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The basic structure of quantum mechanics was delineated in the early days of the theory and has not been modified since. Still it is interesting to ask whether that basic structure can be altered or generalized. In the last decade Bender et al have shown that one of the fundamental assumptions of quantum mechanics, that operators are represented by Hermitian matrices, can to an extent be relaxed. In this theory, the parity (P) and time-reversal (T) operators play a role analogous to the Hermitian conjugate. Recently we have extended the realm of `PT quantum mechanics' to include systems that are odd under time-reversal (T^2 = - 1), in the interest of constructing the PT-analogue of the Dirac equation. We find that the fundamental representation of the Dirac equation, which describes relativistic fermions, remains unchanged in the generalization to the non-Hermitian theory. Higher dimensional representations, which ordinarily decouple into pairs of Dirac fermions in Hermitian quantum mechanics, here describe new types of particles with extremely compelling properties. Most notably we have constructed a toy model representing two generations of massless neutrinos that nonetheless undergo flavor oscillations; furthermore this model is Lorentz invariant and unitary in time. The Standard Model requires that the neutrino be massive in order to accomodate the observed flavor oscillations, thus this toy model represents a significant departure from Standard Model physics.

One of the main science objectives for the Laser Interferometer Space Antenna (LISA) is to quantitatively map the strong field regions around compact objects using Extreme-Mass-Ratio Inspirals (EMRIs). This idea has been shown to be possible in principle, however in practice only inspirals in a Kerr spacetime have been studied in detail. A spacetime mapping algorithm for an EMRI inspiral into a generic compact object is formulated using ideas from integrable systems. I discuss several aspects of the theoretical development required to make the problem tractable. Some recent results about particle orbits around "Bumpy Black" holes are highlighted.

The top quark is the heaviest known type of quark, and possibly the last. Particle physicists sometimes refer to it as the "truth” quark, not always with tongue in cheek. The top quark might be just an ordinary quark, no stranger than the "strange" one, but it might hold the key to major questions of Nature through its connection to the origin of mass, the Higgs boson, and cosmic dark matter. At the Fermi National Accelerator Laboratory outside Chicago, hundreds of these heavy quarks have been observed and some first snapshots of their behavior have been obtained. At the Large Hadron Collider at CERN, millions of the heavy quarks will be produced. This lecture will review current knowledge of the top quark and explain how this knowledge has been obtained through experiments at the giant particle accelerators. Future experiments, which might reveal the top quark's deeper mysteries, will also be described.