Through the Looking Glass

It is frequently possible to produce new Calabi-Yau threefolds from old ones by a process of allowing the complex structure to degenerate to a singular one, and then performing a resolution of singularities. (Some care is needed to ensure that the Calabi-Yau condition be preserved.) There has been speculation that all Calabi-Yau threefolds could be linked in this way, and considerable evidence has been amassed in this direction. We propose here a natural way to relate this construction to the string-theoretic phenomenon known as ``mirror symmetry.'' We formulate a conjecture which in principle could predict mirror partners for all Calabi-Yau threefolds, provided that all were indeed linked by the degeneration/resolution process. The conjecture produces new mirrors from old, and so requires some initial mirror manifold construction---such as Greene-Plesser orbifolding---as a starting point. (Lecture given at the CIRM conference, Trento, June 1994, and at the Workshop on Complex Geometry and Mirror Symmetry, Montr\'eal, March 1995.)Comment: latex2e, 22 pages with 1 figur

The Legality of University-Conducted Dormitory Searches for Internal Disciplinary Purposes

The issue examined is whether those unique characteristics of the university environment that have led to the development of a judicially-sanctioned general regulatory power will automatically render a warrantless disciplinary search reasonable within the terms of the fourth amendment. (LBH

Compactifications of moduli spaces inspired by mirror symmetry

We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some ``extra structure'' described by means of classes in H^2. The expectation that this moduli space is well-behaved in these ``extra structure'' directions leads us to formulate a simple and compelling conjecture about the action of the automorphism group on the K\"ahler cone. If true, it allows one to apply Looijenga's ``semi-toric'' technique to construct a partial compactification of the moduli space. We explore the implications which this construction has concerning the properties of the moduli space of complex structures on a ``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a similarity which might have been noticed between certain work of Mumford and of Mori from the 1970's produces (with hindsight) evidence for mirror symmetry which was available in 1979. [The author is willing to mail hardcopy preprints upon request.]Comment: 25 pp., LaTeX 2.09 with AmS-Font