Publications

Transition metal dichalcogenides possess a unique combination of properties that make them a malleable platform to study and engineer light-matter interactions. On one hand, monolayers of WS2 naturally occur in the semiconducting 1H phase whose optical properties are dominated by excitons emerging from the band edges at the K valley. On the other hand, the 1T phase exhibits metallic properties and can be triggered by weak external stimuli. Here we use plasma irradiation to engineer a 1H/1T mixed phase state in WS2 and control the grain size of the 1T patches by tuning the irradiation time. We show that in the mixed phase WS2 a band nesting effect gives rise to new critical points resulting in a low-energy excitonic transition below the A exciton. Compared to standard excitons in WS2, this new resonance shows larger absorption and longer lifetime. The combination of these properties suggests new concepts for exciton-based optoelectronic devices that could stem from the control of phase mixture states in two-dimensional semiconductors.

The existence of spin-currents in absence of any driving external fields is commonly considered an exotic phenomenon appearing only in quantum materials, such as topological insulators. We demonstrate instead that equilibrium spin currents are a rather general property of materials with non negligible spin-orbit coupling (SOC). Equilibrium spin currents can be present at the surfaces of a slab. Yet, we also propose the existence of global equilibrium spin currents, which are net bulk spin-currents along specific crystallographic directions of materials. Equilibrium spin currents are allowed by symmetry in a very broad class of systems having gyrotropic point groups. The physics behind equilibrium spin currents is uncovered by making an analogy between electronic systems with SOC and non-Abelian gauge theories. The electron spin can be seen as the analogous of the color degree of freedom and equilibrium spin currents can then be identified with diamagnetic color currents appearing as the response to an effective non-Abelian magnetic field generated by SOC. Equilibrium spin currents are not associated with spin transport and accumulation, but they should nonetheless be carefully taken into account when computing transport spin currents. We provide quantitative estimates of equilibrium spin currents for several systems, specifically metallic surfaces presenting Rashba-like surface states, nitride semiconducting nanostructures and bulk materials, such as the prototypical gyrotropic medium tellurium. In doing so, we also point out the limitations of model approaches showing that first-principles calculations are needed to obtain reliable predictions. We therefore use Density Functional Theory computing the so-called bond currents, which represent a powerful tool to understand the relation between equilibrium currents, electronic structure and crystal point group.

In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear systems for the individual tensor train cores. This approach can avoid the curse of dimensionality that threatens both the algorithmic and sample complexities of the recovery problem. Specifically, for Markov models, we prove that the tensor cores can be recovered with a sample complexity that scales logarithmically in the dimensionality. Finally, we illustrate the performance of the method with several numerical experiments.

Game theory has been an effective tool in the control of disease spread and in suggesting optimal policies at both individual and area levels. In this AMS Notices article, we focus on the decision-making development for the intervention of COVID-19, aiming to provide mathematical models and efficient machine learning methods, and justifications for related policies that have been implemented in the past and explain how the authorities' decisions affect their neighboring regions from a game theory viewpoint.

We propose new variants of the sketch-and-project method for solving large scale ridge regression problems. First, we propose a new momentum alternative and provide a theorem showing it can speed up the convergence of sketch-and-project, through a fast sublinear convergence rate. We carefully delimit under what settings this new sublinear rate is faster than the previously known linear rate of convergence of sketch-and-project without momentum. Second, we consider combining the sketch-and-project method with new modern sketching methods such as Count sketch, SubCount sketch (a new method we propose), and subsampled Hadamard transforms. We show experimentally that when combined with the sketch-and-project method, the (Sub)Count sketch is very effective on sparse data and the standard Subsample sketch is effective on dense data. Indeed, we show that these sketching methods, combined with our new momentum scheme, result in methods that are competitive even when compared to the conjugate gradient method on real large scale data. On the contrary, we show the subsampled Hadamard transform does not perform well in this setting, despite the use of fast Hadamard transforms, and nor do recently proposed acceleration schemes work well in practice. To support all of our experimental findings, and invite the community to validate and extend our results, with this paper we are also releasing an open source software package: RidgeSketch. We designed this object-oriented package in Python for testing sketch-and-project methods and benchmarking ridge regression solvers. RidgeSketch is highly modular, and new sketching methods can easily be added as subclasses. We provide code snippets of our package in the appendix.

Linear systems involving contiguous submatrices of the discrete Fourier transform (DFT)matrix arise in many applications, such as Fourier extension, superresolution, and coherent diffraction imaging. We show that the condition number of any such p\times q submatrix of the N\times NDFT matrix is at least exp\bigl( \pi 2\bigl[ min(p,q) - pqN\bigr] \bigr) , up to algebraic prefactors.That is, fixing the shape parameters (\alpha ,\beta ) := (p/N,q/N)\in (0,1)2, the growth ise\rho NasN\rightarrow \infty , the exponential rate being\rho =\pi 2[min(\alpha ,\beta ) - \alpha \beta ]. Our proof uses theKaiser--Bessel transform pair (of which we give a self-contained proof), plus estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized. We warm up with an elementary proof of the above but with half the rate, via a periodized Gaussian trial vector. Using low-rank approximation of the kerneleixt, we also prove another lower bound (4/e\pi \alpha )q, up to algebraic prefactors, which is stronger than the above for small\alpha and\beta . When combined, the bounds are within a factor of two ofthe empirical asymptotic rate, uniformly over (0,1)2, and become sharp in certain regions.However, the results are not asymptotic: they apply to essentially allN,p, andq, and with all constants explicit.

Many high-throughput sequencing data sets in biology are compositional in nature. A prominent example is microbiome profiling data, including targeted amplicon-based and metagenomic sequencing data. These profiling data comprises surveys of microbial communities in their natural habitat and sparse proportional (or compositional) read counts that represent operational taxonomic units or genes. When paired measurements of other covariates, including physicochemical properties of the habitat or phenotypic variables of the host, are available, inference of parsimonious and robust statistical relationships between the microbial abundance data and the covariate measurements is often an important first step in exploratory data analysis. To this end, we propose a sparse robust statistical regression framework that considers compositional and non-compositional measurements as predictors and identifies outliers in continuous response variables. Our model extends the seminal log-contrast model of Aitchison and Bacon-Shone (1984) by a mean shift formulation for capturing outliers, sparsity-promoting convex and non-convex penalties for parsimonious model selection, and data-driven robust initialization procedures adapted to the compositional setting. We show, in theory and simulations, the ability of our approach to jointly select a sparse set of predictive microbial features and identify outliers in the response. We illustrate the viability of our method by robustly predicting human body mass indices from American Gut Project amplicon data and non-compositional covariate data. We believe that the robust estimators introduced here and available in the R package RobRegCC can serve as a practical tool for reliable statistical regression analysis of compositional data, including microbiome survey data.

Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications from astronomy to nanoscale X-ray diffraction imaging. Despite the efforts of a multitude of scientists, from astronomers to mathematicians, there is, as yet, no satisfactory theoretical or algorithmic solution to this class of problems. Written for mathematicians, physicists and engineers working in image analysis and reconstruction, this book introduces a conceptual, geometric framework for the analysis of these problems, leading to a deeper understanding of the essential, algorithmically independent, difficulty of their solutions. Using this framework, the book studies standard algorithms and a range of theoretical issues in phase retrieval and provides several new algorithms and approaches to this problem with the potential to improve the reconstructed images. The book is lavishly illustrated with the results of numerous numerical experiments that motivate the theoretical development and place it in the context of practical applications.

The accurate description of nuclear quantum effects, such as zero-point energy, is important for modeling a wide range of chemical and biological processes. Within the nuclear-electronic orbital (NEO) approach, such effects are incorporated in a computationally efficient way by treating electrons and select nuclei, typically protons, quantum mechanically with molecular orbital techniques. Herein, we implement and test a NEO coupled cluster method that explicitly includes the triple electron-proton excitations, where two electrons and one proton are excited simultaneously. Our calculations show that this NEO-CCSD(eep) method provides highly accurate proton densities and proton affinities, outperforming any previously studied NEO method. These examples highlight the importance of the triple electron-electron-proton excitations for an accurate description of nuclear quantum effects. Additionally, we also implement and test the second-order approximate coupled cluster with singles and doubles (NEO-CC2) method, as well as its scaled-opposite-spin (SOS) versions. The NEO-SOS'-CC2 method, which scales the electron-proton correlation energy as well as the opposite-spin and same-spin components of the electron-electron correlation energy, achieves nearly the same accuracy as the NEO-CCSD(eep) method for the properties studied. Because of its low computational cost, this method will enable a wide range of chemical and photochemical applications for large molecular systems. This work sets the stage for a wide range of developments and applications within the NEO framework.

An asymmetric Bose-Einstein condensation (BEC) dome was observed in a recent experiment on the quantum dimer magnet Yb