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Machine Learning and Scientific Computing
Get PDFThe remarkable success of machine learning methods for tacking problems in computer vision and natural language processing has made them auspicious tools for applications to scientific computing tasks. The present work advances both machine learning techniques by using ideas from numerical analysis, inverse problems, and data assimilation and introduces new machine learning based tools for accurate and computationally efficient scientific computing. Chapters 2 and 3 introduce new methods and analyze existing methods for the optimization of deep neural networks. Chapters 4 and 5 formulate approximation architectures acting between infinite dimensional functions spaces for applications to parametric PDE problems. Chapter 6 demonstrates how to re-formulate GAN(s) so they can condition on continuous data and exhibits applications to Bayesian inverse problems. In Chapter 7, we present a novel regression-clustering method and apply it to the problem of predicting molecular activation energies.</p
Quantum Statistical Mechanics, Noncommutative Geometry, and the Boundary of Modular Curves
Get PDFThe Bost-Connes system is a C*-dynamical system whose partition function, KMS states, and symmetries are related to the explicit class field theory of the field of rational numbers. In particular, its zero-temperature KMS states, when evaluated on certain points in an arithmetic sub-algebra, yield the generators of the maximal abelian extension of the rationals. The Bost-Connes system can be viewed in terms of a geometric picture of 1-dimensional Q-lattices. The GL₂ system is an extension of this idea to the setting of 2-dimensional Q-lattices. A specialization of the GL₂-system introduced in by Connes, Marcolli, and Ramachandran, is related in a similar way to the explicit class field theory of imaginary quadratic extensions.
Inspired by the philosophy of Manin's real multiplication program, we define a boundary version of the GL₂2-system. In this viewpoint we see the projective line under a certain PGL(2,Z) action (which is related to the shift of the continued fraction expansion) as a moduli space characterizing degenerate elliptic curves. These degenerate elliptic curves can be realized as noncommutative 2-tori. This moduli space of the non-commutative tori is interpreted as an invisible boundary of the moduli space of elliptic curves. In fact, we define a family of such boundary GL₂ systems indexed by a choice of continued fraction algorithm. We analyze their partition functions, KMS states, and ground states. We also define an arithmetic algebra of unbounded multipliers in analogy with the GL₂ case. We show that the ground states when evaluated on points in the arithmetic algebra give pairings of the limiting modular symbols introduced by Manin and Marcolli with weight-2 cusp forms.
We also begin the project of extending this picture to the higher weight setting by defining a higher-weight limiting modular symbol. We use as a starting point the Shokurov modular symbols, which are constructed using Kuga modular varieties, which are non-singular projective varieties over the modular curves. We subject these modular symbols to a limiting procedure. We then show, using the coding space setting of Kessenbohmer and Stratmann, that these limiting modular symbols can be written as a Birkhoff ergodic average everywhere.</p
Thermal Behavior of Cuprous Oxide: a Comprehensive Study of Three-Body Phonon Effects and Beyond
Get PDFPhonons, or quantized normal modes of crystal vibrations, are responsible for much of the thermophysical behavior in solid-state systems. This behavior includes properties like thermal expansion, defined as the change in material volume in response to temperature. Typically, materials expand upon heating and contract upon cooling; however, some undergo anomalous or negative thermal expansion (NTE). This study focuses on a material with NTE, cuprous oxide (Cu2O), commonly known as cuprite. Using computational and experimental methods, we identify the underlying mechanisms of the NTE and how these mechanisms relate to temperature-dependent phonon behavior with temperature, using both computational and experimental methods.
Computationally, we interpret temperature-dependent changes in phonon energies with perturbation theory. Assuming that the bonds between atoms behave like simple harmonic oscillators, we model the observed random motion of the atoms around their equilibrium positions with quasi-harmonic (QH) and anharmonic (AH) approximations. Furthermore, the perturbations in the atom position allow us to model phonon energy changes in response to temperatures.
While these models, particularly AH models, have proven accurate in predicting the phonon behavior, experimental methods, like inelastic neutron scattering (INS), remain the gold standard for validation. This study presents INS data from single-crystal cuprite measured on the Wide-Angular Range Chopper Spectrometer (ARCS) at the Oak Ridge National Laboratory (ORNL) Spallation Neutron Source (SNS). We present INS data collected at 10 K, 300 K, 700 K, and 900 K. The post-processing workflow included: (1) binning with the software package Mantid, (2) reducing with a multiphonon background correction for polyatomic crystals, and (3) condensing into a single irreducible wedge in the first Brillouin zone (BZ). From this, we obtain a four-dimensional scattering function S(Q, E). Our AH calculations use the stochastic-Temperature Dependent Effective Potential (sTDEP) and the Machine Learning Interatomic Potential (MLIP) methods. The former method uses perturbation theory to include cubic and quartic AH contributions. The latter uses machine learning (ML), which in principle, includes all orders of AH terms.
This investigation of the NTE of cuprite demonstrates that QH and AH models successfully predict anomalous NTE behavior. However, only AH calculations show the temperature-dependent phonon behavior seen in INS results. This discrepancy likely stems from a fortuitous cancellation of cubic and quartic AH terms giving an apparent success of QH models for the NTE. Ultimately, a correct prediction of thermal expansion with incorrect phonons reinforces the need to look at the role of higher-order terms in the temperature-dependent behavior of this material.
Despite the success of sTDEP at predicting phonon frequency shifts, it could not account for the newly observed diffuse inelastic intensity (DII) in the INS phonon spectra. For this, MLIP was more effective.
This work provides complementary models to explain the origins of the DII, which is likely an emerging category of AH feature best described as a local nonlinear many-body process. We investigate phonon dissipation, the dynamics of systems coupled to their environments, Brownian motion, and discontinuities due to impulse transfer effects. We conclude by addressing the potential applications of the results and their role in future work on thermal lattice dynamics.</p
Thermally Responsive Polymers for Wearable Calorimeters
Get PDFThe measurement of the body core temperature (BCT) can provide insightful health information spanning from hypothermia and heat stroke to inflammations and infections. In addition, the continuous monitoring of the BCT can unlock new possibilities for people’s well-being such counting of burnt calories, prediction of the ovulation period in the female population, and for the assessment of mental health issues. However, the integration of a BCT sensor in wearable devices is extremely challenging, since standard methods cannot combine minimal invasiveness with high measurement accuracy. Dual heat flux (DHF) thermometry is a novel technique that allows the precise estimation of BCT from the measurement of skin temperature. Nevertheless, the limited precision of currently available temperature sensors has not favored the wide spread of devices based on this architecture. In this thesis, we present the fabrication of a fully wearable DHF thermometer realized by integrating new polymers with a remarkable temperature sensitivity. In these particular polymers, an increase in temperature results in a change of the ionic conductivity. In the first part of this work, we focus on the understanding of the ion transport mechanism in these polymers and, in particular, on the nature of the interaction between the functional groups present on the polymer backbone and the conducting species (i.e. metal cations and water molecules). We show that the ion’s coordinating environment is the key to make these materials highly sensitive to temperature. The second part of the thesis tackles the fabrication of a BCT sensor, integrating these temperature responsive polymers in an ultrathin DHF thermometer. Building on the understanding of the nature of the temperature response, we optimize the polymer’s composition to obtain a thermal sensitivity that allows a good precision when measuring the BCT. Finally, we characterize the performance of the fabricated DHF thermometer in different conditions, assessing the sensor’s accuracy and response time
Structure of Entanglement in Fracton Phases of Matter
Get PDFThis thesis discusses recent contributions to the theory of gapped fracton phases of matter, utilizing exactly solvable Hamiltonian models as the primary tool of study. A large component of the work revolves around the notion of a foliation structure, which is a defining feature of the long-range entanglement in certain gapped fracton states. We introduce this concept, identify its presence in a handful of prominent fracton models, and explore its consequences in terms of entanglement entropy and fractional excitations. A second major theme of the thesis is the characterization of gapped fracton states via emergent gauge theories based on discrete subsystem symmetries. We introduce a variety of novel fractonic gauge theories including twisted and fermionic variants, identify their emergence in a bevy of well-known models, and classify them with the use of novel topological invariants. We also establish a link between subsystem symmetry and entanglement renormalization group flow in fractal spin liquids.</p
Biocontrol of Biomolecular Systems: Polyhedral Constraints on Binding's Regulation of Catalysis from Biocircuits to Metabolism
Get PDFOne eventual goal of bioengineering is to build complex biological machines that fully realize the unique potential of biotechnology, namely adaptation, survival, growth, and dominance. In order to do so, not only do we need theoretical understanding and reliable manufacturing of biological parts and components, we also need a systems theory that captures fundamental structures to obtain insight about the space of all possible behaviors when parts are put together. This enables us to understand what can and cannot be achieved. Examples from other engineering disciplines are Turing machines for computers, information channels for communication networks, linear input output systems for electrical circuits, and thermodynamics for heat engines. This work is an attempt at developing a systems theory tailored to biomolecular systems in cells. The results form the following statements.
Biomolecular systems are binding and catalysis reactions. Catalysis determines the direction of change, while binding regulates how the catalysis rates vary with reactant concentrations. Given a binding reaction network, the full range of regulatory profiles can be captured by the reaction orders of catalysis, which in turn is constrained in polyhedral sets determined by the stoichiometry of binding. This constitute a rule, that since cells control catalysis by binding, cells control catalysis rates by regulating reaction orders constrained in polyhedral sets. This rule has ramifications in several directions. On metabolism, by incorporating the constraint that reaction orders of metabolic fluxes, not the fluxes themselves, are controlled, we can predict metabolism dynamics directly from network stoichiometry, e.g. glycolytic oscillations and growth arrests. This is a fully dynamic upgrade of flux balance analysis, a popular constraint-based method to model metabolism. On systems biology, this rule derives a method of biocircuit analysis based on the full range of values that reaction orders can take. This allows discovery of necessary and sufficient conditions for a circuit to achieve a certain function, thus revealing regimes hidden by traditional methods of analysis. It also promotes holistic comparisons of different circuit implementations, e.g. activating versus repressing, thur enabling biocircuit design where we know when a design will work, and when a design will fail. On dynamics and control of biocircuits, reaction order can work as a robust basis for stability, perfect adaptation, multistability, and oscillations. Lyapunov functions and dissipative control theory tailored for biomolecular systems are constructed based on reaction orders. On the mathematics of biology, it relates bioregulation to convex polyhedra, log derivative operator decompositions, and fundamental rules of calculus for positive variables.</p
Harmonic Maps of Riemann Surfaces and Applications in Geometry
Get PDFHarmonic maps are fundamental objects in differential geometry. They play an important role in studying deformations of geometric structures and in various rigidity problems. In this thesis, we present three projects, all of which involve harmonic mappings of Riemann surfaces.
In the first project, we study infinite energy harmonic maps and spacelike maximal surfaces in pseudo-Riemannian manifolds, and give applications to domination for surface group representations and anti-de Sitter geometry. The culminating result is the existence of a new class of anti-de Sitter 3-manifolds and a parametrization of their deformation space.
The second project concerns moduli spaces of harmonic surfaces inside higher dimensional Riemannian manifolds. First, we prove a factorization theorem for harmonic maps. We then use infinite-dimensional transversality theory to prove results about the distribution of certain families of harmonic surfaces inside our moduli spaces.
The final project is motivated by the Labourie conjecture from Higher Teichmueller theory. We prove non-uniqueness results for minimal surfaces in products of hyperbolic surfaces and products of ℝ-trees, and we make a connection to classical minimal surfaces.</p
Fundamental Ways to Probe Gravitational Waves Across Its Spectrum and Propagation
Get PDFIn 2015, the detection of gravitational waves (GWs) from merging black holes by the LIGO Scientific Collaboration and the VIRGO Collaboration opened a new era of observational astronomy. This thesis covers a range of topics on how to test the general theory of relativity using current and future GW detectors --- both ground- and space-based. Starting from general principles, in Chapter 2, we survey how well the so-called parameterized post-Einstein parameters for binary black hole GWs can be constrained by multi-band GW detection, which employs both ground-based detectors (including Einstein Telescope and Cosmic Explorer) and space-based detectors (including the Laser Interferometer Space Antenna and deci-Hertz detectors).
In Chapter 3, we address the limitations of the Fisher Information Matrix approach in testing relativity. Chapter 4 proposes a novel experimental strategy for multi-band GW observation. More specifically, the detection of a stellar-mass binary from the Laser Interferometer Space Antenna can provide forewarning for ground-based observations, e.g., by third-generation detectors. Adjusting optical configurations of ground-based detectors targeting this particular binary can significantly improving our accuracy in testing the "no-hair theorem" of black holes. In Chapter 5, we establish a systematic framework that describes how the propagation of GWs can differ from predictions of general relativity, incorporating both dispersion and birefringence. In Chapter 6, we focus the specific example of massive gravitons and show how the so-called Vainshtein screening of the graviton's mass, by the host galaxy of the source, the Milky way galaxy -- and galaxies in between -- can be extracted from an ensemble of signals.</p
Combinatorial and Algebraic Propeties of Nonnegative Matrices
Get PDFWe study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories.
1. We show the first quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental theorem which has been used within diverse areas of mathematics. The Perron-Frobenius theorem shows that any irreducible nonnegative matrix R will have a largest positive eigenvalue r, and every other eigenvalue λ is such that Reλ < R and |λ| ≤ r. We capture the notion of irreducibility through the widely studied notion of edge expansion φ of R which intuitively measures how well-connected the underlying digraph of R is, and show a quantitative relation between the spectral gap Δ = 1-Reλ/r (where λ ≠ r has the largest real part) of R to the edge expansion φ as follows.
(1/15) • [(Δ(R))/n] ≤ φ(R) ≤ √[2 • Δ(R)].
This also provides a more general result than the Cheeger-Buser inequalities since it applies to any nonnegative matrix.
2. We study constructions of specific nonsymmetric matrices (or nonreversible Markov Chains) that have small edge expansion but large spectral gap, taking us in a direction more novel and unexplored than studying symmetric matrices with constant edge expansion that have been extensively studied. We first analyze some known but less studied Markov Chains, and then provide a novel construction of a nonreversible chain for which
φ(R) ≤ [(Δ(R))/√n],
obtaining a bound exponentially better than known bounds. We also present a candidate construction of matrices for which
φ(R) ≤ 2[(Δ(R))/n]
which is the most beautiful contribution of this thesis. We believe these matrices have properties remarkable enough to deserve study in their own right.
3. We connect the edge expansion and spectral gap to other combinatorial properties of nonsymmetric matrices. The most well-studied property is mixing time, and we provide elementary proofs of the relation between mixing time and the edge expansion, and also other bounds relating the mixing time of a nonreversible chain to the spectral gap and to its additive symmetrization. Further, we provide a unified view of the notion of capacity and normalized capacity, and show the monotonicity of capacity of nonreversible chains amongst other results for nonsymmetric matrices. We finally discuss and prove interesting lemmas about different notions of expansion and show the first results for tensor walks or nonnegative tensors.</p
Female Inventors and Narratives of Innovation in Late Twentieth-Century Computing
Get PDFI examine the history of women’s labor and representation in computer science by studying two distinct categories: women involved in authorial, creative work versus manual, computational labor. Building off the work of historians of technology, I question why we tell the histories we do about the “forgotten women.” The gaps in the histories of computer science innovation are mirrored by shortcomings in the actual practice of computer science: Both the historiography of computer science and the field itself have been shaped by the myth of the lone genius. I trace the shortcomings of this myth throughout the history of modern computer science, finding that narratives of female innovators and movements to incorporate more women into computing only perpetuated connections between individual genius, masculinity, and scientific progress. I explore community-based perspectives from feminist epistemology as possibilities for shifting away from the myth of the lone genius