# Research

For you Simpsons fans out there, I like to think of myself as a *hyperbolic topologist*. The terminology comes from the following Professor Frink quote: *"Well, it should be obvious to even the most dim-witted individual who holds an advanced degree in hyperbolic topology, that Homer Simpson has stumbled into... the third dimension!"*

### Research Interests

- low-dimensional topology
- mapping class groups; geometric and topological group theory
- curve graphs; combinatorics
- hyperbolic geometry; identities on hyperbolic manifolds
- Riemann surfaces; Teichmüller theory (both finite- and infinite-dimensional)
- convex real projective structures; Hitchin representations

### Preprints

- Federica Fanoni, Sebastian Hensel, and Nicholas G. Vlamis.
*Big mapping class groups acting on homology* - Javier Aramayona, Priyam Patel, and Nicholas G. Vlamis.
*The first integral cohomology of pure mapping class groups*

*A pdf link for a preprint indicates that there is a new version available that is not currently posted on the arXiv.

### Publications

- Natalia Pacheco-Tallaj
^{†}, Kevin Schreve, & Nicholas G. Vlamis.*Thurston norms of tunnel number-one manifolds.*J. Knot Theory Ramifications,*.* - Priyam Patel & Nicholas G. Vlamis.
*Algebraic and topological properties of big mapping class groups.**,*no. 7, 4109-4142. - Jonah Gaster, Joshua Evan Greene, & Nicholas G. Vlamis.
*Coloring Curves on Surfaces.*Forum Math. Sigma 6 (2018), e17. (journal | arXiv) - Matthew Durham, Federica Fanoni, & Nicholas G. Vlamis.
*Graphs of curves on infinite-type surfaces with mapping class group actions*. Ann. Inst. Fourier (Grenoble) 68 (2018), no. 6, 2581-2612. - Nicholas G. Vlamis & Andrew Yarmola.
*The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary*. Geom. Dedicata 194 (2018), 81-97.(journal | arXiv) - Ian Biringer & Nicholas G. Vlamis.
*Automorphisms of the compression body graph.*J. London Math. Soc. (2) 95 (2017), no. 1, 94-114.(journal | arXiv) - Nicholas G. Vlamis & Andrew Yarmola.
*Basmajian identities in higher Teichmüller-Thurston theory.*J. Topology 10 (2017), no. 3, 744-764. (journal | arXiv) - Nicholas G. Vlamis.
*Moments of a length function on the boundary of a hyperbolic manifold.*Algebr. Geom. Topol. 15 (2015), no. 4, 1909-1929. - Nicholas G. Vlamis.
*Quasiconformal homogeneity and subgroups of the mapping class group.*Michigan Math. J. 64 (2015), no. 1, 53-75. (journal | arXiv)

(You can also see a list of all my publications/preprints on the arXiv here.)

^{†}Denotes an undergraduate author.

### Notes

*Notes on the topology of mapping class groups*(pdf) [Updated 2/21/2020]

These notes stem from discussions at the AIM workshop "Surfaces of infinite type". In particular, they give proofs that big mapping class groups are not locally compact, not compactly generated, homeomorphic to the irrational numbers, and give an infinite family of mapping class groups that are generated by coarsely bounded sets (and hence have a well-defined quasi-isometry class).

*Update 2/21/2020*: Proposition 18 is now Conjecture 18 (and the following corollary was removed). An error was pointed out to me: please see document for a discussion. Also, added the new and very relevant reference to the paper "Large scale geometry of big mapping class groups" by Mann-Rafi. This paper supercsedes the section of the notes on coarse boundedness, but I still think the explicit examples given in the notes are valuable.

### Recorded Talks

My talk "Algebraic and topological properties of big mapping class groups" at the Groups, Geometry, and Dynamics seminar at UIUC was recorded and can be found here (the link will take you to YouTube).

Other sources for this information: