Probability, Differential Geometry and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
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We will report on a result within the holographic study of conformal geometry initiated by Fefferman and Graham. We will first cover some background and history of this area. Then we will discuss the result which can be contextualized as follows:
In 1999 Graham and Witten showed that one can define a notion of renormalized area for properly embedded minimal submanifolds of Poincare-Einstein spaces. For even-dimensional submanifolds, this quantity is a global invariant of the embedded submanifold. In 2008 Alexakis and Mazzeo wrote a paper on this quantity for surfaces in a 3-dimensional PE manifold, getting an explicit formula and studying its functional properties. We will look at a formula for the renormalized area of a minimal hypersurface of a 5-dimensional Poincare-Einstein space in terms of a Chern-Gauss-Bonnet formula.
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Self-similar solutions and translating solitons are not only special solutions of mean curvature flow (MCF) but a key role in the study of singularities of MCF. They have received a lot of attention. We introduce some examples of self-similar solutions and translating solitons for the mean curvature flow (MCF) and give rigidity results of some of them. We also investigate self-similar solutions and translating solitons to the inverse mean curvature flow (IMCF) in Euclidean space.
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We will discuss stochastic quantization of the Yang-Mills model on two and three dimensional torus. In stochastic quantization we consider the Langevin dynamic for the Yang-Mills model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that the solution has a gauge invariant property in law, which then defines a Markov process on the space of gauge orbits. We will also describe the construction of this orbit space, on which we have well-defined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer.
Watch online via this Zoom link.
We will report on a result within the holographic study of conformal geometry initiated by Fefferman and Graham. We will first cover some background and history of this area. Then we will discuss the result which can be contextualized as follows:
In 1999 Graham and Witten showed that one can define a notion of renormalized area for properly embedded minimal submanifolds of Poincare-Einstein spaces. For even-dimensional submanifolds, this quantity is a global invariant of the embedded submanifold. In 2008 Alexakis and Mazzeo wrote a paper on this quantity for surfaces in a 3-dimensional PE manifold, getting an explicit formula and studying its functional properties. We will look at a formula for the renormalized area of a minimal hypersurface of a 5-dimensional Poincare-Einstein space in terms of a Chern-Gauss-Bonnet formula.
Watch online via this Zoom link.
This talk discusses computing the Morse index for a free boundary minimal submanifold for
more specific problems. We focused on the corresponding problem with fixed boundary conditions
and the association with the Dirichlet-to-Neumann map to Jacobi fields. Also, we will discuss an
application of it.
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In this talk, I will describe some progress towards the construction of the 3D Yang-Mills (YM) measure. In particular, I will introduce a state space of "distributional gauge orbits" which may possibly support the 3D YM measure. Then, I will describe a result which says that assuming that 3D YM theories exhibit short distance behavior similar to the 3D Gaussian free field (which is the expected behavior), then the 3D YM measure may be constructed as a probability measure on the state space. This is based on joint work with Sourav Chatterjee.
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