Clifford algebras
My third research program is in Clifford algebras, Clifford analysis, and their applications. Initially my research in this field focused on fundamental operations in Clifford algebras and their representations and on the approximation of transcendental functions. It now includes applications of Clifford algebras and Clifford analysis.
Numerical Clifford algebra
My work in Clifford algebras began as a coursework Masters project, resulting in my first published paper [1], as well as the GluCat software library and the PyClical Python extension module. Since its first release in 2001, GluCat has been revised 36 times, has been downloaded around 3000 times, and has been incorporated into the openSUSE:Science repository. There is scope to further develop GluCat.
My research in this field has focused on fundamental operations in Clifford algebras and their representations [1], and on the accurate approximation of transcendental functions [2].
Presentations
Practical computation with Clifford algebras, 2002. (Based on the unpublished coursework Masters thesis, UNSW, 2002.)
A generic library of universal Clifford algebra templates, (poster) 2002.
Approximation of the square root and logarithm functions in Clifford algebras: what to do in case of negative eigenvalues? (poster) DWCAA09, 2009.
Approximating functions in Clifford algebras: What to do with negative eigenvalues? AustMS 2009, (AGACSE 2010 short version), (AGACSE 2010 long version).
Visualization as its own reward: The mathematics of conformal chaos, EViMS 2, ANU, 2014.
Publications and preprints
Paul Leopardi, "A generalized FFT for Clifford algebras", Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 11, Number 5, 2005, pp. 663-688. MR 2130632. (Citations). Preprint: UNSW Applied Mathematics Report AMR04/17, March 2004.
Describes algorithms used in the GluCat C++ software library, for the real representations of real Clifford algebras, having the same order of complexity as the generalized FFTs on finite groups.
Paul Leopardi, "Approximating the square root and logarithm functions in Clifford algebras: what to do in the case of negative eigenvalues?", (extended abstract) AGACSE 2010, June 2010.
Describes how the Clifford algebras over the real numbers can be treated as real matrices, except in the case of negative real eigenvalues, when the square root and logarithm functions may take values in a larger Clifford algebra.
Open source software
Numerical Clifford analysis
My joint paper with Ari Stern on the discretization of the Hodge-Dirac operator [2] is one realization of a program on the discretization of Dirac operators, as discussed in one of my earlier papers [1]. There is scope to continue the work on discretization of first order operators, in the development and implementation of numerical methods for finite elements, using (e.g.) FEniCS, and in the investigation of other types of discretization schemes, such as boundary element methods, radial basis functions, and wavelets.
Presentations
Can compatible discretization, finite element methods, and discrete Clifford analysis be fruitfully combined? ICCA 9, 2011.
An abstract Hodge-Dirac operator and its stable discretization, IWOTA 2012, CTAC 2012, ICCA 10, 2014.
Recent work following the discretization of the abstract Hodge-Dirac operator, ICHAA 2024 (edited).
Publications and preprints
Paul Leopardi, "Can compatible discretization, finite element methods, and discrete Clifford analysis be fruitfully combined?", Clifford Analysis, Clifford Algebras and their applications (CACAA), Volume 7, Issue 1, 2012, pp. 57-64. ISSN 2050- 0300 (print), 2050-0319 (online). Preprint: January 2012. Conference paper: 9th International Conference on Clifford Algebras and their Applications (ICCA 9), July 2011. Preprint: Revised July 2011.
Describes work in progress, towards the formulation, implementation and testing of compatible discretization of differential equations, using a combination of Finite Element Exterior Calculus and discrete Geometric Calculus / Clifford analysis.
Paul Leopardi and Ari Stern, "The abstract Hodge-Dirac operator and its stable discretization", SIAM Journal on Numerical Analysis, 54(6), 2016, pp. 3258–3279, (Citations). Preprint: arXiv:1401.1576 [math.NA].
Adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther.
Clifford algebras and constructions for Hadamard matrices
See Combinatorics and Statistics for a description and list of works.