Microlocal Sheaves on Pinwheels

In this thesis, we study the wrapped Fukaya category of the rational homology ball $B_{p,q}$ and the traditional/wrapped microlocal sheaves on its skeleton $L_{p,q}$, called pinwheel. We explicitly calculate both for $q=1$, and show they match in wrapped case.Comment: 109 pages, 25 figures. Comments are welcome

Categorical entropies on symplectic manifolds

In this paper, being motivated by symplectic topology, we study categorical entropy. Specifically, we prove inequalities between categorical entropies of functors on a category and its localization. We apply the inequalities to symplectic topology to prove equalities between categorical entropies on wrapped, partially wrapped, and compact Fukaya categories if the functors are induced by the same compactly supported symplectic automorphisms. We also provide a practical way to compute the categorical entropy of symplectic automorphisms by using Lagrangian Floer theory if their domains satisfy a type of Floer-theoretical duality. Our main examples of symplectic manifolds satisfying the duality conditions are the plumbings of cotangent bundles of sphere along a tree. Moreover, for symplectic automorphisms of Penner type, we prove that our computation of categorical entropy becomes a computation by simple linear algebra.Comment: 38 page