Analysis, Geometry and Dynamics Seminar - UW

We construct the 2D Yang-Mills-Higgs (YMH) measure via stochastic quantization. To this end, we show global well-posedness and uniform-in-time bounds for the associated Langevin dynamics, which is given by the 2D stochastic YMH equations. A key component of our approach is the further development of techniques in stochastic geometric analysis, combining ideas from geometric analysis and stochastic analysis. These methods yield a manifestly gauge-covariant local existence theory, refined estimates for covariant stochastic objects, and a decay mechanism driven by unstable Yang-Mills connections. This is joint work with S. Cao, M. Hairer, and W. Zhao. l

TBA

TBA

TBA

I'll describe recent progress on the existence theory for metrics maximizing Laplace and Steklov eigenvalues on surfaces, and discuss applications to the existence of embedded free boundary minimal surfaces in the 3-ball and nonorientable minimal surfaces in the 4-sphere with any prescribed topology. Based on joint work with Misha Karpukhin, Rob Kusner, Peter McGrath, and Romain Petrides.