We are delighted to announce that, through the diligent efforts of former Chair John Meakin, Michael J. Hopkins, professor at Harvard University and member of the National Academy of Science, will serve as a visiting research professor in our department for each of the next three years. Every semester during this time, Hopkins will spend one week in Lincoln giving a workshop to the graduate students and faculty on a topic of his choosing. Through these workshops, he will expose the members of the department to current trends in algebraic topology and homotopy theory.
The first workshop was held Nov. 12-16, 2012, and the theme was the classification of immersions of compact surfaces via the quadratic refinement of the mod two intersection pairing. His beautiful series of lectures on this topic illustrated the powerful connections between topology and algebra.
His connection to Nebraska goes back to his childhood. Hopkins grew up in Omaha, and attended Westside High School, where he was recently inducted into their Hall of Fame.
As a teenager, he played guitar for a rock band that toured the area. In the summer following his senior year of high school, he had a job driving a truck between the three cities of Lincoln, Fremont and Council Bluffs. One could argue his path toward fame began with this job. As a strategy to stay awake during the long drives, he would read the statement of a theorem from the textbook of a point-set topology class he had taken at UNO, and then he would try to prove it in his head on the road. The first one that stumped him was the Heine-Borel Theorem; its proof eluded him on each of the three-legs of his journey. To this day, Mike calls statements with tricky proofs "three-legged theorems."
The topic of this fall's workshop is related to a very special case of the Kervaire invariant problem, a 45-year-old question concerning framed differentiable manifolds.
The Kervaire invariant is a numerical invariant associated to such manifolds that are of dimension congruent to 2 modulo 4, and it is determined by a refinement of the quadratic form associated to the self-intersection pairing on the Z / 2 homology classes of middle dimension. Roughly, the vanishing of the Kervaire invariant says the manifold is not "exotic.'' The vanishing for all such manifolds except those of dimension 2n – 2 was proven by Browder in 1969. Examples where the invariant does not vanish for manifolds of dimension 6 and 14 (i.e., when n is 3 or 4 in the above notation) were known previously, and examples of non-vanishing in dimensions 30 and 62 (corresponding the cases when n is 5 or 6) were constructed by various mathematicians during the 15 years following Browder's Theorem.
Recently, Mike Hill, Hopkins, and Doug Ravenel settled (nearly) all the remaining cases of the Kervaire invariant problem by proving the invariant vanishes for manifolds of dimension 2n – 2 with n ≥ 8. Roughly speaking, they proved there are no "exotic" manifolds of dimension 254, 510, 1022, etc. Their proof is a tour-de-force of techniques from equivariant stable homotopy theory.
Partly due to his role in settling the Kervaire invariant problem, Hopkins was awarded the NAS Award in Mathematics from the National Academy of Sciences in 2012. Although of recent mintage, this is a highly prestigious award: just one is awarded every four years, and previous recipients include Andrew Wiles and Ingrid Daubechies. (Some of you may recall that Daubechies delivered the 2006 Rowlee Lecture for our department and was a plenary speaker for the 2012 NCUWM.)
Hopkins is also a member of National Academy of Sciences and was awarded the Oswald Veblen Prize in Geometry in 2001. Some of his major previous research accomplishments include proving the Ravenel Conjectures, along with collaborators Ethan Devinatz and Jeff Smith, and proving what is now called the Hopkins-Miller Theorem, with collaborator Haynes Miller.