Scientific Series

In the first part of the seminar I will give a short review of the frame-independent formulation of Newtonian gravity, called Newton-Cartan Gravity, and explain why there is a renewed interest into non-relativistic gravity in general.  In the second part I will discuss, as a particular application,  a recent proposal for an Effective Field Theory describing a massive spin-2 mode (the so-called GMP mode) in the Fractional Quantum Hall Effect.

This project examines the structure of a certain category of representations of a Lie algebra called Whittaker modules. Whittaker modules generalize highest weight modules, and the structure of the category is similar to that of Bernstein-Gelfand-Gelfand’s category O. In particular, Whittaker modules have finite length composition series and all irreducible Whittaker modules appear as quotients of standard Whittaker modules, which are generalizations of Verma modules. Using the localization theory of Beilinson-Bernstein, one obtains a beautiful geometric description of Whittaker modules as twisted sheaves of D-modules on the associated flag variety. I use this geometric setting to develop an algorithm for computing the multiplicities of irreducible Whittaker modules in the composition series of standard Whittaker modules. This algorithm establishes that the multiplicities are determined by parabolic Kazhdan-Lusztig polynomials.

Starting from the well known Laplace-Runge-Lenz vector of the Kepler problem, I will introduce a notion of dynamical (hidden)
symmetries. These are genuine phase space symmetries that stand in contrast to the more familiar symmetries of the configuration space
discussed in truncated versions of Noether's theorem. Proceeding to a relativistic description, I will demonstrate that such symmetries -- encoded
in the so called Killing-Yano tensors -- play a crucial role in the study of rotating black holes described by the Kerr geometry. Even more remarkably, I will show that one such special symmetry is enough to guarantee complete integrability of particle and light motion in general rotating black hole spacetimes in an arbitrary
number of spacetime dimensions. Recent developments on the separability of test fields in these spacetimes will also be discussed.

Diamond anvil cells and shock techniques now permit the study of matter under multimegabar (i.e. several hundred GPa) pressures. The properties of matter in this pressure regime differ drastically from those known at 1 atm. Just how different physics and chemistry are at high pressure and the role that a chemical intuition for bonding and structure can have in understanding matter at high pressures will be explored in this lecture. I will discuss in detail an overlapping hierarchy of responses to increased density, consisting of (a) squeezing out van der Waals space (for molecular crystals); (b) increasing coordination; (c) decreasing the bond length of covalent bonds and the size of anions; and (d) an extreme regime of electrons moving off atoms and new modes of correlation. Examples of the startling chemistry and physics that emerge under such extreme conditions will alternate in this account with qualitative chemical ideas about the bonding involved.

The space-based gravitational wave detector LISA may be able to detect gravitational waves from a first order phase transition at the electroweak scale. Acoustic waves produced during the transition are largely responsible for the resulting signal. I will present results from a large campaign of simulations studying such phase transitions, determining the spectral shape of the gravitational wave power spectrum with unparalleled accuracy. Measuring a cosmological stochastic background could place constraints on the phase transition parameters, such as the nucleation rate and temperature, and therefore provide important information about physics beyond the Standard Model. However, better understanding of the source, as well as the underlying theories of physics beyond the Standard Model, is required before the launch of LISA. I will outline how this understanding can be developed.

In this talk we will review our physical proposal, with D. Persson and R. Volpato, to understand the genus zero property of monstrous moonshine. The latter was proven by Borcherds in 1992 by brute-force computation but has since resisted a conceptual understanding. We embed the Monster VOA of Frenkel-Lepowsky-Meurman into a heterotic string compactification and use physical arguments, i.e. T-dualities and decompactification limits, to understand the genus zero property. We find that the Monster Lie algebra (a Generalized, or Borcherds-Kac-Moody algebra) acts as a sort of "algebra of BPS states". We also sketch an analogous proposal, with S. Harrison and R. Volpato, for Conway moonshine using the type II string. Along the way we construct a new super Borcherds-Kac-Moody algebra on which the Conway group acts faithfully and prove its (twisted) denominator identities, which should be identified with BPS state counts in our string theory. 

Already in their early papers, Kosterlitz and Thouless envisaged the melting of solids by the unbinding of the topological defects associated with translational order: dislocations. Later it was realized that the resulting phases have translational symmetry but rotational rigidity: they are liquid crystals.

We consider the topological melting of solids as a zero-temperature quantum phase transition. In a generalization of particle-vortex duality, the Goldstone modes of the solid, phonons, map onto gauge bosons which mediate long-range interactions between dislocations. The phase transition is achieved by a Bose-Einstein condensation of dislocations, restoring translational symmetry and destroying shear rigidity. The dual gauge fields become massive due to the Anderson-Higgs mechanism. In this sense, the liquid crystal is a "stress superconductor".

We have developed this dual gauge field theory both in 2+1D, where dislocations are particle-like and phonons are vector bosons, and 3+1D where dislocations are string-like and phonons are Kalb-Ramond gauge fields. Focussing mostly on the theoretical formalism, I will discuss the relevance to recent experiments on helium monolayers, which show evidence for a quantum hexatic phase.

References:

2+1D : Physics Reports 683, 1 (2017)

3+1D : Physical Review B 96, 1651115 (2017)

In this talk I will consider the black hole formed as a result of a binary black hole coalescence.  As expected, the final black hole eventually approaches a Kerr solution.  We show numerically, in full general relativity and in the highly non-linear merger regime, how the final black hole approaches equilibrium. In particular we show how the infalling radiation wipes out the deviations from Kerr and the rate at which the multipole moments decay to their asymptotic values.

Topology has in the past decades become an organizing principle in the classification and characterization of phases of matter.  While all possible topological phases of free fermions in the presence of external symmetries have been fully worked out, the inclusion of lattice symmetries relevant to any real-life material provides for an active research area.

In this seminar, I will present a classification of all possible gapped topological phases of non-interacting insulators with lattice symmetries, both in the absence and presence of time-reversal symmetry. This is done using a very simple counting scheme based on the electronic band structure of the materials. Despite the simplicity of the procedure, it is based on (and matches all known predictions of) the far more involved mathematical framework known as K-theory, which establishes the correctness and completeness of the counting scheme. The same straightforward counting can also be used to study transitions between crystalline topological phases. This allows us to list all possible types of such transitions for any given crystal structure, and accordingly stipulate whether or not they give rise to intermediate Weyl semimetallic phases. The presented procedure is ideally suited for the analysis of real, known materials, as well as the prediction of new, experimentally relevant, topological materials.

References:

http://doi.org/10.1103/PhysRevX.7.041069

http://arxiv.org/pdf/1711.04769

Cosmic variance puts hard limits on what we can learn about fundamental physical processes on the largest cosmological scales. A neat trick allows this limit to be circumvented in some cases though, by cross-correlating multiple tracers of the cosmic matter distribution. After giving a few examples of where the multi-tracer technique can be useful, I will outline a new analytic statistical model to describe how the galaxies seen in different surveys (and at different wavelengths) jointly populate the dark matter halo distribution. While the model is necessarily simplified, its flexibility and analytic nature allow rapid exploration of the parameter space relevant to forthcoming surveys in the optical, IR, microwave, and radio. I will show the results of an MCMC fit to low-redshift multi-frequency (radio and optical) luminosity function data, and then discuss several successful (and not-so-successful) consistency tests of the model.

An approximate representation of a finitely-presented group is an assignment of unitary matrices to the generators, such that the defining relations are close to the identity in the normalized Hilbert-Schmidt norm. A group is said to be hyperlinear if every non-trivial element can be bounded away from the identity in approximate representations of the group. Determining whether all groups are hyperlinear is a major open problem, as a non-hyperlinear group would provide a counterexample to the famous Connes embedding problem.

Given the difficulty of the Connes embedding problem, it makes sense to look at an easier problem: how fast does the dimension of approximate representations grow (as a function of how close the defining relations are to the identity) when we require a given set of elements to be bounded away from the identity. These growth rates are called the hyperlinear profile of the group.

In this talk, I will explain our best lower bounds on hyperlinear profile, as well as the connection to entanglement requirements for non-local games (joint work with Thomas Vidick). Time permitting,  I will also mention some other approaches to looking for non-hyperlinear groups, including the recent work of De Chiffre, Glebsky, Lubotzky, and Thom on a group which is not approximable in the unnormalized Hilbert-Schmidt norm.