Computations for Analysis, Geometry and Topology

My research agenda is built on formulating problems in analysis, geometry and topology for a computational realization and conversely, use of computation for formalizing notions in analysis, geometry and topology. The former stems from a motivation to provide solutions for problems in science and engineering using tools of scientific and high performance computing while the latter encapsulates experimentation as a means to deriving theoretical insights. My primary research interests are in structure preserving solutions of ordinary and partial differential equations (PDEs) posed in the language of calculus on manifolds. A compatible or structure preserving discretization is a discrete scheme established to ensure that the discrete world faithfully reproduces essential properties of the continuum. This entails, for example, computing solutions while ensuring underlying physical quantities are conserved, say, velocities in the solution of a flow problem modeled as a PDE. Other examples of structure preservation include discrete conservation laws for energies and symmetries of solutions, geometric quantities such as volume and orientation preservation in the discrete model, and topological properties of the discrete domain over which the problem is modeled such as Betti numbers. This enterprise, thus, intimately conjoins analysis, geometry and topology. A secondary aspect of my interests, consequently, involves applied and computational topology, and discrete differential geometry. Analysis and topology have long been known to be an example of duality in mathematics. As a result, some problems in analysis can be solved by equivalently posing it as one in topology, in itself expressed using algebra. In this setup, I am interested in both finding efficient algorithms for performing such computations and determining engineering and scientific problems that can be modeled using these computational tools. Finally, I am also interested in manifold theory with a view towards computation, again, chiefly in the role of bridging the abstract with the applied via computation. My thesis involves using two different discrete solution approaches for numerically solving PDEs in the generalized setting of calculus on manifolds: finite element exterior calculus (FEEC) and discrete exterior calculus (DEC). In particular, the main problem of my work is in showing convergence of DEC for elliptic PDEs. In addition, my work also includes exploring some connections of these methods to topology and manifold theory; in this area I have shown an efficient way to compute discrete harmonics and extended the applicability of DEC to a larger class of meshes [HKWW11, HKV12].