My research is about generalizations of the concept of functions (say,
from R^n to C) in the frameworks of functional analysis, nonstandard analysis
and algebras of generalized functions. More specifically, the generalized
functions in these frameworks generalize (and often contain a copy of)
the space of Schwartz distributions, familiar to physicists.
Nonstandard analysis (NSA)
Nonstandard analysis is a rigorous foundation for the infinitesimal calculus
used by mathematicians and physicists since the early days of mathematical
analysis, but later rejected and replaced by Weierstrass' "epsilon
delta"-approach, exactly because of the lack of rigour.
NSA was developed in the 1960's by the logician Abraham Robinson as a
direct application of model theory. The use of infinitesimal quantities brings
with it significant simpliciations, intuitions and facilitation for invention
in analysis.
Schwartz distributions
Schwartz distributions are an example of a generalization of the concept of
functions. They originated from physics and contain the Dirac-delta
distribution, which is zero outside the origin and for which
the "area under the curve" is nevertheless equal to one (no classical
function can satisfy these requirements). They are a particularly powerful tool
in the study of linear partial differential equations. Some natural operations
on functions (like the product) cannot be naturally defined on arbitrary
distributions, which makes them sometimes less suited for nonlinear equations.
Although different mathematical foundations for the distributions were developed
in the 1950's, the functional analytic approach of Laurent Schwartz (which
earned him a Fields Medal in 1950) is most widely used.
Algebras of generalized functions
Algebras of generalized functions (introduced under the impetus of the
French mathematician J.-F. Colombeau) are a more recent approach to
generalized functions than distributions. Distributions can in some sense
be viewed as special cases. Moreover, a lot of operations that are not
well-defined on distributions (like multiplication) are well-defined on
the algebras. They find their main application in the study of partial
differential equations with singular data or coefficients and in geometric
problems, e.g. in General Relativity. The algebras are similar to algebras
of nonstandard functions, and their structure is clarified by nonstandard
principles in a "cheap" version of nonstandard analysis (as
Terence Tao calls it).
E. Allaud, A. Delcroix, V. Dévoué, J.-A. Marti, H. Vernaeve, Paradigmatic wellposedness
in some generalized characteristic Cauchy problems, In: Proceedings of the 8th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), Moscow, Russia, August 22-27, 2011. (2013) 476-489. Moscow: People's Friendship University of Russia.
V. Dévoué, J.-A. Marti, H. Vernaeve, J. Vindas, Generalized functions on the closure of an open set. Application to uniqueness of some characteristic Cauchy problem, Novi Sad J. Math. (2016) 46 (2): 163-180.