Physics

Topology illuminates properties of geometric spaces which are independent of scale.  Scale-independent features of physical systems play an important role, for example when deducing the large-scale behavior from a small-scale description.  After an introduction to basic topological ideas, I will discuss two joint results with Mike Hopkins, one an application to string theory and the other an application to condensed matter theory.

Julia is a modern programming language that is (so they say) as easy to use as Python and as fast as Fortran, combined with the expressive power of Lisp. I will introduce the language via examples taken from linear algebra, differential equations, and scientific visualization.

In quantum error correcting codes, there is a distinction
between coherent and incoherent noise. Coherent noise can cause the
average infidelity to accumulate quadratically when a fixed channel is
applied many times in succession, rather than linearly as in the case
of incoherent noise. I will present a proof that unitary single qubit
noise in the 2D toric code with minimum weight decoding is mapped to
less coherent logical noise, and as the code size grows, the coherence
of the logical noise channel is suppressed. In the process, I will
describe how to characterize the coherence of noise using either the
growth of infidelity or the relation between the diamond distance from
identity and the average infidelity. I will explain how coherence in
the noise on physical qubits is transformed by error correction in
stabilizer codes. Then, I will sketch the proof that coherence is
suppressed for the 2D toric code. The result holds even when the
single qubit unitary rotation are allowed to have arbitrary directions
and angles, so long as the angles are below a threshold, and even when
the rotations are correlated. Joint work with John Preskill.

The out-of-time-ordered correlator (OTOC) and entanglement are two physically motivated and widely used probes of the ``scrambling'' of quantum information, which has drawn great interest recently in quantum gravity and many-body physics. By proving upper and lower bounds for OTOC saturation on graphs with bounded degree and a lower bound for entanglement on general graphs, we show that the time scales of scrambling as given by the growth of OTOC and entanglement entropy can be asymptotically separated in a random quantum circuit model defined on graphs with a tight bottleneck. Our result counters the intuition that a random quantum circuit mixes in time proportional to the diameter of the underlying graph of interactions.  It also serves as a more rigorous justification for an argument of [Shor, 1807.04363], that black holes may be very slow scramblers in terms of entanglement generation. Such observations may be of fundamental importance in the understanding of the black hole information problem. The bound we obtained for OTOC is interesting in its own right in that it generalized previous studies of OTOC on lattices to the geometries on graphs and proved it rigorously.   Based on [Harrow-Kong-Liu-Mehraban-Shor, 1906.02219]

It was during his stay in Prague when Einstein started in earnest to develop his ideas about general relativity. I will evoke those days in 1911 and 1912, discuss Einstein's papers on gravitation from that period, emphasize which new concepts and ideas he introduced. I also want to show you how the main themes that preoccupied him then, the principle of equivalence, bending of light, gravitational redshift and frame dragging effects are still alive in contemporary relativity.