Colloquium

Fault-tolerant quantum computers will compute by applying
a sequence of elementary unitary operations, or gates, to an
error-protected subspace.   While algorithms are typically expressed
over arbitrary local gates, there is unfortunately no known theory
that can correct errors for a continuous set of quantum gates.
However, theory does support the fault-tolerant construction of
various finite gate sets, which in some cases generate circuits that
can approximate arbitrary gates to any desired precision.   In this
talk, I will present a framework for approximating arbitrary qubit
unitaries over a very general but natural class of gate sets derived
from the theory of integral quaternions over number fields, where the
complexity of a unitary is algebraically encoded in the length of a
corresponding quaternion.  Then I will explore the role played by
higher-dimensional generalizations of the Pauli gates in various
physical and mathematical settings, from classifying bulk-boundary
correspondences of abelian fractional quantum Hall states to
generating optimal symmetric quantum measurements with surprising
connections to Hilbert's 12th problem on explicit class field theory
for real quadratic number fields.

100 years after the existence of gravitational waves was first postulated by Albert Einstein, the LIGO and Virgo Collaborations detected gravitational waves for the first time on September 14, 2015. The gravitational waves originated from a pair of black holes that merged over one billion years ago.  The merger was so powerful that it shook the very fabric of space and sent a ripple across the Universe that we observed here on Earth at present day. Although this event was but a blip in a sea of data taken by various experiments over the last several decades, it represents a paradigm shift in how we study our Universe.

In this colloquium, I will present some of the history of this great discovery and what it means for our future at the dawn of gravitational wave astronomy.

Recent developments in our understanding of black hole evaporation and the information paradox suggest that effects from quantum gravity are not necessarily hidden at the Planck scale. They might even one day be testable by gravitational wave measurements. To prepare ourselves, we must first understand what quantum gravity really means. Thankfully, we are pre-armed with a deep principle about gravity—that spacetime is really a hologram—and a powerful model for making this idea precise: gauge/gravity duality. The present challenge is to translate our questions about gravity into the natural language of the dual conformal field theory (CFT). I will describe the foundation for such a program that links the integral geometry of a gravitational spacetime to a CFT operator product expansion.

Entanglement is both a central feature of quantum mechanics and a powerful tool for studying quantum systems. Even empty spacetime is a highly entangled state, and this entanglement has the potential to explain puzzling thermodynamic properties of black holes. In order to apply the methods of quantum information theory to problems in gravity we have to confront a more fundamental question: what is a local subsystem, and what are its physical degrees of freedom? I will show that local subsystems in gravity come with new physical degrees of freedom living on the boundary, as well as new physical symmetries. These structures offer us new insight into how spacetime is entangled, and a new perspective on the problem of quantizing gravity.

Winds are driven by the gradients of solar heating. Vertical gradients cause thermal convection on the scale of the troposphere depth (less than 10 km). Horizontal gradients excite motions on a planetary (10000 km) and smaller scales. Weather is mostly determined by the flows at intermediate scale (hundreds of kilometers). Where these flows get their energy from? The puzzle is that three-dimensional small-scale motions cannot transfer energy to larger scales while large-scale planar motions cannot transfer energy to smaller scales. In the talk, I'll describe experimental and observational data that suggest one possible resolution of this puzzle. I also describe some puzzling properties of two-dimensional turbulence including conformal invariance of statistics.

What are the bounds of the AdS/CFT correspondence? Which quantities in conformal field theory have simple descriptions in terms of classical anti-de Sitter spacetime geometry? These foundational questions in holography may be meaningfully addressed via the study of CFT correlation functions, which map to amplitudes in AdS. I will show that a basic building block in any CFT -- the conformal block -- is equivalent to an elegant geometric object in AdS, which moreover greatly streamlines and clarifies calculations of AdS amplitudes. By studying correlators with certain Lorentzian kinematics, one can constrain the space of consistent theories of AdS quantum gravity itself. In particular, by harnessing a recent bound on the rate of onset of chaos in thermal states, I will rule out the existence of certain classes of putative 2D CFTs and their 3D gravity duals, and argue that others exhibit signatures of Regge trajectories of string theory. This may be viewed as a novel, Lorentzian counterpart of the conformal bootstrap, relating dynamical constraints on the development of quantum chaos to the determination of the AdS/CFT landscape.

The coalescence of black hole-black hole (BHBH), black hole-neutron star (BHNS) and neutron star-neutron star (NSNS) systems are among the most promising sources of gravitational waves (GWs) detectable by Advanced LIGO/Virgo and NANOGrav. In addition, distinct observable electromagnetic radiation may accompany these GWs. Such "multi-messenger" sources can be powerful probes of fundamental physics such as the state of matter under extreme conditions, cosmology, as well as our theories of gravity. However, the identification, detection and interpretation of multimessenger signals from such sources requires careful theoretical modeling through the last stages of the compact binary inspiral, during which all approximations to general relativity break down. The only avenue to theoretically understanding these highly non-linear systems is solving the Einstein equations with the aid of supercomputers. This task is far from trivial: ill-posed formulations of the Einstein equations, gauge issues, the presence of singularities, shocks, and large range of length and time scales inherent to these systems pose strong theoretical and computational challenges. In this talk I will review these challenges, describe state-of-the-art numerical relativity techniques that overcome them, and present results from recent supercomputer simulations of binary NSNSs, and BHBHs around magnetized disks. I will conclude by discussing future directions and applications of numerical relativity.

Bott periodicity (1956) is a classical and old result in mathematics.
Its easiest incarnation of which concerns Clifford algebras. It says
that, up to Morita equivalence, the real Clifford algebras Cl_1(R),
Cl_2(R), Cl_3(R), etc. repeat with period 8. A similar result holds
for complex Clifford algebras, where the period is now 2. The modern
way of phrasing Bott periodicity in is terms of K-theory: I will
explain how one computes K-theory, and we will see the 8-fold Bott
periodicity emerge from the computations.

Elliptic cohomology is a fancy version of K-theory which can be
thought of as the K-theory of the loop space. A useful slogan is that
K-theory is to quantum mechanics, what elliptic cohomology is to
string theory. This cohomology theory satisfies a version of Bott
periodicity, with period 576. I will explain where that number 576
comes from, and what physical significance this might have.

I conjecture that the above 576-fold periodicity reflects itself in
the classification of 3d TQFTs. Here, the relevant TQFTs are the ones
associated to the chiral Majorana fermion (a type of abelian
Chern-Simons theory of central charge c=1/2). The claim is that the
theory becomes trivial once the central charge reaches 576·1/2 = 288.
The classification of abelian Chern-Simons theories has been
considered by Belov-Moore (2005), who claimed that the periodicity was
reached at c = 24 and later by Kapustin-Saulina (2010), who claimed
that the periodicity was never reached. Our proposal lies strictly in
between those of Belov-Moore and Kapustin-Saulina.

Almost fifteen years after LIGO started listening to the cosmos, and 100 years after Einstein discovered general relativity, gravitational waves have been detected by ground-based interferometers, opening a new window on the universe. In this talk I will address some of the most exciting areas of research advanced LIGO will allow us to explore in the coming years. Detection and characterization of gravitational wave transients will be discussed, as well as their impact on astrophysics.

In recent decades probing for the subtle indications of new physics in
experimental data has become increasingly difficult. The datasets have gotten
much bigger, the experiments more complex, and the signals ever smaller. Success
stories, like LIGO and Kepler, require a sophisticated combination of statistics
and computation, coupled with an appreciation of both the experimental realities
and the theoretical framework governing the data.

In this talk I will look broadly at data science in physics, and how and why it
has taken an increasingly central role. I'll highlight specifically my current
area of research, radio cosmology: discussing why it is one of the most
challenging areas for data science, and describing my work developing optimal
and efficient statistical methods for turning terabytes of timestreams into
cosmology.