9,901 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

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    This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions

    Exact steady states of minimal models of nonequilibrium statistical mechanics

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    Systems out of equilibrium with their environment are ubiquitous in nature. Of particular relevance to biological applications are models in which each microscopic component spontaneously generates its own motion. Known collectively as active matter, such models are natural effective descriptions of many biological systems, from subcellular motors to flocks of birds. One would like to understand such phenomena using the tools of statistical mechanics, yet the inherent nonequilibrium setting means that the most powerful classical results of that field cannot be applied. This circumstance has fuelled interest in exactly solvable models of active matter. The aim in studying such models is twofold. Firstly, as exactly solvable model are often minimal, it makes them good candidates as generic coarse-grained descriptions of real-world processes. Secondly, even if the model in question does not correspond directly to some situation realizable in experiment, its exact solution may suggest some general principles, which could also apply to more complex phenomena. A typical tool to investigate the properties of a large system is to study the behaviour of a probe particle placed in such an environment. In this context, cases of interest are both an active particle in a passive environment or an active particle in an active environment. One model that has attracted much attention in this regard is the asymmetric simple exclusion process (ASEP), which is a prototypical minimal model of driven diffusive transport. In this thesis, I consider two variations of the ASEP on a ring geometry. The first is a system of symmetrically diffusing particles with one totally asymmetric (driven) defect particle. The second is a system of partially asymmetric particles, with one defect that may overtake the other particles. I analyze the steady states of these systems using two exact methods: the matrix product ansatz, and, for the second model the Bethe ansatz. This allows me to derive the exact density profiles and mean currents for these models, and, for the second model, the diffusion constant. Moreover, I use the Yang-Baxter formalism to study the general class of two-species partially asymmetric processes with overtaking. This allows me to determine conditions under which such models can be solved using the Bethe ansatz

    Techniques for high-multiplicity scattering amplitudes and applications to precision collider physics

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    In this thesis, we present state-of-the-art techniques for the computation of scattering amplitudes in Quantum Field Theories. Following an introduction to the topic, we describe a robust framework that enables the calculation of multi-scale two-loop amplitudes directly relevant to modern particle physics phenomenology at the Large Hadron Collider and beyond. We discuss in detail the use of finite fields to bypass the algebraic complexity of such computations, as well as the method of integration-by-parts relations and differential equations. We apply our framework to calculate the two-loop amplitudes contributing to three process: Higgs boson production in association with a bottom-quark pair, W boson production with a photon and a jet, as well as lepton-pair scattering with an off-shell and an on-shell photon. Finally, we draw our conclusions and discuss directions for future progress of amplitude computations

    UMSL Bulletin 2023-2024

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    The 2023-2024 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1088/thumbnail.jp

    New techniques for integrable spin chains and their application to gauge theories

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    In this thesis we study integrable systems known as spin chains and their applications to the study of the AdS/CFT duality, and in particular to N “ 4 supersymmetric Yang-Mills theory (SYM) in four dimensions.First, we introduce the necessary tools for the study of integrable periodic spin chains, which are based on algebraic and functional relations. From these tools, we derive in detail a technique that can be used to compute all the observables in these spin chains, known as Functional Separation of Variables. Then, we generalise our methods and results to a class of integrable spin chains with more general boundary conditions, known as open integrable spin chains.In the second part, we study a cusped Maldacena-Wilson line in N “ 4 SYM with insertions of scalar fields at the cusp, in a simplifying limit called the ladders limit. We derive a rigorous duality between this observable and an open integrable spin chain, the open Fishchain. We solve the Baxter TQ relation for the spin chain to obtain the exact spectrum of scaling dimensions of this observable involving cusped Maldacena-Wilson line.The open Fishchain and the application of Functional Separation of Variables to it form a very promising road for the study of the three-point functions of non-local operators in N “ 4 SYM via integrability

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    The infrared structure of perturbative gauge theories

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    Infrared divergences in the perturbative expansion of gauge theory amplitudes and cross sections have been a focus of theoretical investigations for almost a century. New insights still continue to emerge, as higher perturbative orders are explored, and high-precision phenomenological applications demand an ever more refined understanding. This review aims to provide a pedagogical overview of the subject. We briefly cover some of the early historical results, we provide some simple examples of low-order applications in the context of perturbative QCD, and discuss the necessary tools to extend these results to all perturbative orders. Finally, we describe recent developments concerning the calculation of soft anomalous dimensions in multi-particle scattering amplitudes at high orders, and we provide a brief introduction to the very active field of infrared subtraction for the calculation of differential distributions at colliders. © 2022 Elsevier B.V

    Backpropagation Beyond the Gradient

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    Automatic differentiation is a key enabler of deep learning: previously, practitioners were limited to models for which they could manually compute derivatives. Now, they can create sophisticated models with almost no restrictions and train them using first-order, i. e. gradient, information. Popular libraries like PyTorch and TensorFlow compute this gradient efficiently, automatically, and conveniently with a single line of code. Under the hood, reverse-mode automatic differentiation, or gradient backpropagation, powers the gradient computation in these libraries. Their entire design centers around gradient backpropagation. These frameworks are specialized around one specific task—computing the average gradient in a mini-batch. This specialization often complicates the extraction of other information like higher-order statistical moments of the gradient, or higher-order derivatives like the Hessian. It limits practitioners and researchers to methods that rely on the gradient. Arguably, this hampers the field from exploring the potential of higher-order information and there is evidence that focusing solely on the gradient has not lead to significant recent advances in deep learning optimization. To advance algorithmic research and inspire novel ideas, information beyond the batch-averaged gradient must be made available at the same level of computational efficiency, automation, and convenience. This thesis presents approaches to simplify experimentation with rich information beyond the gradient by making it more readily accessible. We present an implementation of these ideas as an extension to the backpropagation procedure in PyTorch. Using this newly accessible information, we demonstrate possible use cases by (i) showing how it can inform our understanding of neural network training by building a diagnostic tool, and (ii) enabling novel methods to efficiently compute and approximate curvature information. First, we extend gradient backpropagation for sequential feedforward models to Hessian backpropagation which enables computing approximate per-layer curvature. This perspective unifies recently proposed block- diagonal curvature approximations. Like gradient backpropagation, the computation of these second-order derivatives is modular, and therefore simple to automate and extend to new operations. Based on the insight that rich information beyond the gradient can be computed efficiently and at the same time, we extend the backpropagation in PyTorch with the BackPACK library. It provides efficient and convenient access to statistical moments of the gradient and approximate curvature information, often at a small overhead compared to computing just the gradient. Next, we showcase the utility of such information to better understand neural network training. We build the Cockpit library that visualizes what is happening inside the model during training through various instruments that rely on BackPACK’s statistics. We show how Cockpit provides a meaningful statistical summary report to the deep learning engineer to identify bugs in their machine learning pipeline, guide hyperparameter tuning, and study deep learning phenomena. Finally, we use BackPACK’s extended automatic differentiation functionality to develop ViViT, an approach to efficiently compute curvature information, in particular curvature noise. It uses the low-rank structure of the generalized Gauss-Newton approximation to the Hessian and addresses shortcomings in existing curvature approximations. Through monitoring curvature noise, we demonstrate how ViViT’s information helps in understanding challenges to make second-order optimization methods work in practice. This work develops new tools to experiment more easily with higher-order information in complex deep learning models. These tools have impacted works on Bayesian applications with Laplace approximations, out-of-distribution generalization, differential privacy, and the design of automatic differentia- tion systems. They constitute one important step towards developing and establishing more efficient deep learning algorithms

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    UMSL Bulletin 2022-2023

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    The 2022-2023 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1087/thumbnail.jp
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