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3d-3d Correspondence for Seifert Manifolds
Get PDFIn this dissertation, we investigate the 3d-3d correspondence for Seifert manifolds. This correspondence, originating from string theory and M-theory, relates the dynamics of three-dimensional quantum field theories with the geometry of three-manifolds.
We first start in Chapter II with the simplest cases and demonstrate the extremely rich interplay between geometry and physics even when the manifold is just a direct product. In this particular case, by examining the problem from various vantage points, we generalize the celebrated relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) quantization of Chern-Simons theory and 4) the index theory of the moduli space of flat connections to a completely new set of relations between 1) the "equivariant Verlinde algebra" for a complex group, 2) the equivariant quantum K-theory of the vortex moduli space, 3) quantization of complex Chern-Simons theory and 4) the equivariant index theory of the moduli space of Higgs bundles.
In Chapter III we move one step up in complexity by looking at the next simplest three-manifolds---lens spaces. We test the 3d-3d correspondence for theories that are labeled by lens spaces, reaching a full agreement between the index of the 3d N=2 "lens space theory" and the partition function of complex Chern-Simons theory on the lens space.
The two different types of manifolds studied in the previous two chapters also have interesting interactions. We show in Chapter IV the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory on a lens space, the other is the "equivariant Verlinde formula". We check this relation explicitly for SU(2) and demonstrate that the SU(N) equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg dualities.
In the last chapter, we directly jump to the most general situation, giving a proposal for the 3d-3d correspondence for an arbitrary Seifert manifold. We remark on the huge class of novel dualities relating different descriptions of the "Seifert theory" associated with the same Seifert manifold and suggest ways that our proposal could be tested.</p
Exploring Thermal Phonon Transport from Atomic to Macroscopic Scales for Energy Conversion and Management
Get PDFHeat is one of the most fundamental forms of energy, and the ability to control heat plays a critical role in most current and future energy applications. Recently, interface engineering between heterogeneous solids has provided new approaches to manipulate heat transport at the scales of the energy carriers in solids, i.e. phonons which are quantized lattice vibrations. For example, nanocrystalline materials, which are polycrystalline materials with nanoscale grain sizes, are promising thermoelectric (TE) materials that have achieved substantially improved figure of merits compared to their bulk counterparts. This enhancement is typically attributed to a reduction in lattice thermal conductivity by phonon scattering at grain boundaries. On the other hand, inefficient heat dissipation across interfaces has been a long-standing problem that shortens the lifetime of electronics such as light-emitting diodes.
Despite the importance of interfaces, we still lack a comprehensive understanding of interfacial thermal phonon transport. For instance, the Fresnel coefficients enable the straightforward mathematical description of light as it moves between media of differing dielectric constants. Similarly, interfacial phonon transport can also be characterized by transmission coefficients that vary over the broad phonon spectrum in an analogous manner to Fresnel coefficients for light. However, despite decades of work, the spectral profile of these coefficients and how the profile is influenced by the atomic structure of actual interfaces remains unclear. As a result, the basic phenomenon of interfacial heat transport remains among the most poorly understood transport processes.
To elucidate this process, in this thesis we investigate interfacial thermal phonon transport using both modeling and experiment. The first portion of the thesis examines the impact of frequency-dependent grain boundary scattering in nanocrystalline silicon and silicon-germanium alloys using a novel computational method. We find that the grain boundary may not be as effective as commonly considered in scattering certain phonons, with a substantial amount of heat being carried by low frequency phonons with mean free paths longer than the grain size. Our result will help guide the design of more efficient TEs.
The second part of the thesis focuses on studying heat conduction using the Boltzmann transport equation (BTE), which is the governing equation of energy transport at length scales comparable to phonon mean free paths. The BTE is an integro-differential equation of time, real space, and phase space. Due to its high dimensionality, it is extremely challenging to solve. Here, we develop analytical methods to solve the frequency-dependent BTE, which allow us to obtain simple, closed-form solutions to complex multidimensional problems that have previously been possible to solve only with computationally expensive numerical simulations. We demonstrate that the solution leads to a more accurate measurement of phonon MFP spectra in thermal transient grating experiments.
Finally, we report the first measurements of thermal phonon transmission coefficients at a metal-semiconductor interface using ab-initio phonon transport modeling based on the BTE we develop in the second part and a thermal characterization technique, time-domain thermoreflectance. With our approach, we are able to directly link the atomic structure of an interface to the spectral content of the heat crossing it for the first time. Our work realizes the long-standing goal of directly measuring thermal phonon transmission coefficients and demonstrates a general route to study microscopic processes governing interfacial heat conduction. </p
Development of Ni-Catalyzed Asymmetric Reductive Cross-Coupling Reactions
Get PDFOver the last half century, the development of metal-catalyzed cross-coupling reactions has transformed the toolkit of transformations available to synthetic chemists. From the very beginning of this effort, researchers have studied the application of these reactions to afford enantioenriched products via asymmetric catalysis. A great deal of success has been achieved in this arena, giving rise to an ever-growing number of chiral catalysts for a wide range of transformations. Despite these efforts, inherent difficulties in the reactivity of C(sp3) electrophiles with the most common noble metal catalysts have limited the development of these substrates until more recently. A resurgence of interest in Ni-catalysis has enabled the stereoconvergent cross-coupling of C(sp3) electrophiles with many partners, opening doors to access these challenging chiral products.
Reductive cross-coupling, involving the union of two different electrophiles, has emerged still more recently, and had previously not been employed asymmetrically. Herein we describe our efforts to develop the first Ni-catalyzed asymmetric reductive cross-couplings of C(sp3) halides to afford highly enantioenriched products. In the first such reaction, the coupling of acyl chlorides with benzylic chlorides affords acyclic α-tertiary ketone products. Following this, we describe the coupling of new C(sp3) partners, α-chloronitriles, with challenging Lewis-basic heteroaryl iodides, enabled by the development of a novel PHOX ligand scaffold. Finally, we report the extension of a more general dioxane/TMSCl solvent condition to new asymmetric reductive couplings, including that of heteroaryl iodides with benzylic chlorides, as well as additional preliminary results with new substrate classes.</p
Angle-Resolved Mid-Infrared Spectroscopy of Gyroid Photonic Crystals
No full textGyroid photonic crystals, which can form photonic topological insulators, offer a rich
diversity of physics for fundamental study, in addition to yielding useful applications.
Understanding their band structure is crucial towards advancing both fronts. An
angle-resolved spectroscopy apparatus is constructed to experimentally measure
band structures. Operating at wavelengths from 7.7 to 11.1 μm, it is employed to
study single and double-gyroid photonic crystals, constructed of amorphous silicon
in 5 μm unit cells. Transmittance and reflectance act as indicators of density of
states, and are collected as spectra over angles of incidence ranging from 17 to 30
degrees. Comparison to theory confirms interesting features, such as bandgaps.
Further simulation guides the design of a crystal amenable to the first experimental
measurement of Weyl points at mid-infrared wavelengths
A Model for Energy and Morphology of Crystalline Grain Boundaries with Arbitrary Geometric Character
Get PDFIt has been well-established that interfaces in crystalline materials are key players in the mechanics of a variety of mesoscopic processes such as solidification, recrystallization, grain boundary migration, and severe plastic deformation. In particular, interfaces with complex morphologies have been observed to play a crucial role in many micromechanical phenomena such as grain boundary migration, stability, and twinning. Interfaces are a unique type of material defect in that they demonstrate a breadth of behavior and characteristics eluding simplified descriptions. Indeed, modeling the complex and diverse behavior of interfaces is still an active area of research, and to the author's knowledge there are as yet no predictive models for the energy and morphology of interfaces with arbitrary character. The aim of this thesis is to develop a novel model for interface energy and morphology that i) provides accurate results (especially regarding "energy cusp" locations) for interfaces with arbitrary character, ii) depends on a small set of material parameters, and iii) is fast enough to incorporate into large scale simulations.
In the first half of the work, a model for planar, immiscible grain boundary is formulated. By building on the assumption that anisotropic grain boundary energetics are dominated by geometry and crystallography, a construction on lattice density functions (referred to as "covariance") is introduced that provides a geometric measure of the order of an interface. Covariance forms the basis for a fully general model of the energy of a planar interface, and it is demonstrated by comparison with a wide selection of molecular dynamics energy data for FCC and BCC tilt and twist boundaries that the model accurately reproduces the energy landscape using only three material parameters. It is observed that the planar constraint on the model is, in some cases, over-restrictive; this motivates an extension of the model.
In the second half of the work, the theory of faceting in interfaces is developed and applied to the planar interface model for grain boundaries. Building on previous work in mathematics and materials science, an algorithm is formulated that returns the minimal possible energy attainable by relaxation and the corresponding relaxed morphology for a given planar energy model. It is shown that the relaxation significantly improves the energy results of the planar covariance model for FCC and BCC tilt and twist boundaries. The ability of the model to accurately predict faceting patterns is demonstrated by comparison to molecular dynamics energy data and experimental morphological observation for asymmetric tilt grain boundaries. It is also demonstrated that by varying the temperature in the planar covariance model, it is possible to reproduce a priori the experimentally observed effects of temperature on facet formation.
Finally, the range and scope of the covariance and relaxation models, having been demonstrated by means of extensive MD and experimental comparison, future applications and implementations of the model are explored.</p
Fabrication, Characterization, And Deformation of 3D Structural Meta-Materials
Get PDFCurrent technological advances in fabrication methods have provided pathways to creating architected structural meta-materials similar to those found in natural organisms that are structurally robust and lightweight, such as diatoms. Structural meta-materials are materials with mechanical properties that are determined by material properties at various length scales, which range from the material microstructure (nm) to the macro-scale architecture (μm – mm). It is now possible to exploit material size effect, which emerge at the nanometer length scale, as well as structural effects to tune the material properties and failure mechanisms of small-scale cellular solids, such as nanolattices.
This work demonstrates the fabrication and mechanical properties of 3-dimensional hollow nanolattices in both tension and compression. Hollow gold nanolattices loaded in uniaxial compression demonstrate that strength and stiffness vary as a function of geometry and tube wall thickness. Structural effects were explored by increasing the unit cell angle from 30° to 60° while keeping all other parameters constant; material size effects were probed by varying the tube wall thickness, t, from 200nm to 635nm, at a constant relative density and grain size. In-situ uniaxial compression experiments reveal an order-of-magnitude increase in yield stress and modulus in nanolattices with greater lattice angles, and a 150% increase in the yield strength without a concomitant change in modulus in thicker-walled nanolattices for fixed lattice angles. These results imply that independent control of structural and material size effects enables tunability of mechanical properties of 3-dimensional architected meta-materials and highlight the importance of material, geometric, and microstructural effects in small-scale mechanics.
This work also explores the flaw tolerance of 3D hollow-tube alumina kagome nanolattices with and without pre-fabricated notches, both in experiment and simulation. Experiments demonstrate that the hollow kagome nanolattices in uniaxial tension always fail at the same load when the ratio of notch length (a) to sample width (w) is no greater than 1/3, with no correlation between failure occurring at or away from the notch. For notches with (a/w) > 1/3, the samples fail at lower peak loads and this is attributed to the increased compliance as fewer unit cells span the un-notched region. Finite element simulations of the kagome tension samples show that the failure is governed by tensile loading for (a/w) < 1/3 but as (a/w) increases, bending begins to play a significant role in the failure. This work explores the flaw sensitivity of hollow alumina kagome nanolattices in tension, using experiments and simulations, and demonstrates that the discrete-continuum duality of architected structural meta-materials gives rise to their flaw insensitivity even when made entirely of intrinsically brittle materials
Fast Numerical Methods for Mixed, Singular Helmholtz Boundary Value Problems and Laplace Eigenvalue Problems - with Applications to Antenna Design, Sloshing, Electromagnetic Scattering and Spectral Geometry
Get PDFThis thesis presents a novel class of algorithms for the solution of scattering and eigenvalue problems on general two-dimensional domains under a variety of boundary conditions, including non-smooth domains and certain "Zaremba" boundary conditions - for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the methods for the Zaremba problems on smooth domains concern detailed information, which is put forth for the first time in this thesis, about the singularity structure of solutions of the Laplace operator under boundary conditions of Zaremba type. The new methods, which are based on use of Green functions and integral equations, incorporate a number of algorithmic innovations, including a fast and robust eigenvalue-search algorithm, use of the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities, and newly derived quadrature rules which give rise to high-order convergence even around singular points for the Zaremba problem. The resulting algorithms enjoy high-order convergence, and they can tackle a variety of elliptic problems under general boundary conditions, including, for example, eigenvalue problems, scattering problems, and, in particular, eigenfunction expansion for time-domain problems in non-separable physical domains with mixed boundary conditions
The Physiology and Computation of Pyramidal Neurons
Get PDFA variety of neural signals have been measured as correlates to consciousness. In particular, late current sinks in layer 1, distributed activity across the cortex, and feedback processing have all been implicated. What are the physiological underpinnings of these signals? What computational role do they play in the brain? Why do they correlate to consciousness? This thesis begins to answer these questions by focusing on the pyramidal neuron. As the primary communicator of long-range feedforward and feedback signals in the cortex, the pyramidal neuron is set up to play an important role in establishing distributed representations. Additionally, the dendritic extent, reaching layer 1, is well situated to receive feedback inputs and contribute to current sinks in the upper layers. An investigation of pyramidal neuron physiology is therefore necessary to understand how the brain creates, and potentially uses, the neural correlates of consciousness. An important part of this thesis will be in establishing the computational role that dendritic physiology plays. In order to do this, a combined experimental and modeling approach is used.
This thesis beings with single-cell experiments in layer 5 and layer 2/3 pyramidal neurons. In both cases, dendritic nonlinearities are characterized and found to be integral regulators of neural output. Particular attention is paid to calcium spikes and NMDA spikes, which both exist in the apical dendrites, considerable distances from the spike initiation zone. These experiments are then used to create detailed multicompartmental models. These models are used to test hypothesis regarding spatial distribution of membrane channels, to quantify the effects of certain experimental manipulations, and to establish the computational properties of the single cell. We find that the pyramidal neuron physiology can carry out a coincidence detection mechanism. Further abstraction of these models reveals potential mechanisms for spike time control, frequency modulation, and tuning. Finally, a set of experiments are carried out to establish the effect of long-range feedback inputs onto the pyramidal neuron. A final discussion then explores a potential way in which the physiology of pyramidal neurons can establish distributed representations, and contribute to consciousness.</p
Non-Linear Scale Interactions in a Forced Turbulent Boundary Layer
Get PDFThis thesis explores the dynamics of scale interactions in a turbulent boundary layer through a forcing-response type experimental study. An emphasis is placed on the analysis of triadic wavenumber interactions since the governing Navier-Stokes equations for the flow necessitate a direct coupling between triadically consist scales. Two sets of experiments were performed in which deterministic disturbances were introduced into the flow using a spatially-impulsive dynamic wall perturbation. Hotwire anemometry was employed to measure the downstream turbulent velocity and study the flow response to the external forcing. In the first set of experiments, which were based on a recent investigation of dynamic forcing effects in a turbulent boundary layer, a 2D (spanwise constant) spatio-temporal normal mode was excited in the flow; the streamwise length and time scales of the synthetic mode roughly correspond to the very-large-scale-motions (VLSM) found naturally in canonical flows. Correlation studies between the large- and small-scale velocity signals reveal an alteration of the natural phase relations between scales by the synthetic mode. In particular, a strong phase-locking or organizing effect is seen on directly coupled small-scales through triadic interactions. Having characterized the bulk influence of a single energetic mode on the flow dynamics, a second set of experiments aimed at isolating specific triadic interactions was performed. Two distinct 2D large-scale normal modes were excited in the flow, and the response at the corresponding sum and difference wavenumbers was isolated from the turbulent signals. Results from this experiment serve as an unique demonstration of direct non-linear interactions in a fully turbulent wall-bounded flow, and allow for examination of phase relationships involving specific interacting scales. A direct connection is also made to the Navier-Stokes resolvent operator framework developed in recent literature. Results and analysis from the present work offer insights into the dynamical structure of wall turbulence, and have interesting implications for design of practical turbulence manipulation or control strategies
Engineering, Predicting, and Understanding Nicotinamide Cofactor Specificity
Get PDFOxidoreductases, the enzymes that catalyze the transfer of electrons between molecules, represent the largest group of enzymes in metabolism, and the vast majority of these enzymes use the functionally-equivalent cofactors nicotinamide adenine dinucleotide (NAD) or nicotinamide adenine dinucleotide phosphate (NADP) for the storage and transport of the electrons. Understanding the interactions of these proteins with their cofactors is therefore crucial to the engineering of biological pathways and systems that involve these enzymes. In particular, because cells tightly regulate the levels of oxidized and reduced NAD and NADP, it is often valuable to engineer the specificity of enzymes to better integrate them into particular metabolic contexts.
The first section of this thesis focuses on this specificity, taking as a model system the ketol-acid reductoisomerase (KARI) enzyme family. Prior to the work described here, all known members of the KARI enzyme family displayed a strict specificity for NADP over NAD. However, the use of these enzymes in a constructed pathway for the production of medium-chain alcohols created a clear need for NAD-specific KARIs to improve yields. Chapter 1 briefly summarizes the prior state of the art in nicotinamide cofactor specificity engineering before describing how a previous switching of the KARI from E. coli was extended to create a simple recipe for the specificity reversal of any KARI enzyme. Chapter 2 then uses the insights into cofactor specificity in KARIs afforded by the engineering in Chapter 1 to search databases of KARI sequences and predict naturally NAD-specific KARIs, resulting in the discovery of extremophilic NAD-utilizing KARIs with properties that outstrip those of the best engineered enzymes. Chapter 3 extends this prediction approach into another enzyme family, xylose reductases, and discusses its strengths and limitations across diverse enzyme folds and families.
The next section of the thesis diverges from the topic of cofactor specificity engineering to briefly explore three structural questions which arise from the study of KARIs. Chapter 4 covers an analysis of several new KARI crystal structures, including those obtained during the efforts described in Chapters 1 and 2. From a comprehensive comparison of these structures, two topics are addressed: (1) the effect of insertions and deletions in the cofactor specificity loop on the binding geometry of NAD and NADP, and (2) the conformational motions involved in the binding of cofactor, substrate, and metal ions. Based on a pair of structures from Chapter 4, Chapter 5 experimentally explores the structural evolution of the KARI enzyme family’s two distinct structural classes, replicating in a class I KARI the structural duplication that produced the class II KARI fold, and demonstrating a remarkable retention of enzymatic activity. Chapter 6 explores a curious sensitivity to mutations observed around the adenine moiety of several KARIs, and extends this observation to a range of other NADP- and NAD-dependent enzymes, with implications both for engineering these proteins and for understanding protein evolution more generally.
The third and final section discusses the development of a general method for the cofactor specificity reversal of any NAD(P)-utilizing enzyme. Chapter 7 explains the approach used and how it represents a new paradigm in protein engineering, as well as covering the thorough experimental validation to which the method was subjected. This method was developed into a web applet (CSR-SALAD) for public use, and Chapter 8 discusses the development and use of this tool. This third section represents the culmination of the preceeding work, drawing from an understanding of the sequence and structural determinants of cofactor binding as explored in the preceeding six chapters to create the heuristic picture of specificity which informed the comprehensive engineering approach.</p