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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.

Even the greatest scientists have made some serious blunders. "Brilliant Blunders" concerns the evolution of life on Earth, of the Earth itself, of stars, and of the universe as a whole.

In this talk, astrophysicist Dr. Mario Livio will explore and analyze major errors committed by such luminaries as Charles Darwin, Linus Pauling, and Albert Einstein. Dr. Livio will scrutinize the various types of blunders and attempt to explain how they happen. Blunders are not only inevitable, argues Dr. Livio, but also an integral component of the process of science.

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 findings on quantitative growth patterns have revealed striking generalities across the tree of life, and recurring over distinct levels of organization. Growth-mass relationships in 1) individual growth to maturity, 2) population reproduction, 3) insect colony enlargement and 4)  community production across wholeecosystems of very different types, often follow highly robust near ¾ scaling laws. These patterns represent some of the most general relations in biology, but the reasons they are so strangely similar across levels of organization remains a mystery. The dynamics of these distinct levels are connected, yet their scaling can be shown to arise independently, and free of system-specific properties. Numerous experiments in prebiotic chemistry have shown that minimal self-replicating systems that undergo template-directed synthesis, typically show reaction orders (ie. growth-mass exponents) between ½ and 1. I will outline how modifications to these simplified reaction schemes can yield growth-mass exponents near ¾, which may offer insight into dynamical connections across hierarchical systems.

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.

How well can we predict our future climate? If the flap of a butterfly’s wings can change the course of weather a week or so from now, what hope trying to predict anything about our climate a hundred years hence? In this talk I will discuss the science of climate change from a perspective which emphasises the chaotic (and hence uncertain) nature of our climate system. In so doing I will outline the fundamentals of climate modelling, and discuss the emerging concept of inexact supercomputing, needed - paradoxically perhaps - if we are to increase the accuracy of predictions from these models. Indeed, revising the notion of a supercomputer from its traditional role as a fast but precise deterministic calculating machine, may be important not only for climate prediction, but also for other areas of science such as astrophysics, cosmology and neuroscience.

Almost the first thing we learn as human beings is a sense of spatial awareness: the smaller the Euclidean distance between two objects, the closer they are. As adults, we apply this deeply held intuition to state space. In particular, as philosopher David Lewis made explicit in his seminal 1973 paper on Causation, we presume that one counterfactual world is closer to reality than another if this world resembles reality more than does the other. This intuition has guided the development of physical theory over the years. However, I will argue that our intuition is letting us down very badly. Motivated by results from nonlinear dynamical systems theory, I will argue that the so-called p-adic metric provides a much more physically meaningful measure of state-space distance than does the Euclidean metric, and moreover that the set of p-adic integers, for large p, provides the basis for constructing a realistic, locally causal description of quantum physics which is neither fine tuned nor violates experimenter free will, the Bell theorem notwithstanding. Indeed, using the p-adic metric  in state space, I assert that experimenters (from Aspect onwards) are not actually testing the Bell inequalities at all - not even approximately! A description of cosmological state space based on the set of p-adic integers suggests a new geometric route to the unification of quantum and gravitational physics, consistent with general relativity.

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.