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Abstract: Internal waves are suspected to have a crucial role in everything from the global overturning circulation to the growth of bacteria on the coast, yet they are almost undetectable from the surface. While there are several pathways to the generation and break-down of these waves in the ocean, there are still many unknowns within these processes. The reflection or breaking of internal waves is one such area of ongoing research. By considering the Navier-Stokes equations, we can understand the general attributes of these waves and ultimately study the linear (and nonlinear!) behavior of reflected internal waves off sloping boundaries.

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Abstract: $\Gamma$-convergence is a mode of convergence introduced by De Giorgi in the 1970s relevant when studying sequences of minimization problems. Its importance is described by the “Fundamental Theorem of $\Gamma$-convergence,” which says that minimizers of a sequence of $\Gamma$-convergent functionals will converge to a minimizer of the $\Gamma$-limit functional. We will describe its basic properties in the setting of functions on metric spaces, and we will compare with more elementary notions of convergence for functions, e.g., pointwise and uniform. Through examples, we will show that these elementary modes of convergence are not useful when studying sequences of minimization problems.

Then we will informally peruse many applications of $\Gamma$-convergence, including the Modica-Mortola regularization of the perimeter functional, discrete-to-continuous limits appearing in modeling and numerical analysis, and the rigorous derivation of lower-dimensional models from higher-dimensional models.

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Abstract: The walk-on-spheres (WoS) method is a monte-carlo technique for solving PDEs. First developed by Muller in 1956, it has seen a recent resurgence in computer graphics as it allows for problems to be solved on complicated geometries without the need for meshing. The main theoretical tool behind WoS relies on the stochastic process interpretation of solutions of boundary value problems. In this talk, the basics of the walk-on-spheres method will be discussed, its advantages & disadvantages, as well as some applications.