(781) 283-3495
B.A., Yale University; M.S., Ph.D., New York University, Courant Institute of Mathematical Sciences
SCI 373

Steven Simon
Visiting Lecturer in Mathematics

Research in discrete and metric geometry; geometric and topological combinatorics; algebraic and geometric topology.

My research focuses on topics in discrete geometry and combinatorics. I am especially fascinated by questions - including those concerning equipartitions and transversality for finite measures such as point collections, as well as those regarding geometric incidence for objects like simplicial complexes - which can be approached via topological methods.

Equipartition problems deal with the possibility of dissecting arbitrary bodies by a variety of "nice" geometric regions in such a way that each region contains an equal part of each body. For instance, the 3-D version of the “ham sandwich” theorem asserts that any three masses in space - think of a chunk of bread, a hunk of ham, and a slice of cheese suspended in air - can be simultaneously bisected by a plane - a knife cutting each ingredient in half. The incidence questions I deal with, called Tverberg-type problems, are deep generalizations of Radon's theorem, itself a cornerstone of convex geometry, which in the plane guarantees that any 4 points can be partitioned into two disjoint sets whose convex hulls nonetheless overlap - either two intersecting segments with endpoints the 4 points, or three points determining a triangle containing the remaining point. Surprisingly, by exploiting hidden symmetries, such inherently discrete problems can be solved using the continuous techniques of the seemingly unrelated fields of algebraic and geometric topology. For example, both Radon's theorem and the ham sandwich theorem can be derived as consequences of the classical Borsuk-Ulam theorem: for any continuous map from a sphere to a plane, there must exist some pair of antipodal points which are sent to the same spot.

While I love the exploration of ideas in my research, my true mathematical calling is as a devoted and passionate teacher. In my four years of postdoctoral and two years of graduate teaching, I have come to enjoy teaching a broad spectrum of courses - quantitative reasoning, single and multivariable calculus, differential equations, abstract algebra, and topology. It is a great pleasure to help students explore the inherent beauty of mathematical ideas and to experience the joys of problem solving, whether through dynamic and creative lectures or collaborate student-based work. While my main goals as an instructor are for students to attain thorough conceptual understanding and to improve as intuitive and rigorous thinkers, as a former philosophy major it is also important to me that students appreciate the unique contributions of mathematics at the physical, metaphysical, and epistemological levels - and therefore its essential role in the "examined life" and a complete liberal arts education. I also believe strongly in the benefits of undergraduate research, and I look forward to continued supervision of projects adapted from my own research program, such as those on equipartitions I designed this summer at Wellesley.