We describe an isomorphism of categories conjectured by Kontsevich. If M and M̃ 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 M̃. We prove this equivalence when M is an elliptic curve and M̃ is its dual curve, exhibiting the dictionary in detail.

Let M be a simple 3-manifold with a toral boundary component ∂0M . If Dehn filling M along ∂0M one way produces a toroidal manifold and Dehn filling M along ∂0M another way produces a boundary-reducible manifold, then we show that the absolute value of the intersection number on ∂0M of the two filling slopes is at most two. In the special case that the boundary-reducing filling is actually a so...

The Heisenberg antiferromagnet, which arises from the large U Hubbard model, is investigated on the C60 molecule and other fullerenes. The connectivity of C60 leads to an exotic classical ground state with nontrivial topology. We argue that there is no phase transition in the Hubbard model as a function of U/t, and thus the large U solution is relevant for the physical case of intermediate coup...

Reexamination of general relativistic experimental results shows the universe is governed by Einstein’s static-spacetime general relativity instead of Friedmann-Lemaitre expanding-spacetime general relativity. The absence of expansion redshifts in a static-spacetime universe suggests a reevaluation of the present cosmology is needed. For many decades the Friedmann-Lemaitre spacetime expansion r...

We present an experimental study of velocity statistics for a partial layer of inelastic colliding beads driven by a vertically oscillating boundary. Over a wide range of parameters (accelerations 3-8 times the gravitational acceleration), the probability distribution P (v) deviates measurably from a Gaus-sian for the two horizontal velocity components. It can be described by P (v) ∼ exp(−|v/v ...

In the Starobinsky inflationary model inflation is driven by quantum corrections to the vacuum Einstein equation. We reduce the Wheeler-DeWitt equation corresponding to the Starobinsky model to a Schroedinger form containing time. The Schroedinger equation is solved with a Gaussian ansatz. Using the prescription for the normalization constant of the wavefunction given in our previous work, we s...

We investigate the ratio of gravitational binding energy to rest mass in general relativity. For N pointlike masses, an upper bound on the magnitude of this ratio can be derived using the second law of black hole dynamics. Only as N approaches infinity can the ratio approach one. A configuration that saturates the N = ∞ bound is a thin spherical shell as its radius is taken to zero. This system...

The averaging problem for inhomogeneous cosmologies is discussed in the form of a disputation between two cosmologists, one of them (RED) advocating the standard model, the other (GREEN) advancing some arguments against it. Technical explanations of these arguments as well as the conclusions of this debate are given by BLUE. 1. The conjecture about the average flow The standard model which, on ...

Using a statistical model for the normally deformed states and for their coupling to a member of the superdeformed band, we calculate the ensemble average and the fluctuations of the intensity for decay out of the superdeformed band and of the intraband decay intensity. We show that both intensities depend on two dimensionless variables: The ratio Γ ↓ /Γ S and the ratio Γ N /d. Here, Γ ↓ is the...

A simple and purely algebraic construction of super-energy (s-e) tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely timelike component of these s-e tensors has the mathematical...