David Ayala
gmail account: davidayala.math
office: 464C, KAP

pic Welcome. As of August 2012, I am a postdoc in mathematics at University of Southern California supported in part by an NSF grant. Previously, I was a postdoc at Harvard University as well as the University of Copenhagen. I received my PhD through Stanford University under the supervision of Ralph Cohen.
See my Curriculum Vitæ (5 pages) for specifics.


Research:
My background is in algebraic topology with an emphasis on the homotopy theory of manifolds. My research concerns the differential topology of locally defined entities, such as manifolds, and their moduli as informed by conceptual and computational techniques from (derived) algebraic geometry and homotopy theory. Slightly more specifically, my work characterizes invariants of things like manifolds or links which are obtained from algebraic or categorical data -- the goal is to facilitate the construction of a wealth of robust invariants, equipped with desired local-to-global functorialities, with an eye toward calculations among them. Instances of such local invariants include sheaves, cosheaves, motives, and topological field theories; and proposed examples of such invariants include the Turaev-Viro TFT invariants, versions of Heegaard Floer homology, Rozansky-Witten and Khovanov homology, and finite type knot invariants.


Teaching:
I am not teaching this academic year. In the past I have taught two calculus courses, a basic topology course, and a course on the homotopy theory of manifolds, as well as various teaching assistantships. See my CV for specifics.

I supervised a masters project of Casper Guldberg's (University of Copenhagen) on basic constructions in quasi-categories (29 pages).
I supervised a side PhD project of Emanuele Dotto's (University of Copenhagen) where he is used simplicial techniques to consider h-principles with boundary conditions (33 pages).


Other stuff:
Some juggling videos
Some photos of outrageous natural features
Some modest comics
Mojave Run


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