About Me
I received my Ph.D. from Rutgers University in 2013 under the supervision of Feng Luo. From 2013-2016, I was a postdoc at Purdue University, and I am now a postdoc at the University of California, Santa Barbara.
My research interests lie broadly in the fields of low-dimensional topology, hyperbolic geometry, and geometric group theory. Much of my previous work and future projects are in line with a recent trend in low-dimensional topology directed at obtaining effective (i.e. constructive) proofs of various existence results. Such proofs can offer concrete topological and geometric information about manifolds that existence results alone cannot. Often the questions concern a bridge between algebra and topology/geometry, an example being the connection between effective separability properties of the fundamental groups of hyperbolic manifolds and quantitative aspects of promoting immersed sets in such manifolds to embeddings in finite covers. My projects have expanded into related fields of mathematics, including right-angled Artin groups and cube complexes, Teichmüller theory, the mapping class groups of surfaces, the combinatorics of 3-manifolds, as well as the representation theory of surfaces and 3–manifolds. More recently, I have also become interested in the mapping class groups of infinite type surfaces, referred to as "big mapping class groups".
Here is my CV (last updated 10/30/18).
My research interests lie broadly in the fields of low-dimensional topology, hyperbolic geometry, and geometric group theory. Much of my previous work and future projects are in line with a recent trend in low-dimensional topology directed at obtaining effective (i.e. constructive) proofs of various existence results. Such proofs can offer concrete topological and geometric information about manifolds that existence results alone cannot. Often the questions concern a bridge between algebra and topology/geometry, an example being the connection between effective separability properties of the fundamental groups of hyperbolic manifolds and quantitative aspects of promoting immersed sets in such manifolds to embeddings in finite covers. My projects have expanded into related fields of mathematics, including right-angled Artin groups and cube complexes, Teichmüller theory, the mapping class groups of surfaces, the combinatorics of 3-manifolds, as well as the representation theory of surfaces and 3–manifolds. More recently, I have also become interested in the mapping class groups of infinite type surfaces, referred to as "big mapping class groups".
Here is my CV (last updated 10/30/18).
Papers and Preprints
0. PhD thesis: Quantifying Algebraic Properties of Surface Groups and 3-manifold Groups.
1. On a Theorem of Peter Scott. Proc. Amer. Math. Soc. 142 (2014) pp 2891-2906.
2. Residual Finiteness Growths of Virtually Special Groups. Math. Z. 279 (2015), no. 1-2, pp 297-310. (with K. Bou-Rabee and M.F. Hagen)
3. On the Residual Finiteness Growths of Particular Hyperbolic Manifold Groups. Geom Dedicata. 185 (2016) Issue 1, pp 87-103.
4. Quantifying separability in virtually special groups. Pacific J. Math. 284-1 (2016), pp 103-120. (with M.F. Hagen)
5. Zariski closures and Subgroup Separability. Selecta Math. 23 (2017) Issue 3, pp 2019-2027. (with L. Louder and D.B. McReynolds)
6. Building hyperbolic metrics suited to closed curves and applications to lifting simply. Math. Res. Lett. 24 (2017), no. 3, pp 593-617.
(with T. Aougab, J. Gaster, and J. Sapir)
7. Algebraic and topological properties of big mapping class groups. (with N. Vlamis) Accepted to Algebr. Geom. Topol.
8. The first integral cohomology of pure mapping class groups. (with J. Aramayona and N. Vlamis) Submitted.
9. Effective virtual and residual properties of some arithmetic hyperbolic 3-manfiolds. (with J. Deblois and N. Miller) Submitted.
10. Covers of surfaces, Kleinian groups, and the curve complex. (with T. Aougab and S. Taylor) Preprint.
11. A combinatorial characterization of hyperbolic 3-manifolds: Thurston's Lego (with D. Cooper) In preparation.
Here is a link to a video of my talk at the Effective and Algorithmic Methods in Hyperbolic Geometry and Free Groups Workshop. (May 2016)
Here is a link to a video of my talk at the GGD/GEAR Seminar at UIUC (October 2014).
1. On a Theorem of Peter Scott. Proc. Amer. Math. Soc. 142 (2014) pp 2891-2906.
2. Residual Finiteness Growths of Virtually Special Groups. Math. Z. 279 (2015), no. 1-2, pp 297-310. (with K. Bou-Rabee and M.F. Hagen)
3. On the Residual Finiteness Growths of Particular Hyperbolic Manifold Groups. Geom Dedicata. 185 (2016) Issue 1, pp 87-103.
4. Quantifying separability in virtually special groups. Pacific J. Math. 284-1 (2016), pp 103-120. (with M.F. Hagen)
5. Zariski closures and Subgroup Separability. Selecta Math. 23 (2017) Issue 3, pp 2019-2027. (with L. Louder and D.B. McReynolds)
6. Building hyperbolic metrics suited to closed curves and applications to lifting simply. Math. Res. Lett. 24 (2017), no. 3, pp 593-617.
(with T. Aougab, J. Gaster, and J. Sapir)
7. Algebraic and topological properties of big mapping class groups. (with N. Vlamis) Accepted to Algebr. Geom. Topol.
8. The first integral cohomology of pure mapping class groups. (with J. Aramayona and N. Vlamis) Submitted.
9. Effective virtual and residual properties of some arithmetic hyperbolic 3-manfiolds. (with J. Deblois and N. Miller) Submitted.
10. Covers of surfaces, Kleinian groups, and the curve complex. (with T. Aougab and S. Taylor) Preprint.
11. A combinatorial characterization of hyperbolic 3-manifolds: Thurston's Lego (with D. Cooper) In preparation.
Here is a link to a video of my talk at the Effective and Algorithmic Methods in Hyperbolic Geometry and Free Groups Workshop. (May 2016)
Here is a link to a video of my talk at the GGD/GEAR Seminar at UIUC (October 2014).
Upcoming Talks:
Classical and quantum 3-manifold topology, Melbourne, Australia, December 2018
5th Annual Graduate Student Conference in Algebra, Geometry, and Topology, Temple University, June 2019
Combinatorial and algebraic aspects of geometric structures, Chiang Mai University, Thailand July 2019
5th Annual Graduate Student Conference in Algebra, Geometry, and Topology, Temple University, June 2019
Combinatorial and algebraic aspects of geometric structures, Chiang Mai University, Thailand July 2019
Miscellaneous
Here is a link to my Math Genealogy.