B.A., Yale University; M.S., Ph.D., Courant Institute (New York University)
Visiting Lecturer in Mathematics
Research in geometric combinatorics, algebraic topology, and especially the surprising ways the latter can be used to solve problems of the former.
A large class of problems in geometric combinatorics concerns equipartitions - the ability to dissect objects by various geometric regions, each of which contains an equal part of each given object. For instance, the “Ham Sandwich Theorem” asserts that any three measures (think of a slice of bread, a piece of ham, and another slice of bread, each of arbitrary shape) on three-dimensional space can be bisected by a single plane (a knife which simultaneously cuts each ingredient in half). In some cases, such problems can be seen almost entirely in terms of the seemingly unrelated field of algebraic topology. For example, the Ham Sandwich Theorem arises from the antipodal symmetry of a sphere (if you interchange each point with its diametric opposite, you wind up with the same sphere), and hence from real projective space. Along with some other topics, my research focuses on how to realize more general symmetries, especially those given by group representations, as naturally corresponding measure symmetries (equipartitions), with the link between the two symmetries provided by associated algebraic invariants.
My views on mathematics are greatly influenced by having majored in philosophy for my first two years of college, and particularly by studying Plato's metaphysics. Many mathematical ideas, such as those of symmetry, are intrinsically beautiful, and indeed underly our perception of aesthetics. Moreover, owing to its logical rigor, mathematics is perhaps the most epistemologically secure knowledge one can hope for, and at the same time serves as a very accurate model of the physical world. In this profound sense, mathematics concerns the study of logically necessary beauty, fundamental to the human understanding of nature, and as such is unique in its contributions to the ideals of a liberal arts education. It is this perspective, along with mathematics’ inherent dynamism and creativity, that I hope to convey to all of my students.
I enjoy teaching a great range of mathematical subjects, and have so far taught courses in abstract algebra, topology, differential equations, multivariable calculus, and quantitative reasoning. I have also designed and led a summer enrichment program in group theory for advanced high school students. My most basic goal in any course is provide my students with a solid intuition that makes the subject organic and provides a base from which deep conceptual understanding is possible.
When I am not teaching or conducting research, I enjoy listening to music on my record player, hiking, and swimming, but most especially spending time with my wife.