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The mini-proceedings of the STRONG2020 Virtual Workshop "Space-like and Time-like determination of the Hadronic Leading Order contribution to the Muon $g-2$", November 24--26 2021, are presented. This is the first workshop of the STRONG2020 WP21: JRA3-PrecisionSM: Precision Tests of the Standard Model (http://www.strong-2020.eu/joint-research-activity/jra3-precisionsm.html). The workshop was devoted to review of the working group activitity on: $(\it i)$ Radiative Corrections and Monte Carlo tools for low-energy hadronic cross sections in $e^+ e^-$ collisions; $(\it ii)$ Annotated database for $e^+e^-$ into hadrons processes at low energy; $(\it iii)$ Radiative Corrections and Monte Carlo tools for $\mu$-$e$ elastic scattering., Comment: 43 pages, 21 contributions. Editors: A. Kup\'s\'c and G. Venanzoni

In this paper we continue the work begun in 2002 on the identification of the analytical expressions of Feynman integrals which require the evaluation of multiple elliptic integrals. We rewrite and simplify the analytical expression of the 3-loop self-mass integral with three equal masses and on-shell external momentum. We collect and analyze a number of results on double and triple elliptic integrals. By using very high-precision numerical fits, for the first time we are able to identify a very compact analytical expression for the 4-loop on-shell self-mass integral with 4 equal masses, that is one of the master integrals of the 4-loop electron g-2. Moreover, we fit the analytical expressions of some integrals which appear in lattice perturbation theory, and in particular the 4-dimensional generalized Watson integral., 13 pages, 1 figure, LaTex; v2: some rephrasing of text; v3: reference added, minor modifications; v4: checks of lattice integrals up to 2400 digits; some modifications of text; version accepted for publication in IJMPA, needs the document class ws-ijmpa.cls

We describe a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace's transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions., Comment: 55 pages, 5 figures, LaTex

The master differential equations in the external square momentum p^2 for the master integrals of the two-loop sunrise graph, in n-continuous dimensions and for arbitrary values of the internal masses, are derived. The equations are then used for working out the values at p^2 = 0 and the expansions in p^2 at p^2 =0, in (n-4) at n to 4 limit and in 1/p^2 for large values of p^2 ., 28 pages, Latex, 2 encapsulated Postscript figures with epsfig, mistype corrected in Eq.(56)