<p>Unlike matrix rank, hypermatrix rank is not lower semi-continuous. As a result, optimal low rank approximations of hypermatrices may not exist. Characterizing hypermatrices without optimal low rank approximations is an important step in implementing algorithms with hypermatrices. The main result of this thesis is an original coordinate-free proof that real 2 by 2 by 2 tensors that are rank three do not have optimal rank two approximations with respect to the Frobenius norm. This result was previously only proved in coordinates. Our coordinate-free proof expands on prior results by developing a proof method that can be generalized more readily to higher dimensional tensor spaces. Our proof has the corollary that the nearest point of a rank three tensor to the second secant set of the Segre variety is a rank three tensor in the tangent space of the Segre variety. The relationship between the contraction maps of a tensor generalizes, in a coordinate-free way, the fundamental relationship between the rows and columns of a matrix to hypermatrices. Our proof method demonstrates geometrically the fundamental relationship between the contraction maps of a tensor. For example, we show that a regular real or complex tensor is tangent to the 2 by 2 by 2 Segre variety if and only if the image of any of its contraction maps is tangent to the 2 by 2 Segre variety. </p>

This work includes various proofs of cohomology vanishing for logarithmic de Rham complex of unitary local system defined on an open algebraic complex manifold, which has a projective compactification by normal crossing divisor

We study log varieties, over a field of characteristic zero, which are generically logarithmically smooth and fs in the Kummer normally log étale topology. As an application, we prove an analog of Abramovich-Temkin-Wlodarczyk’s log resolution of singularities of fs log schemes in the Kummer fs setting.<br>

<p>This dissertation concerns the construction of some examples of complex algebraic varieties giving insight into certain questions in Hodge theory. </p>

<p>See abstract. </p>

<p>A Hodge module on an algebraic variety may be viewed as a variation of Hodge structure  with singularities. Given an irreducible variety $X$, for any polarized variation of Hodge structure $\bold{H}$ on a smooth open subvariety $U\subset X,$ there exists a unique Hodge module $\cM \in HM_{X}(X)$ that extends $\bH.$ Conversely, for any Hodge module $\cM \in HM_{X}(X)$ with strict support on $X,$ there exists a polarized variation of Hodge structure $\bH$ on a smooth open subset $U \subset X$ such that $\cM \vert _{V} \cong \bH.$ In this thesis, we first study the singularities of a Hodge module $\cM \in HM_{X}(X)$ by using Morihiko Saito's theory of $S$-sheaves and duality. Then using local cohomology and the theory of mixed Hodge modules, we study the Hodge structure of $H^{i}(X, DR(\cM))$  when $X$ is a projective variety. Finally, we consider a variation of Hodge structure $\bH$ on $U$ as a Hodge module $\cN \in HM(U)$ on $U,$ and study the local cohomology of the complex $Gr^{F}_{p}DR(j_{!}\cN) \in D^{b}_{coh}(\cO_{X}),$ where $j: U \hookrightarrow X$ is the natural map.</p>

<p>See abstract. </p>

<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>