As part of his bi-annual visit to the UNL Department of Mathematics, Michael Hopkins delivered a public lecture, “Mathematical Invariants: How to Know the Answer in Advance,” on Thursday, Nov. 14, 2013.The audience included members of the math department, undergraduates, university administrators, and even elementary, middle and high school students who filled the Union Auditorium. His talk was live-streamed over the web and a recording of it remains available for viewing at http://www.math.unl.edu/events/special/hopkins2013.
Mike worked the audience with the stage presence of a professional comedian. Members of the audience, including an enthusiastic young boy sitting front and center, were incorporated into his lively and highly interactive presentation. The energy and wit Mike displayed surely served as an inspiration to the many young people in audience, and all stereotypes of mathematicians as being dull or socially awkward were demolished during his performance.
Mike started out demonstrating the idea of an invariant by using the notion of parity to dispel the mystery of a popular card trick, as acted out to great comic effect by members of the audience called up onto the stage. He proceeded to explore the Euler characteristic of surfaces. Mike recruited members of the audience to subdivide various balloons and balls (surfaces of genus 0) into “pastures,” by drawing fence posts connected with fences on them. The volunteers reported back the value of the number of “pastures” minus the number of “fences” plus the number of “fence posts”; it will not be a surprise to any mathematician to learn that the answer was two in each case. Mike introduced the idea of surfaces of higher genus and their Euler numbers by using bleach bottles and a torus-shaped balloon, among other items.
All of this transitioned into a discussion of the much more sophisticated notion of “surgery of surfaces.” In this context, surgery refers to modifying a surface by cutting out a (typically short) cylinder and replacing it with two discs. For example, Mike illustrated how both a torus and a Klein bottle can be transformed into a sphere by doing surgery. Poincare believed he had a proof that every surface was equivalent to a sphere via repeated surgery maneuvers, but his proof was false. Indeed, much like the Euler characteristic, it is now known that the Kervaire invariant is an obstruction for transforming certain (stably framed) surfaces (and higher dimensional manifolds) into each other via surgery. For example, Boy’s surface (an immersion of the projective plane into three space, as depicted in a well-known sculpture at Oberwolfach) has Kervaire invariant one, and hence it cannot be transformed into a sphere via surgery.
Mike even managed to touch a bit upon his recent and seminal work with Mike Hill and Doug Ravenel. The three of them recently settled a 50-year-old question about the Kervaire invariant:
Theorem (Hill-Hopkins-Ravenel): If M is a smooth, stably framed manifold of Kervaire invariant one, then the dimension of M is 2, 6, 14, 30, 62 or 126.
At the end of the talk, Mike fielded a variety of interesting questions from diverse members of the audience. Thanks to the live streaming of the event, Mike was even able to respond to a question posed by former UNL graduate student Courtney Gibbons, who viewed the talk live from her new home at Hamilton College.
- Mark Walker