**Random groups at density act non-trivially on a CAT(0) cube complex.** [arXiv] To appear in *Transactions of the AMS*.

*For random groups in the Gromov density model at , we construct walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities and , respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density .*

**Graphs with Many Hamiltonian Paths**. With Erik Carlson, Willem Fletcher, Chi Nguyen, Xingyi Zhang. [arXiv] Submitted.

*A graph is hamiltonian-connected if every pair of vertices can be connected by a hamiltonian path, and it is hamiltonian if it contains a hamiltonian cycle. Every hamiltonian-connected graph is hamiltonian, however we also construct families of nonhamiltonian graphs with `many’ hamiltonian paths, where ‘many’ is interpreted with respect to the number of pairs of vertices connected by hamiltonian paths. We then consider minimal graphs that are hamiltonian-connected; we show that any order- graph that is hamiltonian-connected graphs must have edges, and we construct an infinite family of graphs realizing this minimum.*

**Property (T)** **in -gonal random groups***. Glasgow Mathematical Journal*, 64 (3): 734-738, 2022. [arXiv]

*The -gonal models of random groups are defined as the quotients of free groups on n generators by cyclically reduced words of length k. As tends to infinity, this model approaches the Gromov density model. In this paper we show that for any fixed , if positive -gonal random groups satisfy Property (T) with overwhelming probability for densities , then so do -gonal random groups, for any . In particular, this shows that for densities above , groups in -gonal models satisfy Property (T) with probability 1 as approaches infinity.*

**Knot projections with a single multi-crossing.** With Colin Adams, Thomas Crawford, Benjamin DeMeo, Michael Landry, Alex Tong Lin, Seojung Park, Saraswathi Venkatesh, Farrah Yhee. *Journal of Knot Theory and Its Ramifications*. Volume 24 (3) 2015. [arXiv]

*Introduced recently, an -crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an übercrossing projection, a knot projection with a single -crossing. Such a projection is necessarily composed of a collection of loops emanating from the crossing. We prove the surprising fact that all knots have a special type of übercrossing projection, which we call a petal projection, in which no loops contain any others. The rigidity of this form allows all the information about the knot to be concentrated in a permutation corresponding to the levels at which the strands lie within the crossing. These ideas give rise to two new invariants for a knot : the übercrossing number , and petal number . These are the least number of loops in any übercrossing or petal projection of , respectively. We relate and to other knot invariants, and compute for several classes of knots, including all knots of nine or fewer crossings.*

**Complementary regions of multi-crossing projections of knots.**[arXiv]

*An increasing sequence of integers is said to be universal for knots if every knot has a reduced regular projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. Adams, Shinjo, and Tanaka have, in a work, shown that and (where is a positive integer greater than ), among others, are universal. In a forthcoming paper, Adams introduces the notion of a multi-crossing projection of a knot. An -crossing projection is a projection of a knot in which each crossing has strands, rather than strands as in a regular projection. We then extend the notion of universality to such knots. These results allow us to prove that is a universal sequence for both n-crossing knot projections, for all . Adams further proves that all knots have an -crossing projection for all positive . Another proof of this fact is included in this paper. This is achieved by constructing -crossing template knots, which enable us to construct multi-crossing projections with crossings of any multiplicity.*

## Other Papers

*Mathematics of the Fon people.* With T Helke. In progress.

Topic Proposal: Cubulated Groups