[Dottorcomp] Avviso di seminario presso Unimi

[Stefano Lisini]

SEMINARIO DI ANALISI MATEMATICA
https://sites.google.com/view/seminarioanalisiunimi/

Nell’ambito del seminario di Analisi Matematica del Dipartimento di
Matematica “Federigo Enriques"

Giovedì 23 Gennaio alle 16:30

presso l'Aula C del Dipartimento di Matematica dell'Università  di Milano
si terrà il seguente seminario:

Paolo Giordano
(University of Vienna)

A Grothendieck topos of generalized functions

Abstract: We present a new approach to generalized functions,
so-called generalized smooth functions (GSF). GSF are set-theoretical
maps defined on, and taking values in the non-Archimedean ring (i.e.
with real, infinite and infinitesimal numbers) of Robinson-Colombeau,
and form a concrete category which unifies and extends Schwartz
distributions and Colombeau generalized functions. The calculus of
these generalized functions is closely related to classical analysis,
with point values, the usual rules for differentiation and
integration, free composition and hence non linear operations. We have
classical theorems such as: intermediate value theorem, mean value
theorems, extreme value theorem, several forms of Taylor formula,
local and global inverse and implicit function theorems, a suitable
sheaf property; Multidimensional integration with convergence
theorems; A theory of non-Archimedean locally convex topological
vector spaces of GSF; A theory of singular nonlinear ODE with Banach
fixed point theorem, Picard-Lindelof theorem, maximal set of
existence, Gronwall theorem, flux properties, continuous dependence on
initial conditions, full compatibility with classical smooth
solutions; Calculus of variations with: Euler-Lagrange equations, the
necessary Legendre condition, Jacobi’s theorem on conjugate points and
Noether’s theorem. Using GSF, we can also prove a Picard-Lindelof
theorem for nonlinear singular PDE in normal form. Therefore, all
these results also apply to Schwartz distributions. Finally, we can
define a concrete site and hence a Grothendieck topos of sheaves of
generalized functions which contains the sheaves of Schwartz
distributions and Colombeau generalized functions and all smooth
manifolds. We hence present the planned future developments of this
theory.

References

[1] A. Lecke, L. Luperi Baglini, P. Giordano, The classical theory of
calculus of variations for generalized functions. Accepted in Advances
in Nonlinear Analysis. DOI: 10.1515/anona-2017-0150.

[2] P. Giordano, M. Kunzinger, Inverse Function Theorems for
Generalized Smooth Functions. Chapter in "Generalized Functions and
Fourier Analysis", Volume 260 of the series Operator Theory: Advances
and Applications pp. 95-114.

[3] P. Giordano, M. Kunzinger, H. Vernaeve, Strongly internal sets and
generalized smooth functions. Journal of Mathematical Analysis and
Applications, volume 422, issue 1, 2015, pp. 56-71.

[4] P. Giordano, M. Kunzinger, A convenient notion of compact set for
generalized functions. Proceedings of the Edinburgh Mathematical
Society, Volume 61, Issue 1, February 2018, pp. 57-92.

[5] P. Giordano, home page: www.mat.univie.ac.at/~giordap7/