Joshua D. Cruz, Graduate Student
I am a graduate student of Les Saper. For my thesis, I have been calculating L^{2} cohomology groups of incomplete metrics coming from singular complex varieties. For example, one would like to show that the L^{2} cohomology of a small neighborhood of an isolated singular point is the cohomology of the link in low degrees and zero in high degrees. This work is an interesting example of the interplay between analysis and topology.
I also work in applied topology. I wrote a paper in 2016 with Chad Giusti, Vladimir Itskov, and Bill Kronwell on convex codes, a concept coming from neuroscience which describe the neural firing patterns of (e.g.) placeholder cells. More recently, I've been working on applied sheaf theory, much of which was in collaboration with Justin Curry.  Contact Info:
Office Location:  274G Physics  Email Address:    Office Hours:
 Help Room Hours: Monday 68pm in Carr 132
 Education:
BS  Washington State University  2013 
 Research Interests:
Current projects:
Some Results on Max IntersectionComplete Codes, Decomposing Vineyards with Sheaf Theory
I am a student of Les Saper, with broad interests in algebraic topology and complex geometry.
I am also interested in many other fields of mathematics, including geometric analysis, functional analysis, representation theory, stochastic analysis, and applied topology, especially persistent homology.
 Keywords:
Applied Topology • Sheaf theory • Topology
