The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general timedependent one-dimensional harmonic oscillator, whose Newton equation q̈+ω(t)q = 0 cannot be solved in general. We follow the time-evolution of an initial e...

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL2(K) representation variety, where K = C or R, answering a question of W. Goldman. We also prove the existence of faithful representations into PU(2, 1) with certain nonintegral Toledo invariants.

The crepant resolution conjecture states that the Gromov–Witten invariants of an orbifold X should be determined in a precise way by the Gromov–Witten invariants of a crepant resolution of its coarse moduli space. We compute the Gromov–Witten invariants of the stack symmetric square of P 2 and compare them with the Gromov– Witten invariants of its crepant resolution, Hilb 2 P 2 (which were comp...

Redshift obeys adiabatic invariance. From this fact it follows that not only the individual photons are dimmed by a factor of z + 1 (if z is the redshift in percent), but the photon flux is reduced by the same factor once more. Hence the luminosity of any source is in the presence of redshift dimmed by a factor of (z+1). This model-independent result possibly "explains away" the excess dimming ...

A major challenge facing adiabatic quantum computing is that algorithm design and error correction can be difficult for adiabatic quantum computing. Recent work has considered addressing this challenge by using coherently controlled adiabatic evolutions in the place of classically controlled evolution. An important question remains: what is the relative power of controlled adiabatic evolution t...

We consider wave generation by turbulent convection in a plane parallel, stratified atmosphere that sits in a gravitational field, g. The atmosphere consists of two semi-infinite layers, the lower adiabatic and polytropic and the upper isothermal. The adiabatic layer supports a convective energy flux given by mixing length theory; Fe ""'pv~, where pis mass density and v8 is the velocity of the ...