Research

Areas of Interest

My interests lie in low-dimensional topology and geometry, and particularly deformations of geometric structures on 3-manifolds and their character varieties. I am also interested in topological properties of spatial embeddings of graphs, and especially knotting and linking of random embeddings.

My Ph.D. advisor was Steven Kerckhoff. My thesis was Singular hyperbolic structures on pseudo-Anosov mapping tori. Chapter 4 has been superseded by the similarly titled paper below. Chapter 5 has some results on veering triangulations which may be of interest Much of what I think about with regards to 3-manifolds is about various geometric structures on fibered 3-manifolds. Fibered 3-manifolds are formed by mapping tori of surfaces: take your favorite surface S and take the product with a closed interval. This now gives a cylinder whose boundaries are two copies of the surface S. If you glue these two boundary components together using a homeomorphism of S, you obtain a 3-manifold that is often a hyperboblic 3-manifold. I am interested in studying these and other structures on these manifolds using representations into Lie groups or geometric triangulations.

I also study properties of spatial embeddings of graphs. By a graph, we mean a set of vertices or nodes connected by edges. Abstract graphs come up in many areas of mathematics and computer science, but these graphs can also be realized in 3-dimensional space by placing the vertices as points in space and connecting the points together by arcs whenever there is an edge connecting the corresponding vertices. Sometimes these graphs contain knots or links in them, and I am interested in the properties of such knots and links. I have advised a handful of students on projects related to this topic either as independent research or as part of the Rose-Hulman Mathematics REU

If you are a current student interested in studying either of these topics as an independent study course or as a thesis, feel free to contact me. I also have a handful of other ideas for projects in topology and geometry, including Apollonian circle packings and mapping class groups.

Papers

with Yasmin Aguillon, Eric Burkholder, Xingyu Cheng, Spencer Eddins, Emma Harrell, Elijah Leake, Pedro Morales, Linking number of monotonic cycles in random book embeddings of complete graphs, J. Knot Theory Ramif. 32 (2023), 2350043. Link

Deformations of reducible SL(n,C) representations of fibered 3-manifold groups, Osaka J. Math 59 (2022), 515-527. Link

Random and polygonal spatial graphs, Encyclopedia of Knot Theory. Chapman and Hall 2020.

with Erica Flapan, Ryo Nikkuni, Stick number of non-paneled knotless spatial graphs, New York J. Math. 26 (2020), 836-852. Link

with Erica Flapan, Linking number and writhe in random linear embeddings of graphs, J. Math. Chem. 54 (2016), 1117-1133. Link

Hyperbolic structures from Sol on pseudo-Anosov mapping tori, Geom. Topol. 20 (2016), 437-468. Link

with Jason Bustamante, Jared Federman, Joel Foisy, Kevin Matthews, Kristen McNamara, Emily Stark, Kirsten Trickey, Intrinsically linked graphs in projective space, Algebr. Geom. Topol. 9 (2009), 1255-1274. Link

with L. G. de Pillis, K. Renee Fister, W. Gu, Tiffany Head, Kenny Maples, Todd Neal, Anand Murugan, Optimal Control of Mixed Immunotherapy and Chemotherapy of Tumors, J. Biol. Syst. 16 (2008), 51-80. Link

with J. B. Nowak, J. A. Neuman, L.G. Huey, D. J. Tanner, J. S. Holoway, T. B. Ryerson, G. J. Frost, S. A. McKeen, F. C. Fehsenfeld, A chemical ionization mass spectrometry technique for airborne measurements of ammonia, J. Geophys. Res. 112 (2007), D10S02. Link

with L. G. de Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, T. Neal, A. Murugan, Chemotherapy for Tumors: An Analysis of the Dynamics and a Study of Quadratic and Linear Optimal Controls, Math. Biosci. 209 (2007), 292-315. Link

with Hans Chaumont, Erica Flapan, Woohyung Lee, Sarah Rundell, An infinite family of almost unknotted θ4 graphs, (in progress, draft available upon request).

Miscellaneous

Circle Inversions and Applications to Euclidean Geometry: A supplement that I co-wrote with Shlomo Libeskind for his book Euclidean and Transformational Geometry : Deductive Inquiry back when I was an undergraduate student. The webpage that hosted the supplement no longer seems to exist, and I have noticed an incomplete version floating around that is missing the Contents and Bibliography, so I am posting the latest version here.