Theory of Singularities and Real and Complex Algebraic Geometry

Equisingular approximability of real and complex analytic singularities

I am interested in the question of whether real and complex analytic singularities can be approximated by algebraic singularities in such a way that that preserves essential properties connected to the resolution of singularities. I have shown that an arbitrary real or complex analytic singularity can be approximated up to an arbitrary degree by algebraic singularities that have the same Hilbert-Samuel function and preserve other algebro-geometric properties of interest such as the Cohen-Macaulay type, and the equivalence class under local homeomorphisms. The Hilbert-Samuel function is a measure of singularity and is used in the resolution of singularities. Currently I am trying to show if such an approximation can be found which preserves these properties throughout a given resolution tower.

Publications:

A. Patel. Algebraic approximation of Cohen-Macaulay algebras. J. Algebra 625: 66-81, 2023.
A. Patel. Finite determinacy and approximation of flat maps. Proc. Amer. Math. Soc., 151(1):201–213, 2023.
J. Adamus, A. Patel. Equisingular algebraic approximation of real and complex analytic germs. J. Singul. 20:289-310, 2020.

Locally bounded rational functions and continuous rational functions on singular domains.

Rational continuous functions on singular real algebraic varieties have various interesting properties. In particular a continuous rational function defined on a singular real algebraic set can become non-rational when restricted to a subvariety contained in the singular locus. There is a natural link between this class of functions and the class of locally bounded rational functions as any continuous rational function on a singular real algebraic variety can be exteded to a locally bounded rational function on the ambient space. The behaviour of locally bounded rational functions on singular real algebraic varieties is currently an open area of research which has the potential of revealing much interesting behaviour.

Applied Numerical Linear Algebra

Parallel preconditioning for iterative methods for the solution of large and sparse linear systems

I work on developing parallel and asynchronous algorithms for the construction of preconditioners for iterative methods for the solution of large and sparse linear systems. Given that we are in the era of exa-scale computing where modern supercomputers have thousands of cores, the need for parallel algorithms for preconditioning for the solution of large and sparse linear systems has become more urgent as these form the core of solver packages for problems arising from many application domains. One of the challenges in developing such algorithms for modern supercomputers is the cost of communication and synchronization. This is the main motivation of my interest in asynchronous and communication avoiding parallel algorithms for preconditioning.

Publications:

E. Chow, A. Patel. Fine-grained parallel incomplete LU factorization. SIAM journal on Scientific Computing 37.2: C169–C193, 2015.
A. Patel, E. Boman, S. Rajamanickam, E. Chow. Cross Platform Fine Grained ILU and ILDL Factorizations Using Kokkos. Center for Computing Research Summer Proceedings (2015), A.M. Bradley and M.L. Parks, eds., Technical Report SAND2016-0830R, Sandia National Laboratories, pp. 159-177, 2015

Applications of High Performance computing to scientific computing

I am also interested in the application of HPC to scientific computing problems arising from application domains. One of my previous projects (at Georgia Tech) involved the parallelization of the Hartree-Fock method of computational quantum chemistry. The emphasis was on the development of a parallel algorithm and implementation for distributed memory targeted at large supercomputers.

Publications:

X. Liu, A. Patel, and E. Chow. A new scalable parallel algorithm for Fock matrix construction. IEEE 28th international parallel and distributed processing symposium (IPDPS), 2014.