It is known that the ring of invariants of any two-row group is Cohen-Macaulay. This result inspired the conjecture that the ring of invariants of any two-row group is a complete intersection. In this thesis, we study this conjecture in the case where the ground field is the prime field $\mathbb{F}_p$. We prove that all Abelian reflection two-row $p$-groups have complete intersection invariant rings. We show that all two-row groups with \textit{non-normal} Sylow $p$-subgroups have polynomial invariant rings. We also show that reflection two-row groups with \textit{normal} reflection Sylow $p$-subgroups have polynomial invariant rings. As an interesting application of a theorem of Nakajima about hypersurface invariant rings, we rework a classical result which says that the invariant rings of subgroups of $\text{SL}(2,\,p)$ are all hypersurfaces. In addition, we obtain a result that characterizes Nakajima $p$-groups in characteristic $p$, namely, if the invariant ring is generated by norms, then the group is a Nakajima $p$-group. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-29 15:08:40.705

Prompted by the success of heterotic line bundle model building on Complete Intersection Calabi Yau (CICY) manifolds and the new developments regarding a generalization thereof, I analyze the possibility of model building on generalized CICY (gCICY) manifolds.  Ultimately this is realized on two examples of gCICYs, one of which topologically equivalent to a CICY and one inequivalent to any previously studied examples.  The first chapter is dedicated to reporting background information on CICYs and gCICYs.  The mathematical machinery of CICYs and their generalizations are introduced alongside explicit constructions of two examples.  The second chapter introduces the reader to heterotic line bundle model building on CICYs and gCICYs.  In the setting of gCICYs, similar to regular CICYs, model building is accomplished in two steps: first the larger $E_{8}$ gauge group is broken to an $SU( 5 )$ grand unified theory  through a line bundle model.  Then the GUT is broken using Wilson line symmetry breaking, for which the presence of a freely acting discrete symmetry must be established.  To that end, I proceed to show that the two previous examples benefit from a $\mathbb{Z}_{2}$ freely acting discrete symmetry.  Utilizing this symmetry I construct 20 and 11 explicit models for the two gCICY examples respectively, by scanning over a finite range of line bundle charges. / Ett av de största problemen i modern teoretisk fysik är att hitta en teori för kvantgravitation.För en konsekvent kvantteori gravitation skulle vara en väsentlig del i fysikens pussel, och koppla samman gravitationsfysiken för planeter och galaxer, som beskrivs av allmänna relativitetsteorin, till fysiken för partiklar, beskrivet av kvantfältteori.Bland de mest lovande teorierna finns strängteorin som föreslår att ersätta partiklar med strängar som materiens grundläggande beståndsdel.Förutom att lösa kvantgravitationproblemet hoppas teoretiska fysiker genom strängteorin att förenkla beskrivningen av partikelfysik.Detta skulle ske genom att ersätta hela partikelzoo med ett enda objekt: strängen.Olika vibrationer i strängen skulle motsvara olika partiklar och interaktioner mellan strängar skulle motsvara interaktioner mellan partiklar.För att vara motsägelsefri kräver dock strängteori att det finns minst sex fler dimensioner än de vi kan uppleva.En av strategierna som för närvarande studeras för att förlika extra dimensioner med och moderna experiment kallas ``kompaktifiering'' eller ``compactification'' på engelska.Strategin föreslår att dessa extra dimensioner ska vara kompakta och så små att de är osynliga för observationer.Interesant nog påverkar geometrin i det sexdimensionella kompakta rummet i stor utsträckning fysiken som strängteorin producerar: olika rum skulle producera olika partiklar och olika grundläggande naturkrafter.I den här uppsatsen studerar jag två exempel på sådana sexdimensionella rum som kommer från en uppsättning av rum som kallas `` generaliserade CICYs'' som nyligen har upptäckts.Med hjälp av de tekniker som liknar de som har utvecklats för andra liknade rum, visar jag att vissa aspekter av en strängteori kompaktifierad på generaliserade CICY återspeglar de som mäts genom moderna partikelfysikexperiment.

Of many modern constructions in geometry Calabi Yau manifolds hold special relevance in theoretical physics. These manifolds naturally arise from the study of compactification of certain string theories. In particular Calabi Yau manifolds of dimension three, commonly known as threefolds, are widely used for compactifications of heterotic string theories. Among the many constructions, that of complete intersection Calabi Yau manifolds (CICY) is generally regarded to be the simplest. Furthermore, CICY threefolds have been proven to exist only in finite number. In the following text CICY manifolds will be analyzed, with particular attention to threefolds. A general description of some of their topological quantities and their calculation is offered. Lastly, a proof of the finiteness of CICY threefolds is given.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18766号 / 理博第4024号 / 新制||理||1580(附属図書館) / 31717 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 伊藤 哲史, 教授 上田 哲生, 教授 雪江 明彦 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

My thesis is mostly concerned with algebraic and combinatorial aspects of the theory of hyperplane arrangements. More specifically, I study the Orlik-Terao algebra of a hyperplane arrangement and the broken circuit complex of a matroid. The Orlik-Terao algebra is a useful tool for studying hyperplane arrangements, especially for characterizing some non-combinatorial properties. The broken circuit complex, on the one hand, is closely related to the Orlik-Terao algebra, and on the other hand, plays a crucial role in the study of many combinatorial problem: the coefficients of the characteristic polynomial of a matroid are encoded in the f-vector of the broken circuit complex of the matroid. Among main results of the thesis are characterizations of the complete intersection and Gorenstein properties of the broken circuit complex and the Orlik-Terao algebra. I also study the h-vector of the broken circuit complex of a series-parallel network and relate certain entries of that vector to ear decompositions of the network. An application of the Orlik-Terao algebra in studying the relation space of a hyperplane arrangement is also included in the thesis.