Talks

Some numbers mean things, and some numbers do things. Making--and breaking--that distinction was central to renowned mathematician John von Neumann’s implementation of Alan Turing’s Universal Machine in 1945-56. In this lecture, you will learn about the unlikeliest place on earth to build such a device and how this vital 5-kilobyte step in the digital revolution was sparked by a collision of ideas between mathematicians and engineers. Combining soldering guns with science, Von Neumann and his Electronic Computing Instrument tackled previously intractable problems ranging from thermonuclear explosions, stellar evolution, and long-range weather forecasting to cellular automata, network optimization, and the origins of life. In this highly visual and informative presentation, George Dyson will impart the full story - from the people to their processors - and where our digital directions through history may lead us next.

We investigate the entanglement spectra of topological insulators which have gapless edge states on their spatial boundaries. In the physical energy spectrum, a subset of the edge states that intersect the Fermi level translates to discontinuities in the trace of the single-particle entanglement spectrum, which we call a `trace index'. We find that any free-fermion topological insulator that exhibits spectral flow has a non-vanishing trace index, which provides us with a new description of topological invariants. In addition, we identify the signatures of spectral flow in the single-particle and many-body entanglement spectrum; in the process we present new methods to extract topological invariants and establish a connection between entanglement and quantum Hall physics in translationally invariant and disordered systems.

Time-reversal invariant band insulators can be separated into two categories: `ordinary' insulators and `topological' insulators. Topological band insulators have low-energy edge modes that cannot be gapped without violating time-reversal symmetry, while ordinary insulators do not. A natural question is whether more exotic time-reversal invariant insulators (insulators not connected adiabatically to band insulators) can also exhibit time-reversal protected edge modes. In 2 dimensions, one example of this is the fractional spin Hall insulator (essentially a spin-up and spin-down copy of a fractional quantum Hall insulator, with opposite effective magnetic fields for each spin). I will discuss another family of strongly interacting insulators, which exist in both 2 and 3 dimensions, that can have time-reversal protected edge modes. This gives a new set of examples of `fractional' topological insulators.