  In the last three years a new concept -- the concept of wall crossing has
emerged. The current situation with wall crossing phenomena, after papers of
Seiberg-Witten, Gaiotto-Moore-Neitzke, Vafa-Cecoti and seminal works by
Donaldson-Thomas, Joyce-Song, Maulik-Nekrasov-Okounkov-Pandharipande, Douglas,
Bridgeland, and Kontsevich-Soibelman, is very similar to the situation with
Higgs Bundles after the works of Higgs and Hitchin -- it is clear that a
general "Hodge type" of theory exists and needs to be developed. Nonabelian
Hodge theory did lead to strong mathematical applications -- uniformization,
Langlands program to mention a few. In the wall crossing is is also clear that
some "Hodge type" of theory exists -- Stability Hodge Structure (SHS). This
theory needs to be developed in order to reap some mathematical benefits ---
solve long standing problems in algebraic geometry. In this paper we look at
SHS from the perspective of Landau--Ginzburg models and we look at some
applications. We consider simple examples and explain some conjectures these
examples suggest.
