Fusion energy holds the promise of a clean, sustainable and safe energy source for the future. While research in this field has been ongoing for the last half century, much work remains before it may prove a viable source of energy. In this talk, I discuss some of the scientific and engineering challenges remaining in fusion energy, and the role of applied mathematics and scientific computation in helping overcome these obstacles. I then introduce some of the mathematical models used in studying fusion stability and refueling, and how solutions to those models may be approximated. Of particular interest in such approximation techniques is the ability of the relevant numerical methods to scale up to resolutions necessary for accurately modeling the underlying physics. To that end, I discuss some of my recent work in the development of fully implicit solution approaches for magnetic fusion modeling, presenting both numerical results and theoretical analysis demonstrating the benefit of these approaches over traditional methods.

In the second part of the talk, I will focus on the mathematical aspects of integral
equation methods (IEM) and fast multipole methods (FMM). I will discuss why integral
equation methods are preferred for many problems as well as their limitations. Using
a simple example in one dimensional space, I will discuss the fundamental ideas of
the tree code and fast multipole methods, and how the ideas can be applied to more
general problems.

Consider $b\times n$ real matrices such that every $b\times b$ minor has nonnegative determinant. Since there are polynomial relations between these minors, not every pattern of zero vs. strictly positive is achievable. In a widely circulated prepreprint, Alex Postnikov gave many ways to index the patterns that are.

I'll describe a new indexing, by ``bounded juggling patterns'', which will require a brief foray into the mathematics of juggling (with demonstrations). It turns out that many of the natural concepts from the matrix picture have been known to jugglers for 20 years.

Unbounded juggling patterns form a group, the affine Weyl group, and thereby index the Schubert varieties on the (infinite-dimensional) affine flag manifold. I'll explain how the complicated finite-dimensional geometry of Postnikov's stratification is induced from what is actually much more familiar infinite-dimensional geometry.

We discuss how to cut up a pea into finitely many pieces that
can be rearranged to form a ball the size of the sun (but don't try this at home!), and how this leads to the notion of amenability. We give also a proof of Tarski's Theorem, a gem in mathematics.

In the past decade methods from Riemannian geometry and Banach space theory have become a central tool in the design and analysis of approximation algorithms for a wide range of NP hard problems. In the reverse direction, problems and methods from theoretical computer science have recently led to
solutions of long standing problems in metric geometry. This talk will illustrate the connection between these fields through the example of the Sparsest Cut problem. This problem asks for a polynomial time algorithm which computes the Cheeger constant of a given finite graph. The Sparsest Cut problem is known to be NP hard, but it is of great interest to devise
efficient algorithms which compute the Cheeger constant up to a small multiplicative error. We will show how a simple linear programming formulation of this problem leads to a question on bi-Lipschitz embeddings of finite metric spaces into $L_1$, which has been solved by Bourgain in 1986. We will then proceed to study a quadratic variant of this approach which
leads to the best known approximation algorithm for the Sparsest Cut problem. The investigation of this ``semidefinite relaxation" leads to
delicate questions in metric geometry and isoperimetry, in which the
geometry of the Heisenberg group plays an unexpected role.