Random constraint satisfaction problems encode many interesting questions in random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. I will discuss the question of the number of solutions to random regular NAE-SAT. This involves understanding the condensation regime where the model undergoes what is known as a one step replica symmetry breaking transition. We expect these approaches to extend to a range of other models in the same universality class.
I will review some recent results regarding the general problem of characterizing algebraic surfaces of the form $F(x,y,z)=0$ that can contain $\Theta(n^2)$ points of a $n \times n \times n$ Cartesian product.
Assume that the derived Fukaya category of a symplectic manifold admits a collection of triangular generators. By definition, this means that any other Lagrangian submanifold which is an object of this category can be decomposed in terms of exact triangles involving the generators. The purpose of the talk is to explain why such a decomposition requires a certain non-trivial amount of “energy”. The notion of energy that appears here is an extension of Hofer's energy.