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
and spaces of left-invariant linear
or circular orders
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.
(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
will be updated here as we go along.
Papers and preprints
- Unboundedness of some higher Euler classes. (new preprint draft)
- 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.
Not for publication:
Illumination and Security
with Alex Wright.
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.
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)
Large scale geometry of homeomorphism groups (Young Geometric Group Theory, feb. 2016)
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.
Mnicourse at MIT, March 2017
Courses at UC Berkeley: (note: assignments and solutions have been removed. E-mail me if you are an instructor looking for resources)
At the University of Chicago
(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).
(2011) Math 112 Studies in Mathematics (number theory).
(2010-2011) Math 131-132-133 (single-variable 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
- I was also involved for many years wtih the U Chicago Directed Reading Program.
- And you may have also seen me at Mathcamp!
- S. Ellia's undergraduate honors thesis (spring 2016). See her amazing website
Seminars and activities
Below are the webpages (including notes or readings) for some seminars, reading groups, and the Grad Student Summer school that I organized.
Upcoming travel and invited talks
(for past schedule, see here