Soit X une courbe projective lisse et irréductible munie d'une involution σ. Dans cette thèse, nous étudions les fibrés vectoriels invariants and anti-invariants sur X sous l'action induite par σ. On introduit la notion de modules σ-quadratiques et on l'utilise, avec GIT, pour construire ces espaces de modules, puis on en étudie certaines propriétés. Ces espaces de modules correspondent aux espaces de modules de G-torseurs parahoriques sur la courbe X/σ , pour certains schémas en groupes parahoriques G de type Bruhat-Tits, qui sont twistés dans le cas des anti-invariants. Nous développons les systèmes de Hitchin sur ces espaces de modules et on les utilise pour dériver une classification de leurs composantes connexes en les dominant par des variétés de Prym. On étudie aussi le fibré déterminant sur les espaces de modules des fibrés vectoriels anti-invariants. Dans certains cas, ce fibré en droites admet certaines racines carrées appelées fibrés Pfaffiens. On montre que les espaces des sections globales des puissances de ces fibrés en droites (les espaces des fonctions thêta généralisées) peuvent être canoniquement identifier avec les blocs conformes associés aux algèbres de Kac-Moody affines twistées de type A(2). / Let X be a smooth irreducible projective curve with an involution σ. In this dissertation, we studythe moduli spaces of invariant and anti-invariant vector bundles over X under the induced action of σ. We introduce the notion of σ-quadratic modules and use it, with GIT, to construct these moduli spaces, and than we study some of their main properties. It turn out that these moduli spaces correspond to moduli spaces of parahoric G-torsors on the quotient curve X/σ, for some parahoric Bruhat-Tits group schemes G, which are twisted in the anti-invariant case.We study the Hitchin system over these moduli spaces and use it to derive a classification of theirconnected components using dominant maps from Prym varieties. We also study the determinant of cohomology line bundle on the moduli spaces of anti-invariant vector bundles. In some cases this line bundle admits some square roots called Pfaffian of cohomology line bundles. We prove that the spaces of global sections of the powers of these line bundles (spaces of generalized theta functions) can be canonically identified with the conformal blocks for some twisted affine Kac-Moody Lie algebras of type A(2).

In the present work, we study the motion of homogeneous, isotropic, incompressible fluid with viscosity depending on the pressure. The motion is studied in a infinite domain under a surface load by a prescribed pressure on one of the boundaries. This so called free boundary is due to the exerted pressure deformed and is also subject of study. After an acceptable simplification and prescribing suitable boundary conditions, the problem is solved at first analytically using the perturbation method, then numerically by means of the spectral methods. The effort is to observe any differences in the motion in comparison to commonly used Newtonian fluid.

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.

We provide explicit equations for moduli spaces of Drinfeld shtukas over the projective line with Γ(N), Γ_1(N) and Γ_0(N) level structures, where N is an effective divisor on P^1 . If the degree of N is big enough, these moduli spaces are relative surfaces. We study how the moduli space of shtukas over P^1 with Γ_0(N) level structure, Sht^{2,tr}(Γ_0(N)), can be used to provide a notion of motivic modularity for elliptic curves defined over function fields. Elliptic curves over function fields are known to be modular in the sense of admitting a parametrization from a Drinfeld modular curve, provided that they have split multiplicative reduction at one place. We conjecture a different notion of modularity that should cover the curves excluded by the reduction hypothesis. We use our explicit equations for Sht^{2,tr}(Γ_0(N)) to verify our modularity conjecture in the cases where N = 2(0) + (1) + (∞) and N = 3(0) + (∞).

京都大学 / 0048 / 新制・論文博士 / 博士(理学) / 乙第12910号 / 論理博第1546号 / 新制||理||1590(附属図書館) / 32120 / ソウル大学大学院数学科 / (主査)教授 中島 啓, 教授 小野 薫, 教授 向井 茂 / 学位規則第4条第2項該当 / Doctor of Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・論文博士 / 博士(理学) / 乙第13201号 / 論理博第1560号 / 新制||理||1635(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 宍倉 光広, 教授 泉 正己, 教授 國府 寛司 / 学位規則第4条第2項該当 / Doctor of Science / Kyoto University / DFAM

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In this thesis we study symmetries of quantum field theory visible only at the non-perturbative level, which arise from large deformations of the integration contour in the path integral. We exposit the recently-developed theory of qq-characters that organizes such symmetries in the case of N = 2 supersymmetric gauge theories in four dimensions. We sketch the physical origin of such observables from intersecting branes in string theory, and the mathematical origin as certainequivariant integrals over Nakajima quiver varieties. We explain some of the main applications, including the derivation of Seiberg-Witten geometry for quiver gauge theories and the relations to quantum integrable systems.

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