Seminar

Abstract: A wide variety of partial differential equations that are of interest in industrial applications cannot be solved in closed form. Even those equations for which we do have a well-understood theory may not be solved over domains that aren’t nice enough. The finite element method is a powerful algorithm that provides solutions to both these issues. This talk will provide a very high-level overview of the theory FEM. How do we approximate solutions that we do not know a priori? How can we estimate our error?

Abstract: The spectrum of the Laplacian operator is an important object in the analysis of PDEs which depends on the domain and on the boundary conditions. The smallest ("principal") eigenvalue admits a useful variational characterization in terms of the Rayleigh quotient of the operator. We can adapt inverse iteration, an iterative technique for computing eigenvalues of symmetric PD matrices, to the infinite-dimensional setting.

Abstract: Machine learning is the study of how artificial intelligence mimics the human ability to learn. This is a gentle and interactive introduction to the types of questions involved in machine learning and some of the algorithms used to address those questions.

Abstract: In the late 1800s, Oliver Heaviside popularized a technique for solving differential equations by treating derivatives and integrals as variables. Heaviside was able to derive correct results, but did not rigorously justify his methods. In the early 1900s, many mathematicians attempted to formalize Heaviside’s work by use of integral transforms. These attempts were successful enough to make their way into many undergraduate differential equations curricula.