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We discuss Shnirelman’s Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos.

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel., NSF (DMS-1464693), NSF (DMS-1953852), DOE (DE-SC0009988), DOE (DE-SC0010010)

We give a uniform description of the decomposition of the unipotent variety of a classical group in arbitrary characteristic into pieces (considered in a non-uniform way in the earlier parts of this series). (C) 2009 Elsevier Inc. All rights reserved.

Scalable spaces are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality. They are also characterized by particularly nice behavior from the point of view of quantitative homotopy theory. Among other results, we show that spaces which are formal but not scalable provide counterexamples to Gromov’s long-standing conjecture on distortion in higher homotopy groups.

© 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved. A characteristic of the defocusing cubic nonlinear Schrödinger equation (NLSE), when defined so that the space variable is the multidimensional square (hence, rational) torus, is that there exist solutions that start with arbitrarily small Sobolev norms and evolve to develop arbitrarily large modes at later times; this phenomenon is recognized as a weak energy transfer to high modes for the NLSE [Colliander et al., Invent. Math., 181 (2010), pp. 39{113] and [R. Carles and E. Faou, Discrete Contin. Dyn. Syst., 32 (2012), pp. 2063{2077]. In this paper, we show that when the system is considered on an irrational torus, energy transfer is more difficult to detect.

© 2016, Springer Science+Business Media New York. We give a characterization of modified (in the sense of Zwegers) mock theta functions, parallel to that of ordinary theta functions. Namely, modified mock theta functions are characterized by their analyticity properties, elliptic transformation properties, and by being annihilated by certain second order differential operators.

© 2020, The Author(s). Haplotype reconstruction of distant genetic variants remains an unsolved problem due to the short-read length of common sequencing data. Here, we introduce HapTree-X, a probabilistic framework that utilizes latent long-range information to reconstruct unspecified haplotypes in diploid and polyploid organisms. It introduces the observation that differential allele-specific expression can link genetic variants from the same physical chromosome, thus even enabling using reads that cover only individual variants. We demonstrate HapTree-X’s feasibility on in-house sequenced Genome in a Bottle RNA-seq and various whole exome, genome, and 10X Genomics datasets. HapTree-X produces more complete phases (up to 25%), even in clinically important genes, and phases more variants than other methods while maintaining similar or higher accuracy and being up to 10× faster than other tools. The advantage of HapTree-X’s ability to use multiple lines of evidence, as well as to phase polyploid genomes in a single integrative framework, substantially grows as the amount of diverse data increases.

Let G G be a connected reductive group defined over a finite field F q \mathbf {F}_q and let L L be a Levi subgroup (defined over F q \mathbf {F}_q ) of a parabolic subgroup P P of G G

This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow.

© 2019 Duke University Press. All rights reserved. A combination of direct and inverse Fourier transforms on the unitary group U(N) identifies normalized characters with probability measures on N-tuples of integers. We develop the N → ∞ version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with the law of large numbers and the central limit theorem for global behavior of corresponding random N-tuples. As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon- Okounkov conjecture (which predicts asymptotic Gaussianity and the exact form of the covariance) for a family of non-simply-connected polygons. Another application is a central limit theorem for the U(N) quantum random walk with random initial data.

© 2020 ACM. We consider the bin packing problem with d different item sizes si and item multiplicities ai, where all numbers are given in binary encoding. This problem formulation is also known as the one-dimensional cutting stock problem. In this work, we provide an algorithm that, for constant d, solves bin packing in polynomial time. This was an open problem for all d ≥ 3. In fact, for constant d our algorithm solves the following problem in polynomial time: Given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q. Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times, and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

Consider equipping an alphabet $$\mathcal {A}$$ A with a group action which partitions the set of words into equivalence classes which we call patterns. We answer standard questions for Penney’s game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.

© 2018 Elsevier Inc. We study a notion of pre-quantization for b-symplectic manifolds. We use it to construct a formal geometric quantization of b-symplectic manifolds equipped with Hamiltonian torus actions with nonzero modular weight. We show that these quantizations are finite dimensional T-modules.

© 2016, Springer-Verlag Berlin Heidelberg. Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras Vk(g0) ⊂ Vk(g) , corresponding to an embedding of a maximal equal rank reductive subalgebra g0 into a simple Lie algebra g, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when Vk(g) decomposes finitely as a Vk(g0) -module.

© 2016 J. Weed, V. Perchet & P. Rigollet. Motivated by online advertising auctions, we consider repeated Vickrey auctions where goods of unknown value are sold sequentially and bidders only learn (potentially noisy) information about a good's value once it is purchased. We adopt an online learning approach with bandit feedback to model this problem and derive bidding strategies for two models: stochastic and adversarial. In the stochastic model, the observed values of the goods are random variables centered around the true value of the good. In this case, logarithmic regret is achievable when competing against well behaved adversaries. In the adversarial model, the goods need not be identical. Comparing our performance against that of the best fixed bid in hindsight, we show that sublinear regret is also achievable in this case. For both the stochastic and adversarial models, we prove matching minimax lower bounds showing our strategies to be optimal up to lower-order terms. To our knowledge, this is the first complete set of strategies for bidders participating in auctions of this type.

In this paper, we propose an alternative construction of a certain class of Deformed Double Current Algebras. We construct them as spherical subalgebras of symplectic reection algebras in the Deligne category. They can also be thought of as ultraproducts of the corresponding spherical subalgebras in finite rank. We also provide new presentations of DDCA of types A and B by generators and relations.

In this paper we show that the convolution product of "almost characters" of a connected reductive group over a finite field is given by "structure constants" whose leading coefficients can be interpreted in K-theoretic terms and in particular are natural numbers. I wish to thank M. Finkelberg and V. Ostrik for comments on an earlier version of this paper.

© 2018, Springer International Publishing AG, part of Springer Nature. We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for G= GL(n) and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group.

Abstract Persistent greenhouse gas (GHG) emissions threaten global climate goals and have prompted consideration of climate controls supplementary to emissions mitigation. We present MARGO, an idealized model of optimally-controlled climate change, which is complementary to both simpler conceptual models and more complicated Integrated Assessment Models. The four methods of controlling climate damage—mitigation, carbon dioxide removal (CDR), adaptation, and solar radiation modification (SRM)—are not interchangeable, as they enter at different stages of the causal chain that connects GHG emissions to climate damages. Early and aggressive mitigation is necessary to stabilize GHG concentrations below a tolerable level. While the most cost-beneficial and cost-effective pathways to reducing climate suffering include deployments of all four controls, the quantitative trade-offs between the different controls are sensitive to value-driven parameters and poorly-known future costs and damages. Static policy optimization assumes perfect foresight and obscures the active role decision-makers have in shaping a climate trajectory. We propose an explicit policy response process wherein climate control policies are re-adjusted over time in response to unanticipated outcomes. We illustrate this process in two ‘storyline’ scenarios: (a) near-term increases in mitigation and CDR are deficient, such that climate goals are expected to slip out of reach; (b) SRM is abruptly terminated after 40 years of successful deployment, causing an extremely rapid warming which is amplified by an excess of GHGs due to deterred mitigation. In both cases, an optimized policy response yields substantial benefits relative to continuing the original policy. The MARGO model is intentionally designed to be as simple, transparent, customizable, and accessible as possible, addressing concerns about previous climate-economic modelling approach

This paper contains an exposition of the classical work of Schoenberg and Gantmacher, Krein on the theory of totally positive matrices. It also contains a generalization of this theory to the Lie theoretic context and an application to the recent results of Fock and Goncharov on Teichmüller theory.

We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.

We introduce end-to-end inverse design for multi-channel imaging, in which a nanophotonic frontend is optimized in conjunction with an image-processing backend to extract depth, spectral and polarization channels from a single monochrome image. Unlike diffractive optics, we show that subwavelength-scale “metasurface” designs can easily distinguish similar wavelength and polarization inputs. The proposed technique integrates a single-layer metasurface frontend with an efficient Tikhonov reconstruction backend, without any additional optics except a grayscale sensor. Our method yields multi-channel imaging by spontaneous demultiplexing: the metaoptics front-end separates different channels into distinct spatial domains whose locations on the sensor are optimally discovered by the inverse-design algorithm. We present large-area metasurface designs, compatible with standard lithography, for multi-spectral imaging, depth-spectral imaging, and “all-in-one” spectro-polarimetric-depth imaging with robust reconstruction performance (≲ 10% error with 1% detector noise). In contrast to neural networks, our framework is physically interpretable and does not require large training sets. It can be used to reconstruct arbitrary three-dimensional scenes with full multi-wavelength spectra and polarization textures.

A complete understanding of biological processes requires synthesizing information across heterogeneous modalities, such as age, disease status, or gene expression. Technological advances in single-cell profiling have enabled researchers to assay multiple modalities simultaneously. We present Schema, which uses a principled metric learning strategy that identifies informative features in a modality to synthesize disparate modalities into a single coherent interpretation. We use Schema to infer cell types by integrating gene expression and chromatin accessibility data; demonstrate informative data visualizations that synthesize multiple modalities; perform differential gene expression analysis in the context of spatial variability; and estimate evolutionary pressure on peptide sequences.

© 2020 American Mathematical Society. Let G be a reductive group over C. Assume that the Lie algebra g of G has a given grading (gj ) indexed by a cyclic group Z/m such that g0 contains a Cartan subalgebra of g. The subgroup G0 of G corresponding to g0 acts on the variety of nilpotent elements in g1 with finitely many orbits. We are interested in computing the local intersection cohomology of closures of these orbits with coefficients in irreducible G0-equivariant local systems in the case of the principal block. We show that these can be computed by a purely combinatorial algorithm.

© 2018 SPIE. We propose a method to use artificial neural networks to approximate light scattering by multilayer nanoparticles. We find the network needs to be trained on only a small sampling of the data in order to approximate the simulation to high precision. Once the neural network is trained, it can simulate such optical processes orders of magnitude faster than conventional simulations. Furthermore, the trained neural network can be used solve nanophotonic inverse design problems by using back-propogation - where the gradient is analytical, not numerical.

© 2020 Society for Industrial and Applied Mathematics. We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we show how this leads to an increase in compression for a fixed accuracy. We devise an alternating algorithm to represent a given tensor in the multiresolution format and prove local convergence guarantees. In two dimensions, we provide examples that show that this approach can beat the Eckart-Young theorem, and for dimensions higher than two, we achieve higher compression than the tensor-train format on six real-world datasets. We also provide results on the closedness and stability of the tensor format and discuss how to perform common linear algebra operations on the level of the compressed tensors.

For 75 years it has been known that radiative heat transfer can exceed far-field blackbody rates when two bodies are separated by less than a thermal wavelength. Yet an open question has remained: what is the maximum achievable radiative transfer rate? Here we describe basic energy-conservation principles that answer this question, yielding upper bounds that depend on the temperatures, material susceptibilities, and separation distance, but which encompass all geometries. The simple structures studied to date fall far short of the bounds, offering the possibility for significant future enhancement, with ramifications for experimental studies as well as thermophotovoltaic applications., United States. Army Research Office. Institute for Soldier Nanotechnologies (Contract No. W911NF-07-D0004), United States. Office of Naval Research. Multidisciplinary University Research Initiative for Complex and Robust On-chip Nanophotonics (grant no. FA9550-09-1-0704), National Science Foundation (U.S.) (Grant No. DMR-1454836)

© 2020 The Electrochemical Society ("ECS"). Published on behalf of ECS by IOP Publishing Limited. We introduce a framework for analyzing and designing EIS inversion algorithms. Our framework stems from the observation of four features common to well-defined EIS inversion algorithms, namely (1) the representation of unknown distributions, (2) the minimization of a metric of error to estimate parameters arising from the chosen representation, subject to constraints on (3) the complexity control parameters, and (4) a means for choosing optimal control parameter values. These features must be present to overcome the ill-posed nature of EIS inversion problems. We review three established EIS inversion algorithms to illustrate the pervasiveness of these features, and show the utility of the framework by resolving ambiguities concerning three more algorithms. Our framework is then used to design the generalized EIS inversion (gEISi) algorithm, which uses Gaussian basis function representation, modality control parameter, and cross-validation for choosing the optimal control parameter value. The gEISi algorithm is applicable to the generalized EIS inversion problem, which allows for a wider range of underlying models. We also considered the construction of credible intervals for distributions arising from the algorithm. The algorithm is able to accurately reproduce distributions which have been difficult to obtain using existing algorithms. It is provided gratis on the repository https://github.com/suryaeff/gEISi.git.

For any n,r∈N, we construct an algebra Prn via generators and quadratic relations, and show that it deforms the W–algebra of glnr with respect to a nilpotent with Jordan block decomposition r+…+r. We introduce a surjective morphism from half of the quantum toroidal algebra of gln to Prn, and show that the action of the quantum toroidal algebra on the K–theory groups of the moduli spaces of parabolic sheaves factors through Prn.

We study the degenerations of asymptotically conical Ricci-flat Kähler metrics as the Kähler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kähler metrics converge to a incomplete smooth Ricci-flat Kähler metric away from a compact subvariety. As a consequence, we construct singular Calabi–Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi–Yau manifolds.

© 2019 Elsevier Inc. We present a numerical study of electrokinetic transport in porous media, focusing on the role of heterogeneity in a porous microstructure on ion concentration polarization and over-limiting current. For simplicity, the porous medium is modeled as a network of long, thin charged cylindrical pores, each governed by one-dimensional effective transport equations. For weak surface conduction, when sufficiently large potential is applied, we demonstrate that electrokinetic transport in a porous network can be dominated by electro-convection via internally induced flow loops, which is not properly captured by existing homogenized models. We systematically vary the topology and “accessivity” of the pore network and compare with simulations of traditional homogenized parallel-pore (capillary-bundle) models, in order to reveal the effects of regular and hierarchical connectivity. Our computational framework sheds light on the complex physics of electrokinetic phenomena in microstructures and may be used to design porous media for applications, such as water desalination and purification by shock electrodialysis.

We explore an application of all-pay auctions to model territorial annexation. Specifically, in the model we consider the expected resource, production, and military power are public information, but actual resource levels are private knowledge. We consider the resource transfer at the end of such a competition which deprives the weaker country of some fraction of its original resources. We derive the quasi-equilibria strategies for two country conflicts under different scenarios.

We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups, generalized Taft algebras with non-nilpotent skew primitive generators, and an example of non-Cartan type.

© COPYRIGHT SPIE. Downloading of the abstract is permitted for personal use only. Various scatterers such as rough surfaces or nanostructures are typically used to enhance the low efficiency of Raman spectroscopy (surface-enhanced Raman scattering). In this work, we find fundamental upper bounds on the Raman enhancement for arbitrary-shaped scatterers, depending only on its material constants and the separation distance from the molecule. According to our metric, silver is optimal in visible wavelengths while aluminum is better in the near-UV region. Our general analytical bound scales as the volume of the scatterer and the inverse sixth power of the distance to the active molecule. For periodic scatterers, a second bound with surface-area scaling is presented. Simple geometries such as spheres and bowties are shown to fall short of the bounds. However, using topology optimization based inverse design, we obtain surprising structures maximizing the Raman enhancement. These optimization results shed light to what extent our bounds are achievable., National Science Foundation (Grants 1453218, 1709212), Air Force Offce of Scientific Research (Contract FA9550-17-1-0093), Army Research Office (Contract W911NF-13-D-0001)

© 2017 Elsevier Inc. We present a fast and accurate algorithm to solve Poisson problems in complex geometries, using regular Cartesian grids. We consider a variety of configurations, including Poisson problems with interfaces across which the solution is discontinuous (of the type arising in multi-fluid flows). The algorithm is based on a combination of the Correction Function Method (CFM) and Boundary Integral Methods (BIM). Interface and boundary conditions can be treated in a fast and accurate manner using boundary integral equations, and the associated BIM. Unfortunately, BIM can be costly when the solution is needed everywhere in a grid, e.g. fluid flow problems. We use the CFM to circumvent this issue. The solution from the BIM is used to rewrite the problem as a series of Poisson problems in rectangular domains—which requires the BIM solution at interfaces/boundaries only. These Poisson problems involve discontinuities at interfaces, of the type that the CFM can handle. Hence we use the CFM to solve them (to high order of accuracy) with finite differences and a Fast Fourier Transform based fast Poisson solver. We present 2-D examples of the algorithm applied to Poisson problems involving complex geometries, including cases in which the solution is discontinuous. We show that the algorithm produces solutions that converge with either 3rd or 4th order of accuracy, depending on the type of boundary condition and solution discontinuity.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement We show that topology optimization (TO) of metallic resonators can lead to ∼102 × improvement in surface-enhanced Raman scattering (SERS) efficiency compared to traditional resonant structures such as bowtie antennas. TO inverse design leads to surprising structures very different from conventional designs, which simultaneously optimize focusing of the incident wave and emission from the Raman dipole. We consider isolated metallic particles as well as more complicated configurations such as periodic surfaces or resonators coupled to dielectric waveguides, and the benefits of TO are even greater in the latter case. Our results are motivated by recent rigorous upper bounds to Raman scattering enhancement, and shed light on the extent to which these bounds are achievable.

Let D be a fixed connected component of a reductive algebraic group over an algebraically closed field. We use a variant of the Springer representation to define and study a partition of D into finitely many Strata, one of which is the open set of regular elements. We show that the Strata of D are indexed by a set defined purely in terms of the Weyl group and its automorphism defined by D.

We present a photonics topology optimization (TO) package capable of addressing a wide range of practical photonics design problems, incorporating robustness and manufacturing constraints, which can scale to large devices and massive parallelism. We employ a hybrid algorithm that builds on a mature time-domain (FDTD) package Meep to simultaneously solve multiple frequency-domain TO problems over a broad bandwidth. This time/frequency-domain approach is enhanced by new filter-design sources for the gradient calculation and new material-interpolation methods for optimizing dispersive media, as well as by multiple forms of computational parallelism. The package is available as free/open-source software with extensive tutorials and multi-platform support., Department of Defense (DoD), Simons Foundation, Georgia Electronic Design Center at the Georgia Institute of Technology

© The Author(s) 2017. Porous electrode theory, pioneered by John Newman and collaborators, provides a macroscopic description of battery cycling behavior, rooted in microscopic physical models. Typically, the active materials are described as solid solution particles with transport and surface reactions driven by concentration fields, and the thermodynamics are incorporated through fitting of the open circuit potential. However, this approach does not apply to phase separating materials, for which the voltage is an emergent property of inhomogeneous concentration profiles, even in equilibrium. Here, we present a general framework, "multiphase porous electrode theory", based on nonequilibrium thermodynamics and implemented in an open-source software package called "MPET". Cahn-Hilliard-type phase field models are used to describe the active materials with suitably generalized models of interfacial reaction kinetics. Classical concentrated solution theory is implemented for the electrolyte phase, and Newman's porous electrode theory is recovered in the limit of solid solution active materials with Butler-Volmer kinetics. More general, quantum-mechanical models of faradaic reactions are also included, such as Marcus-Hush-Chidsey kinetics for electron transfer at electrodes, extended for concentrated solutions. The full model and implementation are described, and a variety of example calculations are presented to illustrate the novel features of the software compared to existing battery models.

Simultaneous multilayer coating techniques are widely known, but their industrial application remains limited to narrow market sectors. One barrier to adoption may be the mismatch between industries that are familiar with such processes but have no need, and industries that have need but are not familiar. Moreover, there are application-specific technical challenges to developing multilayer-coating processes. In this article, we describe our resolution of two specific issues in simultaneous multilayer coating of all-aqueous highly loaded slurries for new and emerging energy applications. The first issue is particle jamming (associated with shear-thickening) of the highly loaded slurries in the die internals, which we alleviated by adding small amounts of viscosity modifiers without reducing the solid loading. The second issue is a Marangoni-driven surface instability that resembles top layer de-wetting, which we solved by carefully selecting surfactants to tune the dynamic surface tensions of each slurry. Both issues were resolved early in a step-wise development, saving significant development cost which in our case was driven by expensive materials.

We revisit the role of the local solvent structure on the activity coefficients of electrolytes within a nonlocal dielectric function approach. We treat the concentrated electrolyte as a dielectric medium and suggest an interpolation formula for its nonlocal dielectric response. The water dielectric response is approximated based on MD simulations and experimental data, that gives strong over-screening and oscillations in the potential, which are absent in the standard “primitive model” predictions. We obtain mathematically tractable closed-form expressions for the activity coefficients, in reasonable agreement with experimental data.