The game of Subset Take-Away is an impartial combinatorial game posed by David Gale in 1974. The game can be played on various discrete structures, including but not limited to graphs, hypergraphs, polygonal complexes, and partially ordered sets. While a universal winning strategy has yet to be found, results have been found in certain cases. In 2003 R. Riehemann focused on Subset Take-Away on bipartite graphs and produced a complete game analysis by studying nim-values. In this work, we extend the notion of Take-Away on a bipartite graph to Take-Away on particular hypergraphs, namely oddly-uniform hypergraphs and evenly-uniform hypergraphs whose vertices satisfy a particular coloring condition. On both structures we provide a complete game analysis via nim-values. From here, we consider different discrete structures and slight variations of the rules for Take-Away to produce some interesting results. Under certain conditions, polygonal complexes exhibit a sequence of nim-values which are unbounded but have a well-behaved pattern. Under other conditions, the nim-value of a polygonal complex is bounded and predictable based on information about the complex itself. We introduce a Take-Away variant which we call “Take-As-Much-As-You-Want”, and we show that, again, nim-values can grow without bound, but fortunately they are very easily described for a given graph based on the total number of vertices and edges of the graph. Finally we consider Take-Away on a specific type of partially ordered set which we call a rank-complete poset. We have results, again via an analysis of nim-values, for Take-Away on a rank-complete poset for both ordinary play as well as misère play.

Thesis advisor: Amir H. Hoveyda / Chapter 1. Mechanistic Insights and Factors Influencing Polytopal Rearrangements in Stereogenic-at-Ru Carbenes. Herein, the mechanistic elucidation of the stereochemical inversion in stereogenic-at-Ru carbene complexes through olefin metathesis (OM) and non-olefin metathesis (non-OM) based polytopal rearrangements is provided. Our investigations involve the isolation and characterization of previously hypothesized higher-energy (e.g., endo-anti) and lower-energy (e.g., exo-anti) diastereomers, and their interconversion under thermal and/or acid-catalyzed conditions is demonstrated. Furthermore, our computational efforts highlighting the importance of the anionic ligands, due to their critical role in trans influence, dipolar interactions, and e-e repulsions, in polytopal rearrangements are reported. Finally, the positive influence of H-bonding in OM and non-OM processes is also rationalized. (a) Khan, R. K. M.; Zhugralin, A. R.; Torker, S.; O'Brien, R. V.; Lombardi, P. J. and Hoveyda, A. H. "Synthesis, Isolation, Characterization, and Reactivity of High-Energy Stereogenic-at-Ru Carbenes: Stereochemical Inversion Through Olefin Metathesis and Other Pathways," J. Am. Chem. Soc. 2012, 134, 12438-12441. (b) Torker, S.; Khan, R. K. M. and Hoveyda, A. H. "The Influence of Anionic Ligands on Stereoisomerism of Ru Carbenes and Their Importance to Efficiency and Selectivity of Catalytic Olefin Metathesis Reactions," J. Am. Chem. Soc. 2014, 136, 3439-3455. Chapter 2. Highly Z- and Enantioselective Ring-Opening/Cross-Metathesis of Enol Ethers Through Curtin-Hammett Kinetics. The first instances of Z- and enantioselective Ru-catalyzed olefin metathesis are presented. Ring-opening/cross-metathesis (ROCM) reactions of oxabicyclic alkenes and enol ethers and a phenyl vinyl sulfide are promoted by 0.5-5.0 mol % of enantiomerically pure stereogenic-at-Ru complexes with an aryloxy chelate tethered to the N-heterocyclic carbene. Products are formed efficiently and with exceptional enantioselectivity (up to >98:2 enantiomer ratio). Surprisingly, the enantioselective ROCM reactions proceed with high Z selectivity (up to >98% Z). Moreover, reactions proceed with the opposite sense of enantioselectivity versus aryl olefins, which afford E- isomers exclusively. DFT calculations and deuterium-scrambling experiments, indicating fast interconversion between endo- and exo-Fischer carbene diastereomers, support a Curtin-Hammett situation. On this basis, models accounting for the stereoselectivity levels and trends are provided. Furthermore, the correlation of Fischer carbene character to the observed chemoselectivity in ROCM with enol ethers is also disclosed. Finally, a general proposal for the substrate-controlled Z selectivity in OM is also discussed. (a) Khan, R. K. M.; O'Brien, R. V.; Torker, S.; Li, B. and Hoveyda, A. H. "Z- and Enantioselective Ring-Opening Cross-Metathesis with Enol Ethers Catalyzed by Stereogenic-at-Ru Carbenes: Reactivity, Selectivity, and Curtin-Hammett Kinetics," J. Am. Chem. Soc. 2012, 134, 12774-12779. (b) Torker, S.; Koh, M. J.; Khan, R. K. M. and Hoveyda, A. H. "Origin of Z selectivity in Olefin Metathesis Reactions of Certain Terminal Alkenes Catalyzed by Typically E-Selective Ru Carbenes," manuscript submitted. Chapter 3. A New Class of Highly Efficient Ru Catalysts for Z-Selective Olefin Metathesis. Herein, we outline a general design for Z-selective OM, which led to the development of a new class of stereogenic-at-Ru carbene complexes (Ru4-9). Furthermore, we demonstrate that the newly developed dithiolate complexes Ru4b and Ru5 efficiently promote high activity and selectivity in ROMP reactions of norbornene and cyclooctene. Notably, the catechothiolate Ru4b catalyzes Z-selective ROCM with a broad scope of alkenes involving various functional groups (e.g., alcohols, enol ethers, vinyl sulfides, amides, heterocycles, and conjugated 1,3-dienes). More importantly, we disclose that the catecholate complex Ru4a is kinetically non-selective in OM and readily decomposes in the presence of mildly acidic moieties (e.g., alcohols and CDCl3). Subsequently, Ru9 is developed to efficiently promote highly Z-selective CM of a diol cross-partner with a wide range of alkene substrates. Most remarkably, the aforementioned protocol is employed in two natural product syntheses and the OM-based Z-selective cracking of oleic acid, which is unprecedented with existing Ru-carbenes and Mo/W-alkylidenes. (a) Khan, R. K. M.; Torker, S. and Hoveyda, A. H. "Readily Accessible and Easily Modifiable Ru-Based Catalysts for Efficient and Z-Selective Ring-Opening Metathesis Polymerization and Ring-Opening Cross-Metathesis," J. Am. Chem. Soc. 2013, 135, 10258-10261. (b) Koh, M. J.; Khan, R. K. M.; Torker, S. and Hoveyda, A. H. "Broadly Applicable Z- and Diastereoselective Ring-Opening/Cross-Metathesis Catalyzed By a Dithiolate Ru Complex," Angew. Chem., Int. Ed. 2014, 53, 1968-1972. (c) Khan, R. K. M. ; Torker, S. and Hoveyda, A. H. "Reactivity and Selectivity Differences Between Catecholate and Catechothiolate Ru Complexes. Implications Regarding Design of Stereoselective Olefin Metathesis Catalysts," J. Am. Chem. Soc. 2014, 136, 14337-14340. (d) Koh, M. J.; Khan, R. K. M.; Torker, S.; Yu, M.; Mikus, M. S. and Hoveyda, A. H. "Synthesis of High-Value Alcohols, Aldehydes and Acids by Catalytic Z-Selective Cross-Metathesis" manuscript submitted. / Thesis (PhD) — Boston College, 2014. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Chemistry.

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We consider the distribution of the zero and typical cells of a (homogeneous) Poisson hyperplane tessellation. We give a direct proof adapted to our setting of the well known Complementary Theorem. We provide sharp bounds for the tail distribution of the number of facets. We also improve existing bounds for the tail distribution of size measurements of the cells, such as the volume or the mean width. We improve known results about the generalised D.G. Kendall's problem, which asks about the shape of large cells. We also show that cells with many facets cannot be close to a lower dimensional convex body. We tacle the much less study problem of the number of facets and the shape of small cells. In order to obtain the results above we also develop some purely geometric tools, in particular we give new results concerning the polytopal approximation of an elongated convex body.