Nicholas Touikan

Contact

Postal service

Dr. Nicholas Touikan
Stevens Institute of Technology
Department of Mathematical Sciences
1 Castle Point Terrace
Hoboken, New Jersey
07030, USA

Email

nicholas.touikan@gmail.com

Telephone

+1 201 216 8596

In person

My Office is Kidde 229

Bio

I am currently a Teaching Associate Professor at the Department of Mathematical Sciences of the Stevens Institute of Technology. I dislike injustice.

Research Interest

My field of specialty is Geometric Group Theory, that is the study of groups as geometric objects. More specifically I study algorithmic problems and equations in discrete groups that have large scale non-positive curvature. I also study connections and analogies between group theory and low dimensional topology.

At Stevens I am also an active member of the Calculus Project, a research effort to determine how to effectively use new teaching technologies and promote reasoning based learning.

Published or accepted articles

  1. Magnus pairs in, and free conjugacy separability of, limit groups (with Lars Louder), accepted by Geometriae Dedicata. (arXiv, slides)
  2. Detecting geometric splittings in finitely presented groups, accepted by Transactions of the American Mathematical Society. (arXiv)
  3. Strong accessibility for finitely presented group (with Lars Louder), Geometry & Topology, Volume 21 (2017), 1805-1835. (journal, arXiv)
  4. Multipass automata and group word problems (with Tullio Ceccherini-Silberstein, Michel Coornaert, Francesca Fiorenzi, Paul E. Schupp), Theoretical Computer Science, Volume 600 (2015), 19-33. (journal, arxiv)
  5. On the one-endedness of graphs of groups, Pacific Journal of Mathematics, Volume 278 (2015), Issue 2, 463-478. (journal, arXiv, slides)
  6. Bulitko's Lemma for acylindrical splittings, Journal of Algebra, Volume 406 (2014), 251-271. (journal, arXiv)
  7. The fully residually F quotients of F*< x,y >, Groups, Geometry and Dynamics, Volume 6 (2012), Issue 1, 155-220. (journal, arXiv)
  8. The Solvability Problem for Quadratic Equations over Free Groups is NP-Complete (with Olga Kharlampovich, Igor Lysenok, and Alexei Myasnikov), Theory of Computing systems Volume 47 (2012), Issue 1, 250-258. (journal, arXiv, slides)
  9. The equation w(x, y) = u over free groups: an algebraic approach, Journal of group theory, Volume 12 (2009), Issue 4, 611-634. (journal, arXiv)
  10. A Fast Algorithm for Stallings' Folding Process, International Journal of Algebra and Computation, Volume 16 (2006), Issue 6, 1031-1045. (journal, arXiv)

Preprints (submitted)

  • Deciding Isomorphy using Dehn fillings: the splitting case (with François Dahmani) (arXiv, slides)
  • On geodesic ray bundles in hyperbolic groups (arXiv)

Work in progress

  • With Inna Bumagin, I am writing a paper on Makanin-Razborov diagrams for relatively hyperbolic is being written. (slides)
  • With Alex Taam, I am writing a paper showing quasi-isometric rigidity of certain graphs closed surface groups with cyclic edge groups. (slides)
  • I am also writing up a note on how to use the fast Stallings's Folding algorithm for graphs of finite groups. (slides)
  • With Mark Hagen, I am working on a collapsing technique for cube complexes. These collapses can be thought of high dimensional "Stallings's unfoldings". This method gives a fully cubical proof of Stallings's Theorem on groups with infinitely many ends and promises to have many more applications.