Meeting Summary: The mathematical counterpart of the ubiquitous Boltzmann-Gibbs (BG) weight is the Large Deviation Theory. Its main result proves also that the total entropy of the system is extensive. When we focus on complex systems, many of them appear to exhibit a qexponential weight (consistent with the extensivity of the nonadditive entropy Sq of the total system) which recovers the BG one for q=1. This has been first verified in a probabilistic model (see Ruiz and Tsallis, Physics Letters A 377, 491 (2013)), and preliminary also in the conservative standard map (U. Tirnakli, C. Tsallis and N. Ay, 2019). The aim of the one-week SFI workshop involving Tirnakli, Ay and Tsallis is to implement and analytical proof of a q-generalized Large Deviation Theory. Such a result would constitute an important step forward for nonextensive statistical mechanics, along the lines of the q-generailzed Central Limit Theorem achieved by Umarov, Tsallis, Gell-Mann, and Steinberg a decade ago at SFI.