Categorical Mirror Symmetry: The Elliptic Curve

We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\widetilde{M}$ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on $M$ and a suitable version of Fukaya's category of Lagrangian submanifolds on $\widetilde{M}.$ We prove this equivalence when $M$ is an elliptic curve and $\widetilde{M}$ is its dual curve, exhibiting the dictionary in detail.Comment: harvmac, 29 pages (big); updated with correction

Seidel's Mirror Map for the Torus

Using only the Fukaya category and the monodromy around large complex structure, we reconstruct the mirror map in the case of a symplectic torus. This realizes an idea described by Paul Seidel.Comment: 6 pages, some typos and references fixe

Constructible Sheaves and the Fukaya Category

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $Tw Fuk(T^*X)$ for the $A_\infty$-triangulated envelope of $Fuk(T^*X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $Sh(X)$ quasi-embeds into $Tw Fuk(T^*X)$ as an $A_\infty$-category. Taking cohomology gives an embedding of the corresponding derived categories.Comment: 56 pages; to appear in JAM

A College-Level Course in Logology

The February 1978 issue of Word Ways asked readers for information on recreational linguistics courses taught in college, secondary school, night school or the like. In the spring term of 1978, when I was a junior at the University of Massachusetts, I taught a one-credit colloquium entitled Recreational Logology . In the fall term of 1978, I taught a three-credit course entitled An Introduction to Recreational Logology to 15 freshmen and sophomores, 13 of whom completed it. This met for three hours on Thursday evenings for 13 weeks from September to December, and covered the following topics