Featured Research: Prof. Jason DeBlois
I arrived at Pitt in the fall of 2011. Since then, I have taught service courses ranging from calculus to linear algebra to differential equations, several upper-level undergraduate courses, and graduate courses including the differential geometry, linear algebra, and advanced calculus sequences. One of my favorite courses to teach, which I have now taught three times, is algebraic topology (Math 2701), the core graduate course closest to my research specialty.
My research focus is in topology, which roughly speaking is “the study of shapes.” Those I study most closely are 3-dimensional manifolds, which at small scales look like the 3-dimensional space we are all used to, though at global scales they can have more structure. Imagine taking a cube and identifying each of its faces with the one opposite (you would need an extra dimension to perform these identifications extrinsically) — this is a 3-manifold often called the “3-torus.’’ While topologists usually consider spaces up to a notion of equivalence called homeomorphism, which allows shapes to be stretched or smooshed arbitrarily (though without tearing them), the “geometrization theorem” famously proved by Perelman in the early 2000’s allows 3-manifolds to be studied using much more rigid geometric structures. My own work exploits this, in particular often using hyperbolic geometry, the non-Euclidean geometry with multiple parallels to a given straight line through a given point. This is in some sense the “predominant” geometry among 3-manifolds.
Some of my recent projects have studied sphere packings in hyperbolic geometry, volumes of hyperbolic 3-manifolds, and hyperbolic 3-manifolds that decompose into right-angled polyhedra. I have pursued side interests on Delaunay mesh refinement and inscribed hyperbolic polygons in undergraduate research projects, and I have graduated three PhD students. I brought an NSF grant with me when I arrived and have since been funded by Pitt’s CRDF (Central Research Development Funds).