abstract: Over the last decade, deep neural networks also known as deep learning, combined with large databases and specialized hardware for computation, have made major strides in important areas such as computer vision, computational imaging and natural language processing. However, such frameworks currently suffer from some drawbacks. For example, it is generally not clear how the architectures are to be designed for different applications, or how the neural networks behave under different input perturbations and it is not easy to make the internal representations and parameters more interpretable. In this dissertation, I propose building constraints into feature maps, parameters and and design of algorithms involving neural networks for applications in low-level vision problems such as compressive imaging and multi-spectral image fusion, and high-level inference problems including activity and face recognition. Depending on the application, such constraints can be used to design architectures which are invariant/robust to certain nuisance factors, more efficient and, in some cases, more interpretable. Through extensive experiments on real-world datasets, I demonstrate these advantages of the proposed methods over conventional frameworks. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2019

<div>In the present work, we consider a class of Finsler metrics using the warped product notion introduced by B. Chen, Z. Shen and L. Zhao (2018), with another “warping”, one that is consistent with the form of metrics modeling static spacetimes and simplified by spherical symmetry over spatial coordinates, which emerged from the Schwarzschild metric in isotropic coordinates. We will give the PDE characterization for the proposed metrics to be Ricci-flat and construct explicit examples. Whenever possible, we describe both positive-definite solutions and solutions with Lorentz signature. For the latter, the 4-dimensional metrics may also be studied as Finsler spacetimes.</div>

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of smaller ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the defining equations of Rees rings for a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving that, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of a graded component of the Rees ring in the generic case is an approximate resolution of the same component of the Rees ring in question. We end the paper by giving some examples of explicit generation and concentration degree bounds.

<p>For a prime p>2 and a smooth proper p-adic formal scheme X over O_K where K is a p-adic field of absolute ramification degree e, we study a series of conditions (Cr_s), s>=0 that partially control the G_K-action on the image of the associated Breuil-Kisin prismatic cohomology RΓ_Δ(X/S) inside the A_inf-prismatic cohomology RΓ_Δ(X_Ainf/A_inf). The condition (Cr₀) is a criterion for a Breuil-Kisin-Fargues G_K-module to induce a crystalline representation used by Gee and Liu, and thus leads to a proof of crystallinity of Hⁱ_et(X_CK, Q_p) that avoids the crystalline comparison. The higher conditions (Cr_s) are used in an adaptation of a ramification bounds strategy of Caruso and Liu. As a result, we establish ramification bounds for the mod p representations Hⁱ_et(X_CK, Z/pZ) for arbitrary e and i, which extend or improve existing bounds in various situations.</p>

Itô diffusions that move on level sets of functions in Rn, which we have called level processes, are an overlooked variant of the classical Itô processes. These processes find themselves nestled between the study of regular Itô diffusions in Rn and diffusions which are bound to smooth manifolds. In this thesis we present how to construct these level processes, in both the plane and n-space, with their properties in the plane being examined. We also show how these processes connect to the Itô diffusions on smooth manifolds. In addition, we derive how to affix a given system of Itô diffusion to a level set, given certain constraints. Lastly, we give a brief overview of three numerical schemes for stochastic differential equations and investigate their applicability to the simulation of level processes. For both the probabilistic and numeric sections, reflections on the work done are given and possible extensions, such as the relaxation of the smoothness condition for the level set, are briefly outlined.

<pre>In the first part of this thesis we show that, for a given non-arithmetic closed hyperbolic <i>$</i><i>n</i><i>$</i> manifold <i>$</i><i>M</i><i>$</i>, there exist for each positive integer <i>$</i><i>j</i><i>$</i>, a set <i>$</i><i>M_</i><i>1</i><i>,...,M_j</i><i>$</i> of pairwise nonisometric, strongly isospectral, finite covers of <i>$</i><i>M</i><i>$</i>, and such that for each <i>$</i><i>i,i'</i><i>$</i> one has isomorphisms of cohomology groups <i>$</i><i>H^*(M_i,</i><i>\Zbb</i><i>)=H^*(M_{i'},</i><i>\Zbb</i><i>)</i><i>$</i> which are compatible with respect to the natural maps induced by the cover. In the second part, we prove that hyperbolic <i>$</i><i>2</i><i>$</i>- and <i>$</i><i>3</i><i>$</i>-manifolds which arise from principal congruence subgroups of a maixmal order in a quaternion algebra having type number <i>$</i><i>1</i><i>$</i> are absolutely spectrally rigid. One consequence of this is a partial answer to an outstanding question of Alan Reid, concerning the spectral rigidity of Hurwitz surfaces.</pre>

We give a concise explication of the theory of level set methods for modeling motion of an interface as well as the numerical implementation of these methods. We then introduce the geometry of a camera and the mathematical models for 3D reconstruction with a few examples both simulated and from a real camera. We finally describe the model for 3D surface reconstruction from n-camera views using level set methods.

<p>This work strengthens and extends an algorithm for computing Betti numbers of symmetric semialgebraic sets developed by Basu and Riener in, <em>Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-Algebraic Sets</em>. We first adapt a construction of Gabrielov and Vorobjov in, <em>Approximation of Definable Sets by Compact Families, and Upper Bounds on Homotopy and Homology,</em> for replacing arbitrary definable sets by compact ones to the symmetric case. The original construction provided maps from the homotopy and homology groups of the replacement set to those of the original; we show that for sets symmetric relative to the action of some finite reflection group <em>G</em>, we may construct these maps to be equivariant. This modification to the construction for compact replacement allows us to extend Basu and Riener's theorem on which submodules appear in the isotypic decomposition of each cohomology space to sets not necessarily closed and bounded. Furthermore, by utilizing this equivariant compact approximation, we may obtain a precise description of the aforementioned decomposition of each cohomology space, and not merely the final dimension of the space, from Basu and Riener's algorithm.</p> <p><br></p> <p>    Though our equivariant compact replacement holds for <em>G</em> any finite reflection group, Basu and Riener's results only consider the case of the action the of symmetric group, sometimes termed type <em>A</em>. As a first step towards generalizing Basu and Riener's work, we examine the next major class of symmetry: the action of the group of signed permutations (known as type <em>B</em>). We focus our attention on Vandermonde varieties, a key object in Basu and Riener's proofs. We show that the intersection of a type <em>B</em> Vandermonde variety with a fundamental region of type <em>B</em> symmetry is topologically regular. We also prove a result about the intersection of a type <em>B</em> Vandermonde variety with the walls of this fundamental region, leading to the elimination of factors in a different decomposition of the homology spaces.</p>

<p>The curvatures in Finsler geometry can be defined in similar ways as in Riemannian geometry. However, since there are fewer restrictions on the metrics, many geometric quantities arise in Finsler geometry which vanish in the Riemannian case. These quantities are generally known as non-Riemannian quantities and interact with the curvatures in controlling the global geometrical and topological properties of Finsler manifolds. In the present work, we study general weighted Ricci curvatures which combine the Ricci curvature and the S-curvature, and define a weighted flag curvature which combines the flag curvature and the T -curvature. We characterize Randers metrics of almost isotropic weighted Ricci curvatures and show the general weighted Ricci curvatures can be divided into three types. On the other hand, we show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, generalizing the Gromoll-Meyer theorem in Riemannian geometry.</p>

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