In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in R, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in R provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fa...

We study exact Lagrangian fillings of Legendrian links $D_n$-type in the standard contact 3-sphere. The main result is existence a filling, represented by weave, such that any algebraic quiver mutation associated intersection can be realized as geometric weave mutation. method proof via calculus and construction appropriate 1-cycles whose intersections realize required numbers. In particular, w...

We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over different partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction o...