Talks

We study the problem of learning a Hamiltonian given copies of its Gibbs state at a known inverse temperature. Anshu et al. recently studied the sample complexity (number of copies of the Gibbs state needed) of this problem for geometrically local Hamiltonians. In the high-temperature regime, their algorithm has sample complexity polynomial in the system size, temperature, and accuracy. Their algorithm can also be implemented with polynomial, but suboptimal, time complexity. Here, we study the same question for a more general class of Hamiltonians and present an algorithm that solves this problem with improved sample complexity and time complexity linear in the sample size. Furthermore, we prove a matching lower bound showing that our algorithm's sample complexity is optimal, and hence our time complexity is also optimal. Joint work with Robin Kothari and Ewin Tang.

 

Academic science reflects the context in which science is conducted. Thus, the academic scientific enterprise struggles with privilege, racism, sexism, ableism, and homophobia, in Canada, as around the world. In this talk, I will address the uncomfortable and sometimes difficult conversations we must have in order to get to the real work of identifying and removing barriers that limit access and engagement in the scientific enterprise for the full diversity of humanity. We live within a myth of meritocracy in academia and are failing to achieve our full potential as a sector. Moving on from self-awareness as individuals or organizations, we can start to identify the tools and skill sets that individuals and collectives need in order to create cultures of care that achieve inclusive excellence and the very best scientific outputs.

Academic science reflects the context in which science is conducted.  Thus, the academic scientific enterprise struggles with privilege, racism, sexism, ableism, and homophobia, in Canada, as around the world.  In this talk, I will address the uncomfortable and sometimes difficult conversations we must have in order to get to the real work of identifying and removing barriers that limit access and engagement in the scientific enterprise for the full diversity of humanity.  We live within a myth of meritocracy in academia and are failing to achieve our full potential as a sector.  Moving on from self-awareness as individuals or organizations, we can start to identify the tools and skill sets that individuals and collectives need in order to create cultures of care that achieve inclusive excellence and the very best scientific outputs.  

Zoom Link: https://pitp.zoom.us/j/91715376839?pwd=TnBIM1NNek4zOFNyVWFnMHMySE02Zz09

The lowered accuracy requirements at second order in the mass ratio greatly increases the utility of high-order post-Newtonian calculations at post-adiabatic order. In this talk I will present a methodology for solving the first- and second-order field equations using matched asymptotic expansions. As an application and test of the method, we solve the scalar wave equation sourced by a first order solution to the Regge-Wheeler equation with a particle moving in a circular orbit, and thus provide a construction of an analytic expansions of the Lorenz gauge metric perturbation.

Reconstructing a metric or vector potential that corresponds to a given solution to the Teukolsky equation is an important problem for self-force calculations. Traditional reconstruction algorithms do not work in the presence of sources, and they give rise to solutions in a radiation gauge. In the electromagnetic case, however, Dolan (2019) and Wardell and Kavanagh (2020) very recently showed how to reconstruct a vector potential in Lorenz gauge, which is more convenient for self-force. Their algorithm is based on a new Hertz-potential 2-form. In this talk, I will first show that the electromagnetic Teukolsky formalism takes a simplified form when expressed in terms of differential forms and the exterior calculus. This formalism makes the new Lorenz-gauge construction much more transparent, and it enables an extension to nonzero sources. In particular, I will derive a corrector term, related to the charge current, which when added to the vector potential gives a solution to the Maxwell equations with nonzero source. I will conclude by discussing prospects for extending to the gravitational case.

We identify a set of Hertz potentials for solutions to the vector wave equation on black hole spacetimes. The Hertz potentials yield Lorenz gauge electromagnetic vector potentials that represent physical solutions to the Maxwell equations, satisfy the Teukolsky equation, and are related to the Maxwell scalars by straightforward and separable inversion relations. Our construction, based on the GHP formalism, avoids the need for a mode ansatz and leads to potentials that represent both static and non-static solutions. As an explicit example, we specialise the procedure to mode-decomposed perturbations of Kerr spacetime and in the process make connections with previous results.

Understanding extended sources is an important aspect of the second-order problem in Kerr. Metric reconstruction procedures based on the Teukolsky equation bring challenges at second order due to singularities in the radiation gauge. One approach would be to soften these singularities using an effective source derived from a puncture. I will describe how the corrector-tensor reconstruction algorithm may be applied in a puncture scheme with a spatially compact effective source. To illustrate the method I will use an example in flat spacetime which provides some insights into the structure of the problem in Kerr.

I outline an algorithm for obtaining the first order metric perturbation solving the linearized Einstein equations sourced by a point mass in (a) a no-string gauge and (b) the Lorentz gauge via a variant of the Teukolsky formalism. The aim is to suitably combine the attractive features of this formalism with the milder singularity structure and better localization properties of the solutions in the latter gauges. This part of our work is intended as a starting point for the second order self force problem and relies in an essential way on the recently introduced corrector formalism by Green, Hollands and Zimmerman.

With the publication of the first second-order self-force results, it has become even more clear of the need for fast and efficient calculations to avoid the computational expense encountered when using current methods in the Lorenz gauge. One ingredient for efficient calculation of second-order self-force data will be the use of the highly regular gauge (1703.02836 and 2101.11409) with its weaker divergences along the worldline of the small object. In this talk, we will present steps towards transforming the current Lorenz gauge data into the highly regular gauge to be used for quasicircular orbits in Schwarzschild spacetime. The end result will be a source that can be used as an input into the second-order Einstein equations (see talks by Andrew Spiers and Benjamin Leather). In particular, this will allow us to solve the second-order Teukolsky equation using a point particle source instead of requiring the use of a puncture scheme.