This is joint work with Richard Ketchersid.

Schipperus introduced the countable-finite game in the early 1990s. It is
an infinite game played between two players relative to a set S. In the
presence of choice, it is obvious that player II has a winning strategy
for all S, and it is natural to ask whether choice can be dispensed with.

AD+ is a technical strengthening of AD introduced by Hugh Woodin. It is
open whether AD+ actually follows from AD. All known models of AD come
from certain canonical models produced by the derived model construction.

In these canonical models, we show that every set either embeds the reals
or else is well-orderable.

From this we deduce that, except for the case when S is countable, the
countable-finite game on S is undetermined in these models.

One of the most fundamental problems in complex
geometry is to determine when two bounded domains
in C^n are biholomorphically equivalent. Even for complete
Reinhardt domains, this fundamental problem remains unsolved
for many years. Using the Bergmann function theory,
we construct an infinite family of numerical invariants from
the Bergman functions for complete Reinhardt domains in
C^n. These infinite family of numerical invariants are actually
a complete set of invariants if the domains are pseudoconvex
with C^1 boundaries. For bounded complete Reinhardt domains
with real analytic boundaries, the complete set of numerical
invariants can be reduced dramatically although the
set is still infinite. We shall also discuss the role of the Hilbert 14th
problem in the construction of numerical biholomorphic
invariants of complete Reinhardt domains in C^n.

For a class of second order supersymmetric differential operators,
including the Kramers-Fokker-Planck operator of kinetic theory, we
determine the semiclassical (here the low temperature) asymptotics for the
splitting between the two lowest eigenvalues, with the first one being
0. Specifically, we consider the case when the exponent of the associated
Maxwellian has precisely two local minima and one saddle
point. The splitting is then exponentially small and is related to a
tunnel effect between the minima. We also show that the rate of the return
to equilibrium for the associated heat semigroup is dictated by the first
non-vanishing eigenvalue. This is joint work with Fr\'ed\'eric H\'erau and
Johannes Sj\"ostrand.