology, homology that I ran June 812, 2015 at UC Berkeley.
Lecture notes from my and from Bena Tshishiku's minicourses, and other information are available through the link. 
201415 yearlong BerkeleyStanford joint seminar on cohomology of diffeomorphism groups, coorganized with Sam Nariman.
Upcoming travel and invited talks
(for past schedule, see here )2016:
 February 1519 Young geometric group theory 5 (Karlsruhe, Germany).
 March 22 Cornell topology and geometric group theory seminar
 March 24 CCNY math colloquium.
 April 13 Redbud topology conference. Here's the exercises from my first talk.
 April 910 AMS western sectional meeting, Utah.
 Mid May  mid June, I will be at IMJ (Jussieu), Paris
...planning to visit this workshop on Polish groups and geometry while I'm there  June 2025 Groups acting on manfolds summer school in Rio.
 July 1117 Foliations 2016, Poland.
 September 510 Foliations, Laminations and Contact structures in Munich.
 Fall 2016 MSRI program on geometric group theory.
Speaking at Amenability, coarse embeddability and fixed point properties workshop.  Oct 37 Surfaces in Luminy at CIRM.
 Oct 31 UW Madison colloquium
 Nov. 10 Cornell math colloquium
 Nov. 16 Brown math colloquium
 Nov. 28 University of Michigan topology seminar
 December 1719, at MIT.
2017:
 Jan.Feb. Visiting Stanford University. Colloquium on Jan. 12
 February 27  March 3 Workshop: Groups of dynamical origin
 March (dates to be confirmed) Visiting MIT.
 March 911 Spring Topology and Dynamics Conference
 March 27 U Chicago dynamics seminar
 May 1519 Low dimensional geometry and topology at Vanderbilt
 May 22  June 2 Georgia interntational topology conference
Just for fun...
Here is a template that you can use to make this Escher tessellation on a Thurston wrinklypaper model of hyperbolic space!(and a couple of words of explanation)