Calibrated embeddings in the special Lagrangian and coassociative cases

Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.

Some examples of special Lagrangian tori

A short paper giving some examples of smooth hypersurfaces M of degree n+1 in complex projective n-space that are defined by real polynomial equations and whose real slice contains a component diffeomorphic to an n-1 torus, which is then special Lagrangian with respect to the Calabi-Yau metric on M.

Detection and Classification of Whale Acoustic Signals

This dissertation focuses on two vital challenges in relation to whale acoustic signals: detection and classification.

In detection, we evaluated the influence of the uncertain ocean environment on the spectrogram-based detector, and derived the likelihood ratio of the proposed Short Time Fourier Transform detector. Experimental results showed that the proposed detector outperforms detectors based on the spectrogram. The proposed detector is more sensitive to environmental changes because it includes phase information.

In classification, our focus is on finding a robust and sparse representation of whale vocalizations. Because whale vocalizations can be modeled as polynomial phase signals, we can represent the whale calls by their polynomial phase coefficients. In this dissertation, we used the Weyl transform to capture chirp rate information, and used a two dimensional feature set to represent whale vocalizations globally. Experimental results showed that our Weyl feature set outperforms chirplet coefficients and MFCC (Mel Frequency Cepstral Coefficients) when applied to our collected data.

Since whale vocalizations can be represented by polynomial phase coefficients, it is plausible that the signals lie on a manifold parameterized by these coefficients. We also studied the intrinsic structure of high dimensional whale data by exploiting its geometry. Experimental results showed that nonlinear mappings such as Laplacian Eigenmap and ISOMAP outperform linear mappings such as PCA and MDS, suggesting that the whale acoustic data is nonlinear.

We also explored deep learning algorithms on whale acoustic data. We built each layer as convolutions with either a PCA filter bank (PCANet) or a DCT filter bank (DCTNet). With the DCT filter bank, each layer has different a time-frequency scale representation, and from this, one can extract different physical information. Experimental results showed that our PCANet and DCTNet achieve high classification rate on the whale vocalization data set. The word error rate of the DCTNet feature is similar to the MFSC in speech recognition tasks, suggesting that the convolutional network is able to reveal acoustic content of speech signals.

Augmentations and Rulings of Legendrian Links

For any Legendrian knot in R^3 with the standard contact structure, we show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a normal ruling of the front diagram, generalizing results of Fuchs, Ishkhanov, and Sabloff. We also show that any even graded augmentation must send t to -1.

We extend the definition of a normal ruling from J^1(S^1) given by Lavrov and Rutherford to a normal ruling for Legendrian links in #^k(S^1\times S^2). We then show that for Legendrian links in J^1(S^1) and #^k(S^1\times S^2), the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a normal ruling of the front diagram. For Legendrian knots, we also show that any even graded augmentation must send t to -1. We use the correspondence to give nonvanishing results for the symplectic homology of certain Weinstein 4-manifolds.

Learning from Geometry

Subspaces and manifolds are two powerful models for high dimensional signals. Subspaces model linear correlation and are a good fit to signals generated by physical systems, such as frontal images of human faces and multiple sources impinging at an antenna array. Manifolds model sources that are not linearly correlated, but where signals are determined by a small number of parameters. Examples are images of human faces under different poses or expressions, and handwritten digits with varying styles. However, there will always be some degree of model mismatch between the subspace or manifold model and the true statistics of the source. This dissertation exploits subspace and manifold models as prior information in various signal processing and machine learning tasks.

A near-low-rank Gaussian mixture model measures proximity to a union of linear or affine subspaces. This simple model can effectively capture the signal distribution when each class is near a subspace. This dissertation studies how the pairwise geometry between these subspaces affects classification performance. When model mismatch is vanishingly small, the probability of misclassification is determined by the product of the sines of the principal angles between subspaces. When the model mismatch is more significant, the probability of misclassification is determined by the sum of the squares of the sines of the principal angles. Reliability of classification is derived in terms of the distribution of signal energy across principal vectors. Larger principal angles lead to smaller classification error, motivating a linear transform that optimizes principal angles. This linear transformation, termed TRAIT, also preserves some specific features in each class, being complementary to a recently developed Low Rank Transform (LRT). Moreover, when the model mismatch is more significant, TRAIT shows superior performance compared to LRT.

The manifold model enforces a constraint on the freedom of data variation. Learning features that are robust to data variation is very important, especially when the size of the training set is small. A learning machine with large numbers of parameters, e.g., deep neural network, can well describe a very complicated data distribution. However, it is also more likely to be sensitive to small perturbations of the data, and to suffer from suffer from degraded performance when generalizing to unseen (test) data.

From the perspective of complexity of function classes, such a learning machine has a huge capacity (complexity), which tends to overfit. The manifold model provides us with a way of regularizing the learning machine, so as to reduce the generalization error, therefore mitigate overfiting. Two different overfiting-preventing approaches are proposed, one from the perspective of data variation, the other from capacity/complexity control. In the first approach, the learning machine is encouraged to make decisions that vary smoothly for data points in local neighborhoods on the manifold. In the second approach, a graph adjacency matrix is derived for the manifold, and the learned features are encouraged to be aligned with the principal components of this adjacency matrix. Experimental results on benchmark datasets are demonstrated, showing an obvious advantage of the proposed approaches when the training set is small.

Stochastic optimization makes it possible to track a slowly varying subspace underlying streaming data. By approximating local neighborhoods using affine subspaces, a slowly varying manifold can be efficiently tracked as well, even with corrupted and noisy data. The more the local neighborhoods, the better the approximation, but the higher the computational complexity. A multiscale approximation scheme is proposed, where the local approximating subspaces are organized in a tree structure. Splitting and merging of the tree nodes then allows efficient control of the number of neighbourhoods. Deviation (of each datum) from the learned model is estimated, yielding a series of statistics for anomaly detection. This framework extends the classical {\em changepoint detection} technique, which only works for one dimensional signals. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.

Spic'ing into Existence: Epitaph, Epithet, and the Ethnopoetic Imagination

Quién Es, Quién Somos? Spic’ing into Existence claims a four-fold close-reading: first, analysis of texts: from theoretical meditations to (prison) memoir and film. Second, a half dozen central figures appear, largely Latinx and black American. They cut across a score of registers, socio-economics, ideological reservations, but all are, as Carl Carlton sang, poetry in motion. Writers, poets, theologians, pathologists, artists, comedians, actors, students whose vocation is invocation, the inner surge of their calling. Third, the manuscript draws from a series of historical moments—from radical liberation of the late 60s, to contemporary student activism. Finally, this body of work is movement, in all its social, gestural, and kinesthetic viscera. From this last heading, we peel away layers of what I call the ethnopoet, the fascia undoing that reveals its bio-political anatomy, dressing its bare life with kinship speech. First, the social revolutions of the Civil Rights, Black Power, abolitionism, the Black Panthers and Young Lords, boycotts and jarring artistic performances. These events are superficial not in vain sense, but key epicenters of underground murmurings, the workings of a cunning assailant. She robs not lavish estates, but another day to breathe. Gesturally, as perhaps the interlocutor, lies this author, interspersing his own diatribes to conjure her presence. The final branch is admittedly the most intangible. Kinesthetically, we map the nimbleness, footwork lígera of what I call the ethnopoet. Ethnopoet is no mere aggregate of ethnicity and poetry, but like chemical reaction, the descriptor for its behavior under certain pressures, temperatures, and elements. Elusive and resisting confinement, and therefore definition, the ethnopoet is a shapeshifting figure of how racialized bodies [people of color] respond to hegemonic powers. She is, at bottom, however, a native translator, the plural-lensed subject whose loyalty is only to the imagination of a different world, one whose survival is not contingent upon her exploitation. The native translator’s constant re-calibrations of oppressive power apparatuses seem taxing at best, and near-impossible, at worst. To effectively navigate through these polarized loci, she must identify ideologies that in turn seek “affective liberatory sances” in relation to the dominant social order (43). In a kind of performative contradiction, she must marshall the knowledge necessary to “break with ideology” while speaking within it. Chicana Studies scholar, Chela Sandoval, describes this dual movement as “meta-ideologizing”: the appropriation of hegemonic ideological forms in order to transform them (82). Nuestros padres se subieron encima de La Bestia, y por eso somos pasageros a ese tren. Y ya, dentro su pansa, tenemos que ser vigilantes cuando plantamos las bombas. In Methodology of the Oppressed, Sandoval schematizes this oppositional consciousness around five principle categories: “equal rights,” “revolutionary,” “supremacist,” “separatist,” and “differential.” Taken by themselves, the first four modes appear mutually exclusive, incapable of occupying the same plane, until a fifth pillar emerges. Cinematographic in nature, differential consciousness, as Sandoval defines it, is “a kinetic motion that maneuvers, poetically transfigures, and orchestrates while demanding alienation, perversion, and reformation in both spectators and practitioners” (44). For Sandoval, then, differential consciousness is a methodology that privileges an incredible sense mobility, one reaching artistic sensibilities. Our fourth and final analytic of movement serves an apt example of this dual meaning. Lexically speaking, ‘movement’ may be regarded as a political mobilization of aggrieved populations (through sustained efforts), or the process of moving objects (people or otherwise) from one location to another. Praxis-wise, it is both action and ideal, content and form. Thus, an ethnic poetics must be regarded less as a series of stanzas, shortened lyric, or even arrangement of language, but as a lens through which peripheralized peoples kaleidecope ideological positions in an “original, eccentric, and queer sight” (43). Taking note of the advantages of postponing identifications, the thesis stands its ground on the term ethnopoet. Its abstraction is not dewey-eyed philosophy, but an anticipation of poetic justice, of what’s to come from callused hands. This thesis is divided into 7.5 chapters. The first maps out the ethnopoet’s cartographies of struggle. By revisiting that alleged Tío Tomas, Richard Rodriguez, we unearth the tensions that negatively, deny citizenship to one silo, but on the flipside, engender manifold ways of seeing, hearing, and moving . The second, through George Jackson’s prison memoirs, pans out from this ethnography of power, groping for an apparatus that feigns an impervious prestige: ‘the aesthetic regime of coercion.’ In half-way cut, the thesis sidesteps to spic into existence, formally announcing, through Aime Cesaire, myself, and Pedro Pietri, the poeticization of trauma. Such uplift denies New Age transcendence of self, but a rehearsal of our entrapment in these mortal envelopes. Thirdly, conscious of the bleeding ethnic body, we cut open the incipient corpse to observe her pathologist. Her native autopsies offer the ethnic body’s posthumous recognition, the ethnopoetics ability to speak for and through the dead. Chapter five examines prolific black artists—Beyonce and Kendrick Lamar—to elide the circumvention of their consumption via invoking radical black hi/her-stories, ones fragmenting the black body. Sixth, the paper compares the Black Power Salute of the 1968 Mexico City Olympics to Duke’s Mi Gente Boycott of their Latino Student Recruitment Weekend. Both wielded “silent gestures,” that shrewdly interfered with white noise of numbed negligence. Finally, ‘taking the mask off’ that are her functionalities, the CODA expounds on ethnopoet’s interiority, particularly after the rapid re-calibration of her politics. Through a rerun of El Chavo del Ocho, one of Mexican television’s most cherished shows, we tune into the heart-breaking indigence of barrio residents, only to marvel at the power of humor to, as Friday’s John Witherspoon put it, “fight another day.” This thesis is the tip of my tongue. Y por una vez, déjala que cante.

Moduli Space of (0,2) Conformal Field Theories

In this thesis we study aspects of (0,2) superconformal field theories (SCFTs), which are suitable for compactification of the heterotic string. In the first part, we study a class of (2,2) SCFTs obtained by fibering a Landau-Ginzburg (LG) orbifold CFT over a compact K\"ahler base manifold. While such models are naturally obtained as phases in a gauged linear sigma model (GLSM), our construction is independent of such an embedding. We discuss the general properties of such theories and present a technique to study the massless spectrum of the associated heterotic compactification. We test the validity of our method by applying it to hybrid phases of GLSMs and comparing spectra among the phases. In the second part, we turn to the study of the role of accidental symmetries in two-dimensional (0,2) SCFTs obtained by RG flow from (0,2) LG theories. These accidental symmetries are ubiquitous, and, unlike in the case of (2,2) theories, their identification is key to correctly identifying the IR fixed point and its properties. We develop a number of tools that help to identify such accidental symmetries in the context of (0,2) LG models and provide a conjecture for a toric structure of the SCFT moduli space in a large class of models. In the final part, we study the stability of heterotic compactifications described by (0,2) GLSMs with respect to worldsheet instanton corrections to the space-time superpotential following the work of Beasley and Witten. We show that generic models elude the vanishing theorem proved there, and may not determine supersymmetric heterotic vacua. We then construct a subclass of GLSMs for which a vanishing theorem holds.

Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics

The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.