Talks

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.