Aaron Brown

L. E. Dickson Instructor
University of Chicago
Eckhart 326
awb@uchicago.edu


About Me

I am an L. E. Dickson Instructor in the mathematics department at the University of Chicago. Previously, I was an NSF postdoc at the Pennsylvania State University and a graduate student at Tufts University.


Curriculum Vitae

My C.V. is available here.

Research

I primarily work in smooth hyperbolic dynamics, nonuniform hyperbolicity, and smooth ergodic theory.

My recent work focuses on smooth group actions. I often apply tools from smooth dynamics and smooth ergodic theory to study rigidity phenomenon for actions of large groups. In particular, I am interested in measure rigidity questions and problems related to the rigidity of lattice actions and the Zimmer program.

Descriptions of some recent papers and current projects are below. More details can be found in my Research Statement.

Recent papers

Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T). Joint with David Fisher and Sebastian Hurtado.
We prove Zimmer's conjecture for actions cocompact lattices in the matrix groups \(\mathrm{SL}(n,\mathbb R)\), \(\mathrm{Sp}(n,\mathbb R)\), \(\mathrm{SO}(n,n)\), and \(\mathrm{SO}(n,n+1)\). We also give some partial results for exceptional and non-split Lie groups. The key new ingredient in the proof is the study of smooth ergodic theory for \(\mathbb R^d\) actions, particularly the results from the paper Smooth ergodic theory of \(\mathbb{Z}^d\) actions as applied in the paper Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds below.

Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds. Joint with Federico Rodriguez Hertz and Zhiren Wang.
We study smooth actions of higher-rank lattices on compact manifolds and show that if the dimension is sufficiently small relative to the rank, then there always exists and invariant measure for the action. For actions in manifolds of intermediate dimension we show the existence of a quasi-invariant measure which is a relatively measure preserving extension over a projective action.

Smooth ergodic theory of \(\mathbb{Z}^d\)-actions, parts 1, 2, and 3. Joint with Federico Rodriguez Hertz and Zhiren Wang.
We present a framework in which to study smooth actions of higher-rank abelian groups, possibly acting on non-compact manifolds and acting with discontinuities and singularities. We reprove a number of classical results in smooth ergodic theory and obtain in particular generalizations of the classical Leddrapier-Young entropy formulas. Our main application is to prove a "coarse product structure" and a "coarse Abramov-Rohlin formula" for metric entropy of measures invariant under smooth \(\mathbb{Z}^d\)-actions. The formulas are a critical ingredients in the two papers above.

Global smooth and topological rigidity of hyperbolic lattice actions. Joint with Federico Rodriguez Hertz and Zhiren Wang. To appear, Ann. Math.
We study actions of higher-rank lattices on tori and nilmanifolds and show under suitable lifting conditions that the action is topologically conjugate to an affine action. Assuming the action is Anosov, we then show the action is smoothly conjugate to an affine action. One novelty of our approach is that unlike most results in the literature we do not assume the existence of an invariant measure for the action.

Measure rigidity for random dynamics on surfaces and related skew products. Joint with Federico Rodriguez Hertz. To appear, J. Amer. Math. Soc.
An earlier (permanent preprint) version which assumed some positivity of entropy and is somewhat less technical is here.

Continuity of Lyapunov exponents for cocycles with invariant holonomies. Joint with Lucas Backes and Clark Butler. Under review

Other publications

Smoothness of stable holonomies inside center-stable manifolds.

Smooth stabilizers for measures on the torus. Discrete Contin. Dyn. Syst. 35(2015), 43-58.

Unstable periodic orbits in the Lorenz attractor. Philosophical Transactions of the Royal Society A, 369 (2011), 2345–2353. Joint with Bruce M. Boghosian, Jonas Lätt, Hui Tang, Luis M. Fazendeiro, and Peter V. Coveney.

Constraints on dynamics preserving certain hyperbolic sets. Ergodic Theory and Dynamical Systems, 31 (2011), 719-739.

Nonexpanding attractors: conjugacy to algebraic models and classification in 3-manifolds. Journal of Modern Dynamics, 4 (2010), 517–548.


Teaching

In the 2016-2017 academic year I will be teaching Math 159 in the Fall quarter and Math 202 in the Spring quarter. I will not be teaching during the Winter quarter.

In 2015-2016 I taught Math 161-163, Honors Calculus sequence as well as a section of Math 263, Point set topology.

In 2014-2015 I taught Math 203, 204, and 205, the second year analysis sequence.