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Joshua Cruz
Student

I am a graduate student of Les Saper. For my thesis, I have been calculating L

I have done work in several other fields as well, including mathematical neuroscience, applied sheaf theory, and applied topology more broadly.

^{2}cohomology groups of incomplete metrics coming from singular complex varieties. This work is an interesting example of the interplay between analysis and topology.I have done work in several other fields as well, including mathematical neuroscience, applied sheaf theory, and applied topology more broadly.

### Current Research Interests

For my thesis, I have been calculating L

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.

My last paper studies metric completeness of categories with an interleaving distance. Completeness is an important condition if one wishes to study convergence of probability measures on these spaces. In the paper, I give relatively straightforward categorical conditions for a category with a flow to be metrically complete. I also describe a metric completion of the category using the Yoneda embedding.

^{2}cohomology groups of incomplete metrics coming from singular complex varieties. For example, we hope 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.

My last paper studies metric completeness of categories with an interleaving distance. Completeness is an important condition if one wishes to study convergence of probability measures on these spaces. In the paper, I give relatively straightforward categorical conditions for a category with a flow to be metrically complete. I also describe a metric completion of the category using the Yoneda embedding.

### Current Appointments & Affiliations

### Contact Information

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