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Supersymmetric Scattering Amplitudes and Algebraic Aspects of Holography from the Projective Line
Get PDFIn this thesis, we consider two topics in string theory and quantum field theory which are related by the common appearance of one-dimensional projective geometry. In the first half of the thesis, we study six-dimensional (6D) supersymmetric quantum field theories and supergravity at the leading (tree) approximation and compute the complete S-matrix for these theories as world-sheet integrals over the punctured Riemann sphere. This exploits the analytic structure of tree amplitudes which are rational and holomorphic in the kinematics and naturally related to the geometry of points on the complex projective line. The 6D n-particle S-matrix makes many symmetries and hidden properties manifest and generalizes the well-studied formulas for four-dimensional amplitudes in the form of twistor string theory and the rational curves formalism. While the systems we study are all field theories, they are in essence low-energy effective field theory limits of string theory and M-theory backgrounds. This includes theories such as those with 6D (2,0) supersymmetry which contain U(1) self-dual tensor fields which are difficult to treat from a Lagrangian point of view. Our formulas circumvent this difficulty and allow a generalization and unification of a large class of 6D scattering amplitudes which permit a sensible classical limit, including the abelian world-volume of the M-theory Five-brane. Dimensional reduction to four dimensions is also possible, leading to new formulas for 4D physics from 6D.
In the second half of the thesis, we discuss the projective algebraic and geometric structure of the AdS3/CFT2 correspondence. In the usual statement of this correspondence, two-dimensional conformal field theory (CFT) on the Riemann sphere or a higher-genus surface is holographically dual to features of topological gravity in three dimensions with negative curvature. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. We generalize the AdS (anti-de Sitter space)/CFT correspondence according to this principle using projective geometry over the p-adic numbers, Qp. The result is a formulation of holography in which the bulk geometry is discrete---the Bruhat--Tits tree for PGL(2,Qp)---but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. Parallel to the usual holographic correspondence, semi-classical dynamics of fields in the bulk compute the correlation functions of local operators on the boundary. Beyond correlators on the p-adic line, we propose a tensor network model in which the patterns of entanglement on the boundary are computed by discrete geometries in the bulk. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes. The model is built from tensors based on projective geometry over finite fields, Fp, and correctly computes the Ryu-Takayanagi formula, holographic entanglement and black hole entropy, and multiple interval entanglement inequalities.
In Chapter 2, we present tree-level n-particle on-shell scattering amplitudes of various brane theories with 16 conserved supercharges which are generalizations of Dirac--Born--Infeld theory. These include the world-volume theory of a probe D3-brane or D5-brane in 10D Minkowski spacetime as well as a probe M5-brane in 11D Minkowski spacetime, which describes self interactions of an abelian tensor supermultiplet with 6D (2,0) supersymmetry. We propose twistor-string-like formulas for tree-level scattering amplitudes of all multiplicities for each of these theories, and the amplitudes are written as integrals over the moduli space of certain rational maps localized on the (n-3)! solutions of the scattering equations. The R symmetry of the D3-brane theory is shown to be SU(4) x U(1), and the U(1) factor implies that its amplitudes are helicity conserving. Each of 6D theories (D5-brane and M5-brane) reduces to the D3-brane theory by dimensional reduction. As special cases of the general M5-brane amplitudes, we present compact formulas for examples involving only the self-dual B field with n=4,6,8.
In Chapter 3, we extend this formalism to n-particle tree-level scattering amplitudes of six-dimensional N=(1,1) super Yang--Mills (SYM) and N=(2,2) supergravity (SUGRA). The SYM theory arises on the world volume of coincident D5-branes, and the supergravity is the result of toroidal compactification of string theory. These theories have non-abelian interactions which allow for both even and odd-point amplitudes, unlike the branes of Chapter 2. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of N=(1,1) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2,C) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten--RSV (Roiban, Spradlin, and Volovich) formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional N=(2,2) SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional N=4 SYM on the Coulomb branch.
In Chapter 4, we consider half-maximal supergravity and present a twistor-like formula for the complete tree-level S matrix of chiral 6D (2,0) supergravity coupled to 21 abelian tensor multiplets. This is the low-energy effective theory that corresponds to Type IIB superstring theory compactified on a K3 surface. As in previous chapters, the formula is expressed as an integral over the moduli space of certain rational maps of the punctured Riemann sphere; the new ingredient is an integrand which successfully incorporates both gravitons and multiple flavors of tensors. By studying soft limits of the formula, we are able to explore the local moduli space of this theory, SO(5,21)/(SO(5) x SO(21)). Finally, by dimensional reduction, we also obtain a new formula for the tree-level S-matrix of 4D N=4 Einstein--Maxwell theory.
In Chapter 5, we introduce p-adic AdS/CFT and discuss several physical and mathematical features of the holographic correspondence between conformal field theories on P1(Qp) and lattice models on the Bruhat--Tits tree of PGL(2,Qp), an infinite tree of p+1 valence which has the p-adic projective line as its boundary. We review the p-adic numbers, the Bruhat--Tits tree, and some of their applications to physics including p-adic CFT. A key feature of these constructions is the discrete and hierarchical nature of the tree and the corresponding field theories, which serve as a toy model of holography in which there are no gravitons and no conformal descendants. Standard holographic results for massive free scalar fields in a fixed background carry over to the tree; semi-classical dynamics in the bulk compute correlation functions in the dual field theory and we obtain a precise relationship between the bulk mass and the scaling dimensions of local operators. It is also possible to interpret the vertical direction in the tree a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting and their construction parallels the story in AdS_3 topological gravity.
In Chapter 6, we consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat--Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu--Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein--Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semi-classical states in phase spaces over finite fields, generalizing the CRSS algorithm. These codes and the resulting networks provide a natural bulk geometric interpretation of non-Archimedean notions of entanglement in holographic boundary states.</p
Stable and Radiogenic Isotope Studies of Iron-oxides as Paleoenvironmental and Tectonic Archives
Get PDFGeochemical records of continental weathering environments are limited despite their critical value to understanding how past climates functioned. This thesis seeks to address this limitation by drawing together innovative lines of research in geochronology, stable isotope geochemistry, and chemical weathering. Two distinct projects are described; each project designed to provide new insight into the paleoenvironmental and tectonic history of continental weathering environments. These projects, though distinct in their methods and samples, are unified by their goal: to use the stable, radiogenic, and nucleogenic isotopic composition of iron oxides to provide new constraints on the geologic history of continental weathering environments.
The weathering of Fe-bearing rocks, coupled with the extreme insolubility of iron in moderately acidic to alkaline oxic waters, causes both goethite and hematite to be abundant chemical precipitates in near-surface environments. Goethite is favored in lower temperature and more acidic or alkaline conditions, while hematite precipitates more readily in near-neutral environments. These minerals are found in soils; spring, bog, and stream deposits; oxidized chemical sediments; and hydrothermal deposits. In many cases, substantial crystalline masses occur, which can take the form of nodules, pisoliths, botryoidal, stalactitic, and radiating masses, fibrous needles, pseudomorph, veneers, or as aggregates of flakes, tabular, or anhedral crystals. Time and temperature are arguably the two most fundamental variables we as geologists seek to constrain, and iron oxide deposits can provide a valuable archive of information on low-temperature, near-surface planetary processes.
The first project investigates how the stable oxygen isotopic composition of goethite, when combined with direct He dating on the same texturally resolved scales as stable isotope analyses, can be used to interpret water sources (Chapter 1) and formation temperatures (Chapter 2). The first chapter creates a record of the paleolatitudinal gradient in the oxygen isotope composition of meteoric water. The major finding of this study is the consistency in this gradient over geologic time. This second chapter proposes a new geothermometer using the intracrystalline oxygen isotopic composition of goethite. While stable isotopic compositions of goethite have long been utilized as a tool for reconstructing paleoenvironmental conditions, previous studies have focused on the bulk concentration of stable isotopes within this phase. Since goethite has two structurally non-equivalent oxygen sites, we show it is possible to extract two isotopically unique populations of oxygen, the composition of which we interpret to be dependent on temperature at time of mineral formation. In combination with the ability to directly date goethite by the (U-Th)/He method, we may utilize goethite to constrain both the temperature and timing of goethite formation, providing a valuable archive for information on continental paleoenvironments.
The second project utilized the paired He-Ne chronometer and 4He/3He method in hematite to produce thermal histories of the ancient Kaapvaal Craton over billion-year timescales. We applied these methods to hematite ore hosted within the Transvaal Supergroup in the Griqualand West (Chapter 3) and Transvaal Basin region (Chapter 4) of the ancient Kaapvaal Craton, South Africa. The application of hematite geo- and thermochronometry to these multi-billion year-old deposits represents the most challenging environments these methods have yet been applied to. We found, in some localities, hematite He-Ne ages provided further support of existing indirect age constraints on the timing of ore formation. In other localities, we found hematite He-Ne ages are uncorrelated with known tectono-thermal events. Modeled time-temperature histories indicate the Kaapvaal Craton has experienced exceptionally slow erosion rates over the last billion years, providing further evidence for the extreme tectonic stability of cratonic interiors over geologic timescales. This slow erosion took place over vast intervals of time, during which the craton was undergoing oxidative weathering, offering an additional constraint on understanding the history of atmospheric O2 during Proterozoic time.</p
Special Values of Zeta-Functions for Proper Regular Arithmetic Surfaces
Get PDFWe explicate Flach's and Morin's special value conjectures in [8] for proper regular arithmetic surfaces π : X → Spec Z and provide explicit formulas for the conjectural vanishing orders and leading Taylor coefficients of the associated arithmetic zeta-functions. In particular, we prove compatibility with the Birch and Swinnerton-Dyer conjecture, which has so far only been known for projective smooth X. Further, we derive a direct sum decomposition of Rπ*Z(n) into motivic degree components
Caught in the Middle: Homosexual Guilt, Liminality, and the role of the 'Novel of Identification' in Post-World War, Pre-Stonewall America
Get PDFThe 1950s and 1960s are often regarded as a transitory time period for the American homosexual man, overshadowed by the end of World War II and the tumultuous and radically influential Gay liberation movement beginning in the 1970s. The time period is marked by the publication and increased scrutiny of several influential novels: Christopher Isherwood's A Single Man (1964), Chester Himes's Yesterday Will Make You Cry (1937), James Baldwin's Giovanni’s Room (1956), and Gore Vidal's The City and the Pillar (1948). These novels highlight the tensions of competing forces of continuing repression and increasing acceptance, and complicate and explore the richness and heterogeneity of the gay identity, even before it has fully nucleated in pre-Stonewall America. The novels’ protagonists often have strained relationships both with the conventional society in which they live, but also the homosexual communities that exist around them. The protagonists recast their homosexual relationships as ephemeral, exceptional in nature, or with conventional labels, revealing complex and contradictory ideas about their own sexual identities. The protagonists are forced to come to terms with their identities, a process more complicated than simply coming out to the world and crossing the not-so-singular threshold often associated with the contemporary 'gay closet.' Finally, the novels' often tragically unresolved endings challenge the idea that they may serve as "support" novels for their gay communities; instead, they are better understood as novels of "identification", since they uncompromisingly cover issues that have gone uncovered before this period, identifying the problems that the isolated homosexual may feel, and highlighting and scrutinizing a lack of conventional resolution.</p
A History of Budgetary Politics in the United States
No full textThe sheer carnage seen during the American Civil War and World War I was
unparalleled, with both conflicts originating from decades of brewing tension between
radically incompatible ideologies that quickly erupted from a few small scrimmages to
total warfare. Any numerical analysis of either war would do little justice to the vast
number of casualties maintained by either side and the societal ramifications such loss of
life and collateral damage had on communities in the decades to follow. Yet, it is equally
essential to recognize that war brings about economic revival and resolves domestic and
political economic impasses that were previously deemed insoluble (Luce 1891). Such
dire circumstances compel leaders to generally set aside partisan politics and to institute
policies to fund the war effort
Prodigal
No full textHeavy feet tread the worn stairs of a third-floor walkup. Clinking keys scuffle against a narrow door. The lock gives way eventually, and Zoya looks around her little studio apartment blankly. So that was it, then. Exiled without so much as a by-your-leave
Understanding Lithosphere and Mantle Dynamics with Numerical Models Constrained by Observations
Get PDFNumerical studies play an important role in understanding lithospheric and mantle dynamics. In this thesis, we first develop and use multiphysics geodynamic models to study the evolution of subduction. Our geodynamic models are constrained by different geological and geophysical observations, including topography. We then use 3D numerical simulations of dynamic rupture with off-fault inelastic deformation to study the scaling between damage zone thickness and fault width. Finally, we study the mechanical strength and anisotropy in the continental collision region with flexural models and gravity and topography data.
Topography is valuable data for investigating lithosphere and mantle dynamics and constraining numerical studies. Topography prediction with forward models is well established at plate interiors, while it is still difficult to predict realistic topography at subduction zones. We use multiphysics geodynamic models to tackle this problem. Our models incorporate a true free surface, phase changes, and elasto-visco-plastic rheology. We also include surface processes, water migration and water weakening. We study the influences of different geophysical, petrological, and geochemical processes on topography and subduction zone evolution and show that surface geometry, surface processes, elasticity, and oceanic crust all strongly influence the stress state and deformation within plates, water weakening decouples the overriding plate and the subducting slab at the mantle wedge region and contributes to the initiation of overriding plate failure, and oceanic crust has a similar effect with sediments lubricating the subduction interface. Free slip surface topography and free surface topography have substantial differences, and free surface topography is influenced by different processes by adjusting the force balance. Application to the New Hebrides subduction zone suggests that deformation within a detached slab segment caused by the impact of the slab segment on the strong lower mantle explains the origin of the isolated deep earthquakes in the transition zone beneath the North Fiji Basin, and the difference in the seismic intensities between northern and southern deep earthquake clusters is caused by transition from strong deformation to weak deformation after the impact.
We apply our multiphysics approach to investigate the influence of inherited lithospheric heterogeneity on subduction initiation at the Puysegur Incipient Subduction Zone (PISZ) south of New Zealand. Our predictions fit the morphology of the Puysegur Trench and Ridge and the deformation history on the overriding plate. We show how a new thrust fault forms and evolves into a smooth subduction interface, and how a preexisting weak zone can become a vertical fault inboard of the thrust fault during subduction initiation, consistent with two-fault system at PISZ. The model suggests that the PISZ may not yet be self-sustaining. We propose that the Snares Zone (or Snares Trough) is caused by plate coupling differences between shallower and deeper parts, that the tectonic sliver between two faults experiences strong rotation, and that low density material accumulates beneath the Snares Zone.
We then turn to the scaling between damage zone thickness and fault width. Field observations indicate that damage zone thickness scales with accumulated fault displacement at short displacements but saturates at a few hundred meters for displacements larger than a few kilometers. To explain this transition of scaling behavior, we conduct 3D numerical simulations of dynamic rupture with off-fault inelastic deformation on long strike-slip faults. We find that the distribution of coseismic inelastic strain is controlled by the transition from crack-like to pulse-like rupture propagation associated with saturation of the seismogenic depth. The yielding zone reaches its maximum thickness when the rupture becomes a stable pulse-like rupture. Considering fracture mechanics theory, we show that seismogenic depth controls the upper bound of damage zone thickness on mature faults by limiting the efficiency of stress concentration near earthquake rupture fronts. We obtain a quantitative relation between limiting damage zone thickness, background stress, dynamic fault strength, off-fault yield strength, and seismogenic depth, which agrees with first-order field observations. Our results help link dynamic rupture processes with field observations and contribute to a fundamental understanding of damage zone properties.
Finally, we investigate the interactions between mechanical strength and lithospheric deformations. Variation of lithospheric strength controls the distribution of stress and strain within plates and at plate boundaries. Simultaneously, deformation caused by localized stress and strain reduces the lithospheric strength. We calculate the effective elastic thickness, Te, which is a proxy of lithospheric strength, and its anisotropy at the Zagros-Himalaya belt and surrounding regions. Te varies from < 5 km to over 100 km, and shows good correlations with geological boundaries. Along plate boundaries, mountain belts, and major faults, Te is usually smaller than 30 km. In basins, Te is between 30 - 60 km. In stable cratons, Te is larger than 60 km. In the regions with low and intermediate strength (Te < 60 km), the extent of Te anisotropy is usually large, and the weak direction of Te anisotropy agrees well with the directions of GPS data and crustal stress. In stable cratons, the extent of Te anisotropy is usually small. Our results suggest that mechanical weakening is the dominant mechanism to reduce the lithospheric strength in regions where Te is smaller than 60 km. In stable cratons, the effects of mechanical weakening can be ignored, and only thermal weakening resulting from mantle processes can modify the lithospheric strength substantially.</p
Transport Signatures of Spin-Orbit Coupling in Graphene-Based Materials
Get PDFTopological materials have been a fastest growing research topic in the recent decade. Out of the numerous new phases proposed and/or discovered, "topological insulators" (TIs) are one of the most promising materials that could lead to further advances in high-performance electronics and to applications in quantum computing. Similar to the ordinary semiconductors, TIs have a bulk gap; yet they host robust edge/surface states which are protected from non-magnetic disorder and interactions while the gap remains open. This feature is a manifestation of the non-trivial topology of TIs, the crucial feature that distinguishes them from ordinary semiconductors. Although the search for more topological materials continues, discovered TI currently are limited by practical difficulties that prevent industrialization.
In this thesis, we study graphene, which is the first proposed TI candidate in the history, and its derivatives. With the intrinsic spin-orbital coupling (SOC) on graphene, one can open a topologically nontrivial band gap at the Dirac cones, although the SOC of the carbon atoms is exceedingly small for topological insulation to be observed in experiments. Many proposals exist to enhance the SOC on graphene by doping with adatoms, changing the functionality of the surface, placing graphene on top of other strong SOC materials, etc. However, few proposed TI signatures have been found experimentally. Furthermore, measuring these intrinsic SOCs through magnetoconductance is challenging due to their relatively weak signatures in transport. This work addresses the challenges in transport measurements from both analytical and numerical approaches on various graphene-based materials. Graphene’s Dirac band structure and open geometry underlie its exciting prospects for engineering new physics via impurity-induced spin-orbit coupling. As a tantalizing example, previous theory works predicted a robust quantum-spin-Hall phase in graphene covered with dilute heavy adatoms such as In, Tl, and Os, although experiments to date have not detected the required enhancement of spin-orbit coupling. Motivated by these experiments, we explore the consequences of adatom-generated spin-orbit couplings on magneto-transport in graphene. We attack the problem using diagrammatic techniques and the Landauer-Buttiker transport simulation informed by microscopics, and study various coverages, chemical potentials, and disorder types. We find that the induced spin-orbit couplings can contribute to magneto-conductance differently from conventional intrinsic and Rasbha spin-orbit couplings. Our results provide a possible rationale for the absence of spin-orbit signatures in recent experiments, and also highlight a roadmap for their discovery
in future work.
In addition to the adatom-dedoped graphene, we also study graphene placing on top of strong SOC substrate, WS2, by jointing theory, numerics, and experiment. We demonstrate, in experiment, a clear weak anti-localization (WAL) effect arising from induced Rashba spin–orbit coupling (SOC) in WS2-covered single-layer and bilayer graphene devices. Contrary to the uncovered region of a shared single layer graphene flake, WAL in WS2-covered graphene occurs over a wide range of carrier densities on both the electron and hole sides. </p
Fitting Convex Sets to Data: Algorithms and Applications
Get PDFThis thesis concerns the geometric problem of finding a convex set that best fits a given dataset. Our question serves as an abstraction for data-analytical tasks arising in a range of scientific and engineering applications. We focus on two specific instances:
1. A key challenge that arises in solving inverse problems is ill-posedness due to a lack of measurements. A prominent family of methods for addressing such issues is based on augmenting optimization-based approaches with a convex penalty function so as to induce a desired structure in the solution. These functions are typically chosen using prior knowledge about the data. In Chapter 2, we study the problem of learning convex penalty functions directly from data for settings in which we lack the domain expertise to choose a penalty function. Our solution relies on suitably transforming the problem of learning a penalty function into a fitting task.
2. In Chapter 3, we study the problem of fitting tractably-described convex sets given the optimal value of linear functionals evaluated in different directions.
Our computational procedures for fitting convex sets are based on a broader framework in which we search among families of sets that are parameterized as linear projections of a fixed structured convex set. The utility of such a framework is that our procedures reduce to the computation of simple primitives at each iteration, and these primitives can be further performed in parallel. In addition, by choosing structured sets that are non-polyhedral, our framework provides a principled way to search over expressive collections of non-polyhedral descriptions; in particular, convex sets that can be described via semidefinite programming provide a rich source of non-polyhedral sets, and such sets feature prominently in this thesis.
We provide performance guarantees for our procedures. Our analyses rely on understanding geometrical aspects of determinantal varieties, building on ideas from empirical processes as well as random matrix theory. We demonstrate the utility of our framework with numerical experiments on synthetic data as well as applications in image denoising and computational geometry.
As secondary contributions, we consider the following:
1. In Chapter 4, we consider the problem of optimally approximating a convex set as a spectrahedron of a given size. Spectrahedra are sets that can be expressed as feasible regions of a semidefinite program.
2. In Chapter 5, we consider change-point estimation in a sequence of high-dimensional signals given noisy observations. Our method integrates classical approaches with a convex optimization-based step that is useful for exploiting structure in high-dimensional data.</p
Fast Adaptive Augmented Lagrangian Digital Image Correlation
Get PDFDigital image correlation (DIC) is a powerful experimental technique for measuring full-field displacement and strain. The basic idea of the method is to compare images of an object decorated with a speckle pattern before and after deformation in order to compute the displacement and strain fields. Local Subset DIC and finite element-based Global DIC are two widely used image matching methods; however there are some drawbacks to these methods. In Local Subset DIC, the computed displacement field may not satisfy compatibility, and the deformation gradient may be noisy, especially when the subset size is small. Global DIC incorporates displacement compatibility, but can be computationally expensive. In this thesis, we propose a new method, the augmented-Lagrangian digital image correlation (ALDIC), that combines the advantages of both the local (fast and in parallel) and global (compatible) methods. We demonstrate that ALDIC has higher accuracy and behaves more robustly compared to both Local Subset DIC and Global DIC.
DIC requires a large number of high resolution images, which imposes significant needs on data storage and transmission. We combined DIC algorithms with image compression techniques and show that it is possible to obtain accurate displace- ment and strain fields with only 5 % of the original image size. We studied two compression techniques – discrete cosine transform (DCT) and wavelet transform, and three DIC algorithms – Local Subset DIC, Global DIC and our newly proposed augmented Lagrangian DIC (ALDIC). We found the Local Subset DIC leads to the largest errors and ALDIC to the smallest when compressed images are used. We also found wavelet-based image compression introduces less error compared to DCT image compression.
To further speed up and improve the accuracy of DIC algorithms, especially in the study of complex heterogeneous strain fields at various length scales, we apply an adaptive finite element mesh to DIC methods. We develop a new h-adaptive technique and apply it to ALDIC. We show that this adaptive mesh ALDIC algorithm significantly decreases computation time with no loss (and some gain) in accuracy.</p