UMD Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/3
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Item Logarithmic connections on arithmetic surfaces and cohomology computation(2022) Dykas, Nathan; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)De Rham cohomology is important across a broad range of mathematical fields. The good properties of de Rham cohomology on smooth and complex manifolds are also shared by those schemes which most closely resemble complex manifolds, namely schemes that are (1) smooth, (2) proper, and (3) defined over the complex numbers or other another field of characteristic zero. In the absence of one or more of those three properties, one observes more pathological behavior. In particular, for affine morphisms $X/S$, the groups $\Hop^i(X/S)$ may be infinitely generated. In this case, when $S = \Sp(k), \op{char}(k) > 0$, the \textit{Cartier isomorphism} allows one to view the groups as finite dimensional over a different base: $\OO_{X^{(p)}}$. However when $S$ is a Dedekind ring of mixed characteristic, there is no good substitute for the Cartier isomorphism. In this work we explore a method of calculating the de Rham cohomology of some affine schemes which occur as the complement of certain divisors on arithmetic surfaces over a Dedekind scheme of mixed characteristic. The main tool will be (Koszul) connections on vector bundles, whose primary role is to generalize the exterior derivative $\OO_X \xrightarrow{\D{}} \Omega_{X/S}^1$ to a map $\mathcal{F} \xrightarrow{\nabla} \Omega_{X/S}^1\otimes\mathcal{F}$ defined on more general quasi-coherent modules $\mathcal{F}$. Given an suitable arithmetic surface $X$ and divisor $D$ with complement $U=X\setminus D$, the de Rham cohomology $\Hop^1(U/S)$ is infinitely generated. We use a natural filtration $\op{Fil}^\bullet\OO_U$ to construct a filtration $\op{Fil}^\bullet\Hop^1(U/S)$. We show that associated graded of this filtration is the direct sum of finitely generated modules, and we give a formula to calculate them in terms of the structure sheaf $\OO_D$ of the divisor as well as the different ideal $\mathcal{D}_D \subset \OO_D$ of the finite, flat extension $D/S$.Item Motivic Cohomology of Groups of Order p^3(2018) Black, Rebecca; Brosnan, Patrick; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis we compute the motivic cohomology ring (also known as Bloch's higher Chow groups) with finite coefficients for the two nonabelian groups of order $27$, thought of as affine algebraic groups over $\mathbb{C}$. Specifically, letting $\tau$ denote a generator of the motivic cohomology group $H^{0,1}(BG,\Z/3) \cong \Z/3$ where $G$ is one of these groups, we show that the motivic cohomology ring contains no $\tau$-torsion, and so can be computed as a weight filtration on the ordinary group cohomology. In the case of a prime $p > 3$, there are also two nonabelian groups of order $p^3$. We make progress toward computing the motivic cohomology in the general case as well by reducing the question to understanding the $\tau$-torsion on the motivic cohomology of a $p$-dimensional variety; we also compute the motivic cohomology of $BG$ for general $p$ modulo the $\tau$-torsion classes.Item Combinatorics of K-Theoretic Jeu de Taquin(2010) Clifford, Edward Grant; Tamvakis, Harry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Thomas and Yong [5] introduced a theory of jeu de taquin which extended Schutzenberger's [4] for Young tableaux. The extended theory computes structure constants for the K-theory of (type A) Grassmannians using combinatorial machinery similar to that for cohomology. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P. In this dissertation, we see that the root-system uniform rule is well-defined for certain G/P other than the Grassmannian. This gives rise to combinatorially defined rings which are conjecturally isomorphic to K(G/P). Although we do not prove that these rings are isomorphic to K(G/P), we do produce a ``Pieri rule" for computing the product of a general class with a generating class in the type B combinatorial case. We also investigate some symmetries which support the conjectural isomorphism. Moreover, our results combined with recent work of Buch and Ravikumar [1] imply that this conjecture is in fact true. Lenart [2] gave a Pieri rule for the type A K-theory, demonstrating that the Pieri structure constants are binomial coefficients. In contrast, using techniques of [3], we show that type B Pieri structure constants have no such simple closed forms. References: [1] A. Buch and V. Ravikumar: Pieri rules for the K-theory of cominuscule Grassmannians, arXiv:1005.2605, 2010. [2] C. Lenart: Combinatorial aspects of K-theory of Grassmannians. Ann. Combin. 4 (2000), 67--82. [3] M. Petkovsek and H. Wilf and D. Zeilberger: A=B. A K Peters, Ltd. (1996). [4] M.-P. Schutzenberger: Combinatoire et representation du groupe symetrique. Springer-Verlag Berlin, Lec. Notes in Math. 579 (1977), 59--113. [5] H. Thomas and A. Yong: A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus. Algebra Number Theory 3 (2009), no. 2, 121--148.