Talks

A graph is said to be a (1-dimensional) expander if the second eigenvalue of its adjacency matrix is bounded away from 1, or almost-equivalently, if it has no sparse vertex cuts. There are several natural ways to generalize the notion of expansion to hypergraphs/simplicial complexes, but one such way is 2-dimensional spectral expansion, in which the expansion of vertex neighborhoods (remarkably) witnesses global expansion. While 1-dimensional expansion is known to be achieved by, e.g., random regular graphs, very few constructions of sparse 2-dimensional expanders are known, and at present all are algebraic. It is an open question whether sparse 2-dimensional expanders are "abundant" and "natural" or "rare." In this talk, we'll give some evidence towards abundance: we show that the set of triangles in a random geometric graph on a high-dimensional sphere yields an expanding simplicial complex of arbitrarily small polynomial degree.

In this talk, I will discuss certain homogeneous spaces of the Poincaré group that correspond to the well-known asymptotic regions of asymptotically flat spacetimes: time-, space-, and light-like infinity. I will then show that all of these spaces admit a uniform description as surfaces embedded in a higher-dimensional space. I will conclude with some comments concerning the computation of correlators of conformal carrollian theories from this embedding space picture.

Sparse linear regression is a fundamental problem in high-dimensional statistics, but strikingly little is known about how to solve it efficiently without restrictive conditions on the design matrix. This talk will focus on the (correlated) random design setting, where the covariates are independently drawn from a multivariate Gaussian and the ground-truth signal is sparse. Information theoretically, one can achieve strong error bounds with O(klogn) samples for arbitrary covariance matrices and sparsity k; however, no efficient algorithms are known to match these guarantees even with o(n) samples in general. As for hardness, computational lower bounds are only known with worst-case design matrices. Random-design instances are known which are hard for the Lasso, but these instances can generally be solved by Lasso after a simple change-of-basis (i.e. preconditioning). In the talk, we will discuss new upper and lower bounds clarifying the power of preconditioning in sparse linear regression. First, we show that the preconditioned Lasso can solve a large class of sparse linear regression problems nearly optimally: it succeeds whenever the dependency structure of the covariates, in the sense of the Markov property, has low treewidth -- even if the covariance matrix is highly ill-conditioned. Second, we construct (for the first time) random-design instances which are provably hard for an optimally preconditioned Lasso. Based on joint works with Jonathan Kelner, Frederic Koehler, Dhruv Rohatgi.

The Morita 2-category has as objects associative algebras, 1-morphisms are bimodules and 2-morphisms are given by bimodule homomorphisms. Equivalent objects in this category are exactly Morita equivalent algebras. A vast generalisation of this as a higher category is the so-called higher Morita category, denoted Alg_n. It has two constructions, one due to Haugseng, and one due to Scheimbauer which uses (constructible) factorization algebras. In the latter, Gwilliam-Scheimbauer has proven that every object of Alg_n is n-dualizable. Hence, by the Cobordism Hypothesis, every object gives rise to an n-dimensional (fully extended framed) topological field theory. A natural question to ask is “Which objects of Alg_n are also (n+1)-dualizable?”. This talk is on work in progress (for n=2) to prove a conjecture due to Lurie answering this question. 

Zoom link: https://pitp.zoom.us/j/98595913913?pwd=Vlo1aWtXVlZBVjYxNFlTM2VZY2s3Zz09

Massless particles are always assumed to have Lorentz invariant helicity. However, the generic massless particles allowed by Lorentz invariance and quantum mechanics are "continuous spin particles," whose infinitely many discrete helicity states mix under little group transformations by an amount determined by the spin scale. Previous work at the Perimeter Institute has shown that CSPs have well-behaved soft emission amplitudes, are described by a simple free field theory, and reduce to the photon or graviton at momenta above the spin scale. In this talk, I will show how to couple CSP fields to classical particles. When treating on-shell CSP emission and absorption, the resulting dynamics are causal and readily calculable. Remarkably, they are also unique, depending on only the spin scale. This opens the door to precision experiments which probe the spin scale of the photon and graviton.

Zoom Link:  https://pitp.zoom.us/j/94728811371?pwd=Y3RoRnJuUzNzT0FmRVFiazI5czc5Zz09

Over the past few years, Transformers have revolutionized deep learning, leading to advances in natural language processing and beyond. These models discard recurrence and convolutions, in favor of "self-attention" which directly and globally models interactions within the input context. Despite their success, currently there is limited understanding of why they work. In this talk, I will present our recent results on rigorously quantifying the statistical and representational properties of Transformers which shed light on their ability to capture long range dependencies efficiently. First, I will show how bounded-norm self-attention layers can represent arbitrary sparse functions of the input sequence, with sample complexity scaling only logarithmically with the context length, akin to sparse regression. Subsequently, I will briefly show how this ability of self-attention to compute sparse functions along with its ability to compute averages can be used to construct Transformers that exactly replicate the dynamics of a recurrent model of computation depth $T$ with only $o(T)$ depth. I will conclude the talk with experimental results on synthetic tasks based on learning Boolean functions and automata. Based on joint works with Jordan T. Ash, Ben L. Edelman, Sham M. Kakade, Akshay Krishnamurthy, Bingbin Liu, and Cyril Zhang. 

In this talk I will explore the role of the boundary Cotton tensor in the reconstruction of the solution space of four-dimensional asymptotically AdS and mostly asymptotically flat spacetimes. I will discuss charges from a purely boundary perspective, which emerge in sets of electric and magnetic towers, not necessarily conserved and possibly including subleading components.

On Tuesday, M. Schiavina laid out the theoretical framework for the symplectic reduction of gauge theories in the presence of corners. In this talk I will apply this theoretical framework to Yang-Mills theory on a null boundary and show how a pair of soft charges controls the residual (corner) gauge symmetry after the first-stage symplectic reduction, and therefore the superselection structure of the theory after the second-stage symplectic reduction. I will also discuss the subtleties of the gauge A_u = 0, the interpretation of electromagnetic memory as superselection, and how the nonlinear structure of the non-Abelian theory complicates this picture.

After reviewing the constrained Hamiltonian analysis of geometric actions, the construction is applied to the case of the BMS group in four dimensions, where it yields two plus one dimensional BMS4 invariant field theories. (Based on work done in collaboration with K. Nguyen and R. Ruzziconi)

This talk focuses on classical features of asymptotic QED, i.e. the limit of QED at null and time-like infinity. The BV-BFV formalism allows one to view this as a boundary theory of bulk QED and carries a natural notion of what it means to be a symmetry of the model. I will make the connection between this perspective and the earlier findings of Herdegen (JMP 1996) and Strominger et.al. (JHEP 2014) concerning large gauge symmetries in QED. This is based on a joint paper with Michele Schiavina (CMP 2021).

We construct a Hermitian random matrix model that provides a stable non-perturbative completion of Cangemi-Jackiw (CJ) gravity, a two-dimensional theory of black holes in asymptotically flat spacetimes. The matrix model reproduces, to all orders in the topological expansion, the Euclidean partition function of CJ gravity with an arbitrary number of boundaries. The non-perturbative completion enables the exact computation of observables in flat space quantum gravity which we use to explicitly characterize the Bondi Hamiltonian spectrum. We discuss the implications of our results for the flat space S-matrix and black holes.

Google's TPUs were exclusively designed to accelerate and scale up machine learning workloads, amid the ongoing planet-wide race to build faster specialized hardware for artificial intelligence. But one must surely be able to use this hardware for other challenging computational tasks, right? We explored how to turn a TPU pod (2048 TPU v3 cores) into a dense linear algebra supercomputer to e.g. multiply two matrices of size 1,000,000 x 1,000,000 in just 2 minutes. We then used this power to perform a number of quantum physics and quantum chemistry computations at scale. For instance, we recently completed two largest-ever computations: a Density Functional Theory DFT computation of electronic structure (with N = 248,000 orbitals), and a Density Matrix Renormalization Group DMRG computation (with bond dimension D = 65,000). Cloud-based TPU pods and GPU pods are accessible to anyone and are poised to revolutionize the scientific supercomputing landscape.

Zoom link:  https://pitp.zoom.us/j/97006251134?pwd=WHhLa2pRUno0LzJjbzVnL2tsdGhOUT09