The average quantum physicist on the street believes that a quantum-mechanical Hamiltonian must be Dirac Hermitian (symmetric under combined matrix transposition and complex conjugation) in order to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, for example, which is clearly not Dirac Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and thus it defines a perfectly acceptable quantum mechanics. Evidently, the axiom of Dirac Hermiticity is too restrictive. While the Hamiltonian $H=p^2+ix^3$ is not
Dirac Hermitian, it is PT symmetric; that is, it is symmetric under
combined space reflection P and time reversal T. In general, if a Hamiltonian $H$ is not Dirac Hermitian but exhibits an unbroken PT symmetry, there is a procedure for
determining the adjoint operation under which $H$ is Hermitian. (It is wrong to assume a priori that the adjoint operation that interchanges bra vectors and ket vectors in the Hilbert space of states is the Dirac adjoint. This would be like assuming a priori what the metric $g^{\mu\nu}$ in curved space is before solving
Einstein's equations.)

In the past a number of interesting quantum theories, such as the Lee model and the Pais-Uhlenbeck model, were abandoned because they were thought to have an incurable disease. The symptom of the disease was the appearance of ghost states
(states of negative norm). The cause of the disease was that the
Hamiltonians for these models were inappropriately treated as if they were DiracHermitian. The disease can be cured because the Hamiltonians for these models are PT symmetric, and one can calculate exactly and in closed form the appropriate adjoint operation under which each Hamiltonian is Hermitian. When
this is done, one can see immediately that there are no ghost states and that these models are fully acceptable quantum theories.

Muscles have hierarchical structure that spans several scales. The basic contraction unit is
the sarcomere, having a length of two microns. Myofibrils, typically of a few millimeters
long, are composed of thousands of sarcomeres connected in series. The muscle fiber is
made of a large number of parallel myofibrils coupled by a tissue.

Much is known about the overall mechanical response of an entire muscle fiber. Further,
recent technological advances have revealed the structure of the single sarcomere down
to the molecular level. Nevertheless, there is still a gap in understanding how the
collective behavior of the various scales gives rise to overall behavior. Importantly,
single-sarcomere models can not explain various experimental observations on the
macro-scale.

We present a theoretical framework for predicting the collective behavior of biologically
relevant ensembles of sarcomeres. The analysis is accomplished by transforming the non-
linear dynamics of an assemblage of sarcomeres into a partial differential equation for the
distribution of sarcomere lengths in the presence of stochastic fluctuations and biological
variability. It reproduces the results of previous experiments with no fitting parameters,
explains some puzzling observations and provides insights into damage under cyclic
eccentric loading.

For every prime power q and positive integer g, we let N_q(g) denote the maximum value of #C(F_q), where C ranges over all genus-g curves over F_q. Several years ago Kristin Lauter and I used a number of techniques to improve the known upper bounds on N_q(g) for specific values of q and g. The key to many of our improvements was a numerical invariant attached to non-simple isogeny classes of abelian varieties over finite fields; when this invariant is small, any Jacobian in the given isogeny class must satisfy restrictive conditions. Now Lauter and I have come up with a better invariant, which allows us to make even stronger deductions about Jacobians in isogeny classes. In this talk, I will explain how we have been able to use this new invariant, together with arguments about short vectors in Hermitian lattices over imaginary quadratic fields, to pin down several more values of N_q(g) and to improve the best known upper bounds in a number of other cases.