Associate Professor

Northwestern University

Lunt Hall, B4

awb at northwestern dot edu

I am an Associate Professor in the Mathematics Department at Northwestern University. Previously, I was an Assistant Professor and a Dickson Instructor at the University of Chicago, an NSF postdoc at the Pennsylvania State University, and a graduate student at Tufts University.

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.

**Zimmer's conjecture for
non-uniform lattices and escape of mass**. Joint with David
Fisher and Sebastian Hurtado.

We prove Zimmer's conjecture for actions of general lattice subgroups in
\(\mathbb R\)-spit simple Lie groups.

**Zimmer's conjecture
for actions of \(\mathrm{SL}(m,\mathbb Z)\)**. Joint with
David Fisher and Sebastian Hurtado.

We prove Zimmer's conjecture for actions of (finite index subgroups
of) \(\mathrm{SL}(m,\mathbb Z)\), extending our previous results for
actions by cocompact lattices.

**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 \(\mathrm{SL}(n,\mathbb R)\), \(\mathrm{Sp}(2n,\mathbb R)\),
\(\mathrm{SO}(n,n)\), and \(\mathrm{SO}(n,n+1)\). We also give some
partial results for exceptional and non-split Lie groups. A 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.

For more about this paper:

- Two talks on our work from a conference held at IPAM at UCLA in January 2018: part 1, part 2
- Seminar Bourbaki on our work, presented by Serge Cantat (in French)
- Lecture notes compiled from various minicourses

** 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**. Ann.
of Math. (2) **186** (2017), 913–972.
Joint with Federico Rodriguez Hertz and Zhiren Wang.

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. J. Amer. Math. Soc. **30**
(2017), 1055–1132.

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. To appear, Journal of Modern
Dynamics.

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.