Mathematical Neuroscience Lab

University of Nebraska -- Lincoln

229 Avery Hall

University of Nebraska -- Lincoln

Lincoln, Nebraska 68588

402-472-7233

cgiusti2 "at" standard university e-mail suffix

My research interests currently span two seemingly distant areas. I began my research career working with computational aspects of algebraic topology, and I have recently begun work in mathematical neuroscience. There are a surprising number of ways in which one can apply the tools of algebraic topology to modern problems in neuroscience, and I am regularly stumbling across new ones.

My work in neuroscience is just beginning, but currently focuses on understanding neural coding from a topological perspective. All work here on this topic thus far is joint with Carina Curto and Vladimir Itskov.

My ongoing projects in "pure" algebraic topology involve understanding spaces of knots using cellular models, computations in group cohomology (with Paolo Salvatore and Dev Sinha) and a geometric look at RO(Z/2)-graded cohomology (with Bill Kronholm).

One facet of my dissertation was the construction of a family of combinatorial knot spaces called *plumbers' knots*. These spaces admit an algorithm for classification of components which can be exploited to algorithimcally determine if two topological knots are isotopic. Here are images of representatives of the components of the first two non-trivial spaces of plumbers'
knots.

- "Plumbers' knots" (arXiv:0811.2215v2 [math.AT]) (to appear in Fund. Math.)
- (with P. Salvatore and D. Sinha) "Mod-two cohomology of symmetric groups as a Hopf ring" (arXiv:0909.3292v1 [math.AT]) (J. Top. 2012; doi: 10.1112/jtopol/jtr031)
- "Unstable Vassiliev theory" (arxiv:1107.4717v1 [math.AT]) (submitted)
- (with D. Sinha) "Fox-Neuwirth cell structures and the cohomology of symmetric groups" (arXiv:1110.4137v3 [math.AT]) (to appear in Pub. del Centro De Giorgi)

- My PhD dissertation: "Plumbers' knots and unstable Vassiliev theory".