A major mathematical breakthrough led by an Indiana University Bloomington professor will be the subject of an international conference this month in California.
IU mathematician David Fisher and colleagues’ work on Zimmer’s conjecture – a well-known subject in the field of “pure mathematics” – will be the subject of an Institute for Pure and Applied Mathematics workshop Jan. 22 to 26 at UCLA. Fisher is a professor in the IU Bloomington College of Arts and Sciences’ Department of Mathematics.
The workshop is expected to attract about 100 highly specialized mathematicians from across the globe.
Originally posed over 30 years ago, Zimmer’s conjecture broadly concerns the subject of symmetry, a concept in many fields of mathematics but most familiar in geometry, where a shape appears the same from many angles, such as a reflection.
Symmetry is especially important in the field of physics, and it’s fundamental to how experts describe the rules that guide the physical world. Broadly, Zimmer’s conjecture concerns the challenge of understanding the behavior of abstract structures called lattices, which behave in a geometric way when analyzed in higher dimensional space. It roughly states that certain groups of symmetries of these objects cannot act on certain spaces.
The conjecture was originally posed by Robert J. Zimmer, who was Fisher’s thesis advisor and the current president of the University of Chicago. Although many approaches to the conjecture have been proposed over the years, little progress had been made.
Fisher first tackled Zimmer’s conjecture in 2001, and he came closest to a breakthrough in 2008 while working on a related conjecture. The new discovery, which verifies a large portion of Zimmer’s conjecture, was discovered in collaboration with Aaron Brown and Sebastian Hurtado-Salazar, postdoctoral instructors at the University of Chicago.
Together, Fisher and these colleagues proved for the first time that no “non-trivial” solution for symmetry exists for these types of variables under certain specific conditions. The situation is similar to trying to solve a number equation in which there are too many variables to reach a conclusion. Except instead of numbers, the equation uses complicated geometric objects with many parameters.
“This result brings us much closer to the full conjecture than anything that had been done before,” Fisher said. “I think it’s important that the proof of our result draws together threads of several different subfields of mathematics in a way that is surprising and interesting.”
Specifically, Fisher said the proof combines topics and tools from mathematical fields such as group actions, geometry, dynamics, smooth ergodic theory and functional analysis. The result was able to prove the conjecture for one major class of lattices in essentially any dimension.
The decision of the Institute for Pure and Applied Mathematics to dedicate a workshop to pure mathematics is a notable distinction because the events are typically reserved for applied mathematics. The workshops are also highly competitive, requiring an application and a review by a scientific board.
In addition, Fisher’s work on Zimmer’s conjecture was recently featured in a Bourbaki Seminar, a prestigious lecture series hosted by the Institut Henri Poincare in Paris that highlights the year’s top 16 discoveries in mathematics. The prior year’s series also featured work by a faculty member in the IU Bloomington Department of Mathematics, Ciprian Demeter.
An expert in rigidity theory, Fisher focuses his research on geometric group theory and ergodic theory. He is the recipient of several grants from the National Science Foundation, and he is a fellow of the American Mathematical Society, the Simon’s Foundation and the Radcliffe Institute Fellowship Program at Harvard University. He holds a Ph.D. from the University of Chicago and a bachelor’s degree in mathematics from Columbia University.
The Institute of Pure and Applied Mathematics was founded in 2000 as an NSF Mathematical Sciences Institute. Each year, thousands of visitors attend its workshops and programs. The institute seeks to bring the full range of mathematical techniques to bear on the great scientific challenges of our time.