The prehistory of Algebraic Topology dates back to Euler, Riemann and Betti, who started the idea of attaching various invariants to a topological space. With his simplicial (co)homology theory, Poincaré was the first to give an instance of what in modern terms we would call a contravariant functor H° from the category of (sufficiently nice) topological spaces to the category of cyclic complexes of abelian groups.

In this thesis we investigate the moduli space M3[2] of curves of genus 3 equipped with a symplectic level 2 structure. In particular, we are interested in the cohomology of this space. We obtain cohomological information by decomposing M3[2] into a disjoint union of two natural subspaces, Q[2] and H3[2], and then making S7- resp. S8-equivariantpoint counts of each of these spaces separately. / Målet med denna uppsats är att undersöka modulirummet M3[2] av kurvor av genus 3 med symplektisk nivå 2 struktur. Mer specifikt vill vi hitta informationom kohomologin av detta rum. För att uppnå detta delar vi först upp M[2] i en disjunkt union av två naturliga delrum, Q[2] och H3[2], och räknar därefter punkterna av dessa rum S7- respektive S8-ekvivariant.

Im Mittelpunkt der Untersuchungen stehen Grundlagen für äquivariante motivische Homotopietheorie. Für eine neue Grothendieck-Topologie auf einer Kategorie von äquivarianten glatten k-Schemata werden unstabile und stabile motivische Homotopietheorie entwickelt. Im zweiten Teil der Arbeit wird als Anwendung der stabilen Theorie eine Adams-Spektralsequenz mit motivischer Borel-Kohomologie konstruiert.

Toric face rings are a generalization of Stanley-Reisner rings and affine monoid rings. New problems and results are obtained by a systematic study of toric face rings, shedding new lights to the understanding of Stanley-Reisner rings and affine monoid rings. We study algebra retracts of Stanley-Reisner rings, in particular, classify all the $\mathbb{Z}$-graded algebra retracts. We consider the Koszul property of toric face rings via Betti numbers and properties of the defining ideal. The last chapter is devoted to local cohomology of seminormal toric face rings and applications to singularities of toric face rings in positive characteristics.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18044号 / 理博第3922号 / 新制||理||1566(附属図書館) / 30902 / 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 浅岡 正幸, 教授 加藤 毅, 教授 藤原 耕二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19259号 / 理博第4114号 / 新制||理||1592(附属図書館) / 32261 / 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 入谷 寛, 教授 加藤 毅, 教授 吉川 謙一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20885号 / 理博第4337号 / 新制||理||1623(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 望月 拓郎, 教授 中島 啓, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

This thesis consists of two papers with somewhat different flavours. In Paper I we compute the étale cohomology ring H^*(X,Z/nZ) for X the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim. We also give examples of two distinct number fields whose rings of integers have isomorphic cohomology groups but distinct cohomology ring structures. In Paper II we define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2, 1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a stacky building datum comes from a ramified cover for a finite abelian group scheme over k, generalizing a result of Biswas–Borne. / Denna avhandling består av två artiklar som skiljer sig något i karaktär. I Artikel I beräknar vi den étala kohomologiringen H^*(X,Z/nZ) då X är ringen av heltal av en talkropp K. Som en tillämpning, ger vi ett kriterium i form av en formel för när en invariant definierad av Minhyong Kim är noll eller ej. Vi ger också exempel på två olika talkroppar vars ringar av heltal har isomorfa kohomologigrupper men olika kohomologiringstrukturer. I Artikel II definierar vi stackig byggnadsdata för stackiga övertäckningar i Pardinis anda och visar en ekvivalens av (2,1)-kategorier mellan kategorin av stackiga övertäckningar och kategorin av stackig byggnadsdata. Vi visar att varje stackig övertäckning är en platt rotstack i Olsson och Borne–Vistolis mening och vi ger en intrinsisk beskrivning av den som en rotstack med hjälp av stackig byggnadsdata. När basen S är definierad över en kropp, ger vi ett kriterium för när ett stackigt byggnadsdatum kommer från en ramifierad övertäckning för ett ändligt abelskt gruppschema över k. Detta generaliserar ett resultat av Biswas–Borne.

[pt] O estudo de deformações e degenerações de estruturas de Poisson ocupa posição especial dentro do marco clássico de análise de degenerações de estruturas geométricas. Nesta tese como resultado principal construímos uma deformação não trivial na qual a estrutura quadrática canônica do espaço projetivo complexo n-dimensional é limite contínuo de estruturas Kahlerianas. Além disso, como resultado segundário de estudos de deformações mostramos que uma estrutura Poisson invariante numa variedade tórica com número finito de folhas não pode ser exata na cohomologia Poisson. Nosso estudo também inclui considerações sobre cohomologia Poisson da estrutura quadrática canônica do espaço vetorial complexo n-dimensional. / [en] The study of deformations and degenerations of Poisson structures occupies a special position within the classical framework of analysis of degenerations of geometric structures. In this thesis as the main result we build a non-triavial deformation in which the canonical quadratic structure in CP(n) is a continuous limit of Kahlerian structures. Furthermore, as a secondary result of deformation studies we have shown that an invariant Poisson structure in a toric variety with finite number of leaves cannot be exact in Poisson cohomology. Our study also includes considerations about Poisson cohomology of the canonical quadratic structure of C(n).

This thesis deals with the study of Schubert varieties, which are subsets of flag varieties indexed by elements of Weyl groups. We start by defining Lascoux elements in the Hecke algebra, and showing that they coincide with the Kazhdan-Lusztig basis elements in certain cases. We then construct a resolution (Zw, π) of the Schubert variety Xw for which Rπ*(C[l(w)]) is a sheaf on Xw whose expression in the Hecke algebra is closely related to the Lascoux element. We also define two new polynomials which coincide with the intersection cohomology Poincar\'e polynomial in certain cases. In the final chapter, we discuss some interesting combinatorial results concerning Bell and Catalan numbers which arose throughout the course of this work.