New Simons Collaboration Will Explore Discrete Subgroups of Lie Groups

The Simons Foundation is pleased to announce the launch of the Simons Collaboration on Discrete Subgroups of Lie Groups led by Yair Minsky of Yale University.

Symmetry describes the structures of the world: One symmetry governs the progress of time, while another governs the quantum state of an electron. Continuous symmetries concern things such as motion through space and the rotation of a sphere. Unlike a cube, which needs discrete 90-degree rotations to look like itself again, a sphere can rotate fluidly without breaking its core structure. These symmetries are described by fundamental mathematical objects called Lie groups.

Studying Lie groups goes hand in hand with studying their discrete subgroups, which retain only certain symmetries. For instance, 90-degree rotations form a discrete subgroup of a sphere’s Lie group by including only those spins that preserve the appearance of a cube inscribed in the sphere.

Such mathematical structures are found in everything from quantum physics to general relativity to robotics. Better understanding Lie groups, therefore, could yield powerful new tools for studying the universe.

The new collaboration will investigate this area via three interconnected subfields: randomness, representation theory, and geometry and dynamics.

“These three areas, which are all quite classical, have each developed their own culture and their own ways of asking questions,” Minsky says. “It seems to us that there are ripe signs of connections that we want to try to get to by learning from each other.”

From the geometry and dynamics side, mathematicians have recently nailed down definitions for generalizations of lattices (a specific and useful type of discrete subgroup), which have deep connections to probability theory, supersymmetric quantum field theory and moduli spaces. Representation theory also connects with physical processes like crystallization and phase transitions.

Breakthroughs in randomness have revealed a new approach to Lie groups: Choosing points randomly in a specific manner, called a point process, yields new insights into the discrete subgroups and their geometry. Instead of acting on geometric spaces like the circle, discrete subgroups can also act on probability spaces. This approach carries theorems from the mathematics of randomness into Lie groups.

“Discrete subgroups are sort of a model for random processes, and conversely, random processes are kind of playgrounds in which you can test your beliefs about discrete groups,” Minsky says.

Simons Collaborations in Mathematics and the Physical Sciences bring together groups of outstanding researchers to address topics of fundamental scientific importance. Collaborations receive up to $2 million per year for an initial period of four years, including indirect costs, and may be extended for an additional three years. The collaboration will be funded by grants from Simons Foundation International administered by the Simons Foundation.

The members of the new collaboration are:

Yair Minsky
Director; Yale University

Miklos Abert
PI; Alfréd Rényi Institute of Mathematics

Uri Bader
PI; Weizmann Institute of Science

Alex Eskin
PI; University of Chicago

Mikolaj Fraczyk
PI; University of Chicago

Alexander Goncharov
PI; Yale University

Tom Hutchcroft
PI; California Institute of Technology

Sebastian Hurtado
PI; University of Chicago

Richard Kenyon
PI; Yale University

Or Landesberg
PI; University of Texas at Austin

Elon Lindenstrauss
PI; Hebrew University of Jerusalem

Alex Lubotzky
PI; Weizmann Institute of Science

Andrew Neitzke
PI; Yale University

Gábor Pete
PI; Alfréd Rényi Institute of Mathematics

Anna Wienhard
PI; Max Planck Institute for Mathematics in the Sciences

Tianyi Zheng
PI; University of California, San Diego