Abstracts

Week of March 3, 2024

March 4, 2024
Group Actions, Geometry and Dynamics Ergodic dichotomy for subspace flows in higher rank 4:00pm -
KT205

In rank one, the Hopf-Tsuji-Sullivan dichotomy theorem states the dichotomy for the ergodicity of geodesic flow in terms of the divergence/convergence of the Poincaré series at the critical exponent. Burger-Landesberg-Lee-Oh initiated the study of higher rank version of the Hopf-Tsuji-Sullivan dichotomy for a one-dimensional diagonal flow and the associated Poincaré series. In this talk, we discuss the Hopf-Tsuji-Sullivan dichotomy for higher dimensional flows which we call subspace flows of Weyl chamber flows. Just like the dimension dichotomy for Brownian motions in $\mathbb{R}^n$, the codimension dichotomy of the flow occurs for Anosov homogeneous spaces. This is based on joint work with Hee Oh and Yahui (Amy) Wang.
March 6, 2024
Colloquium On the dimension of limit sets on the real projective plane via stationary measures 4:00pm -
KT207

We consider the (semi)group action of SL(3, R)  on RP^2 as a prime example of a non-conformal, non-linear, and non-strictly contracting action. We study the Hausdorff dimension of a dynamically defined limit set in RP^2 and generalize the classical Patterson-Sullivan formula. A prominent example is Anosov representations in SL(3, R), for which we prove the equality between the Hausdorff dimensions and the affinity exponents of their limit sets. As an application, it reveals a dimension jump in the Barbot component, which is a local generalization of Bowen’s dimension rigidity result. Another example is the Rauzy gasket, for which we confirm a folklore conjecture about the Hausdorff dimension of the gasket. These results originate from a dimension formula of stationary measures on RP^2. This talk is based on the joint works with Yuxiang Jiao, Jialun Li, Disheng Xu.
March 7, 2024
Analysis Scaling thresholds and quantitative criteria for blow-up in defocusing energy-supercritical settings 4:00pm -
KT 201

In this talk, we establish quantitative bounds for the defocusing
energy-supercritical Nonlinear Schr\"odinger equation (NLS) and use
these to give a universal blow-up criteria which goes below the
scaling invariant threshold. These results are in line with a recent
breakthrough construction of finite-time blow-up solutions, and in
particular give the first generic result distinguishing potential
defocusing blow-up phenomena from many of the known examples of
blow-up in the focusing setting. The argument is based on a delicate
refinement of induction on scales techniques. At the end of the talk,
we will briefly describe applications to related models.
March 8, 2024
Friday Morning Seminar Friday Morning Seminar 10:00am -
KT801

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!