Computational complexity theory is the systematic study of computational problems in order to classify them in terms of their inherent logical hardness. Several decades of research have not only given rise to important understanding of limits of computation, but have also developed algorithms which constitute a crucial part of modern life. A formidable challenge in complexity theory is to show non-linear lower bounds for an explicit Boolean function. Our project is motivated by this fundamental problem and in fact we will approach several such questions motivated by understanding the complexity of explicit Boolean functions. Our main objective look at circuit complexity through the lens of extremal combinatorics, a rich and vibrant of branch of combinatorics which studies objects satisfying various constraints. Therefore we aim to develop a systematic methodology which adopts tools of extremal combinatorics to tackle complexity problems. More concretely we attack the problem of lower bounds for depth-3 circuits and specifically attempt to prove sharp lower bounds for the Majority function thus breaking a barrier in this area. We will further extend the techniques used in recent breakthrough on the Sunflower Conjecture and apply it to CNF formula and the structure of their satisfying assignments. We will our new insights on the structure of satisfying assignments to develop new improved algorithms for the satisfiability problem (SAT).

Lie theory pervades all areas of mathematics, by providing natural encodings of symmetries; representation theory and Cartan geometry are two of its most successfull manifestations. CaLiForNIA aims to push the frontier of research in these two key topics, Lie Theory and Cartan Geometry in synergy with the complementary investigations in the areas of quantum groups and more generally non commutative geometry, including physical applications. CaLiFornia goal is to apply the new mathematics originating by the above research to the new and strategic fields of quantum computing and geometric deep learning, top priorities in HorizonEurope.