Dr Dylan Allegretti
I completed a B.S. in mathematics with honors and a B.A. in physics at the University of Chicago in 2010. I then attended Yale University where I received an M.S. and M.Phil. in 2013 and a Ph.D. in 2016. My thesis, The geometry of cluster varieties from surfaces
, was supervised by Alexander Goncharov.
Broadly speaking, my interests lie in mathematical physics, a subject at the intersection of pure mathematics and theoretical physics. I am interested in using math to solve physics problems and using ideas from physics to solve problems of a purely mathematical nature.
A unifying concept in my work so far is the notion of a cluster variety
. This is a kind of geometric object defined using combinatorial ideas from the theory of cluster algebras. The notion of a cluster variety has found applications in several areas, including quantum Teichmüller theory and the study of wall-crossing phenomena in algebraic geometry and mathematical physics. I am currently working with Tom Bridgeland on an ERC-funded project exploring the relationship between cluster varieties and spaces of stability conditions on triangulated categories.