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Martin Lorenz, Temple University
This is the first lecture in a minicourse (probably three lectures) surveying some topics in Galois Theory that are not typically covered in the graduate algebra course (Math 8011/12): inverse Galois theory, Noether's rationality problem, the Chebotarev density theorem,... The Galois Theory portion of Math 8011/12 will be sufficient background for the material presented in this minicourse; so it will be accessible to all students in my current Math 8012 class. No proofs will be given; the goal is to describe some research directions that are of current interest.
Jose Maria Martell, ICMAT, Madrid, Spain
In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the Dirichlet problem for the Laplacian with data in Lebesgue spaces $L^p$ is solvable for some finite $p$. This property is equivalent to the fact that the associated harmonic measure is absolutely continuous, in a quantitative way, with respect to the surface measure on the boundary. In this talk we will study under what circumstances the harmonic measure for a rough domain is a well-behaved object. We will also present some results for the converse, in which case good properties for the domain and its boundary can be proved by knowing that the harmonic measure satisfies a quantitative absolute continuity property with respect to the surface measure. We will describe the two main features appearing in this context: one related to the regularity of the boundary, expressed via its uniform rectifiablity, and another one related to the connectivity of the domain, written in terms of some quantitative connectivity towards the boundary using non-tangential paths. The results that we will present are higher dimensional scale-invariant extensions of the F. and M. Riesz theorem and its converse. That classical result says that, in the complex plane, the harmonic measure is absolutely continuous with respect to the arc-length measure for simply connected domains (a strong connectivity condition) with rectifiable boundary (a regularity condition).
Marius Mitrea, University of Missouri
In this talk I will discuss, in a methodical manner, the process that lets us consider singular integral operators of boundary layer type in a given compact Riemannian manifold M, and then use these to solve boundary value problems in subdomains of M of a general nature, best described in the language of Geometric Measure Theory. The talk is intended for a general audience, and it only requires a basic background in analysis.
Xiaoming Song, Drexel University
We prove large deviation principles for $\int_0^t \gamma(X_s)ds$, where $X$ is a $d$-dimensional Gaussian process and $\gamma(x)$ takes the form of the Dirac delta function $\delta(x)$, $|x|^{-\beta}$ with $\beta\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-\beta_i}$ with $\beta_i\in(0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.
Aaron Fogelson, University of Utah
There are no conferences this week.