We provide algorithms and documention to compute the cohomology of congruence subgroups of the special linear group over the integers when n=3 using the well-rounded retract and the Voronoi decomposition. We define the Sharbly complex and how one acts on a k-sharbly by the Hecke operators. Since the norm of a sharbly is not preserved by the Hecke operators we also examine the reduction techniques described by Gunnells and present our implementation of said techniques for n=3.

<p>This thesis is focused on determinantal rings in 2 different contexts. In Chapter 3 the homological properties of powers of determinantal ideals are studied. In particular the focus is on local cohomology of determinantal thickenings and we explicitly describe the $R$-module structure of some of these local cohomology modules. In Chapter 4 we introduce \textit{generalized diagonal} matrices, a class of sparse matrices which contain diagonal and upper triangular matrices. We study the ideals of minors of such matrices and describe their properties such as height, multiplicity, and Cohen-Macaulayness. </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>

In this thesis we investigate 1-skeleta and their associated cohomology rings. 1-skeleta arise from the 0- and 1-dimensional orbits of a certain class of manifold admitting a compact torus action and many questions that arise in the theory of 1-skeleta are rooted in the geometry and topology of these manifolds. The three main results of this work are: a lifting result for 1-skeleta (related to extending torus actions on manifolds), a classification result for certain 1-skeleta which have the Morse package (a property of 1-skeleta motivated by Morse theory for manifolds) and two constructions on 1-skeleta which we show preserve the Lefschetz package (a property of 1-skeleta motivated by the hard Lefschetz theorem in algebraic geometry). A corollary of this last result is a conceptual proof (applicable in certain cases) of the fact that the coinvariant ring of a finite reflection group has the strong Lefschetz property.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24165号 / 理博第4856号 / 新制||理||1695(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 入谷 寛, 教授 藤原 耕二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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<p> We study two objects: an ideal of a Hecke algebra, and a pairing in Galois cohomology.</p> <p> Let h be the Hecke algebra of cusp forms of weight 2, level n, and a fixed Dirichlet character modulo n generated by all Hecke operators, where n is an odd prime p or a product of two distinct odd primes N and p. We study the Eisenstein I ideal of h. We wrote a computer program to test whether Up - 1 generates this ideal, where Up is the pth Hecke operator in h. We found many cases of n and the character so that Up - 1 alone generates I. On the other hand, we found one example with N = 3 and p = 331 where Up - 1 does not generate I.</p> <p> Let K = Q(μn) be the nth cyclotomic field. Let S be the set of primes above p in K, and let G_K,S be the Galois group of the maximal extension of K unramified outside S. We study a pairing on cyclotomic p-units that arises from the cup product on H1(G_K,S, μp). This pairing takes values in a Gal(K/Q)-eigenspace of the p-part of the class group of K. Sharifi has conjectured that this pairing is surjective. We studied this pairing in detail by imposing linear relations on the possible pairing values. We discovered many values of n and the character such that these relations single out a unique nontrivial possibility for the pairing, up to a possibly zero scalar.</p> <p> Sharifi showed in [S2] that, under an assumption on Bernoulli numbers, the element Up - 1 generates the Eisenstein ideal I if and only if pairing with the single element p is surjective. In particular, in the instances for which we found a unique nontrivial possibility for the pairing, then if Up - 1 generates I, we know that the scalar up to which it is determined cannot be zero.</p> / Thesis / Master of Science (MSc)

Two results are obtained in this work. First, we prove that for a commutative ring embedded in a larger ring, which is not necessarily commutative, its division and rational closures coincide. Second, for an infinite discrete group G, we investigate group cohomology and homology with coefficients in lp(G). We prove that if G is of type FPn, then all its homology and cohomology groups up to n are either zero or infinite dimensional. This generalizes one of the results obtained by Bekka and Valette / Ph. D.

This thesis studies N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories and their mathematical applications. After a brief review of GLSMs, we systematically study nonabelian GLSMs for symplectic and orthogonal Grassmannians, following up a proposal in the math community. As consistency checks, we have compared global symmetries, Witten indices, and Calabi-Yau conditions to geometric expectations. We also compute their nonabelian mirrors following the recently developed nonabelian mirror symmetry. In addition, for symplectic Grassmannians, we use the effective twisted superpotential on the Coulomb branch of the GLSM to calculate the ordinary and equivariant quantum cohomology of the space, matching results in the math literature. Then we discuss 3d gauge theories with Chern-Simons terms. We propose a complementary method to derive the quantum K-theory relations of projective spaces and Grassmannians from the corresponding 3d gauge theory with a suitable choice of the Chern-Simons levels. In the derivation, we compare to standard presentations in terms of Schubert cycles, and also propose a new description in terms of shifted Wilson lines, which can be generalized to symplectic Grassmannians. Using this method, we are able to obtain quantum K-theory relations, which match known math results, as well as make predictions. / Doctor of Philosophy / In this thesis, we study two specific models of supersymmetric gauge theories, namely two-dimensional N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories. These models have played an important role in quantum field theory and string theory for decades, and generated many fruitful results, improving our understanding of Nature by drawing on many branches in mathematics, such as complex differential geometry, intersection theory, quantum cohomology/quantum K-theory, enumerative geometry, and many others. Specifically, this thesis is devoted to studying their applications in quantum cohomology and quantum K-theory. In the first part of this thesis, we systematically study two-dimensional GLSMs for symplectic and orthogonal Grassmannians, generalizing the study for ordinary Grassmannians. By analyzing the Coulomb vacua structure of the GLSMs for symplectic Grassmannians, we are able to obtain the ordinary and equivariant quantum cohomology for these spaces. A similar methodology applies to 3d Chern-Simons-matter theories and quantum K-theory, for which we propose a new description in terms of shifted Wilson lines.

The purpose of this thesis is to develop machinery for calculating Hochschild cohomology groups of certain finite dimensional algebras. So let A be a finite dimensional quotient of a path algebra. A method of modeling the enveloping algebra Ae of A on a computer is presented. Adding the extra hypothesis that A is a monomial algebra, we construct a minimal projective resolution of A over A e. The syzygies for this resolution exhibit an alternating behavior which is explained by the construction of a special sequence of paths from the quiver of A. Finally, a technique for calculating Hochschild cohomology groups from these resolutions is presented. An important application involving an invariant characterization for a certain class of monomial algebras is also included. / Ph. D.