We consider a compact, oriented,smooth Riemannian manifold $M$ (with or without boundary) and wesuppose $G$ is a torus acting by isometries on $M$. Given $X$ in theLie algebra of $G$ and corresponding vector field $X_M$ on $M$, onedefines Witten's inhomogeneous coboundary operator $\d_{X_M} =\d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariantforms on $M$) and its adjoint $\delta_{X_M}$. First, Witten [35] showed that the resulting cohomology classeshave $X_M$-harmonic representatives (forms in the null space of$\Delta_{X_M} = (\d_{X_M}+\delta_{X_M})^2$), and the cohomologygroups are isomorphic to the ordinary de Rham cohomology groups ofthe set $N(X_M)$ of zeros of $X_M$. The first principal purpose isto extend Witten's results to manifolds with boundary. Inparticular, we define relative (to the boundary) and absoluteversions of the $X_M$-cohomology and show the classes haverepresentative $X_M$-harmonic fields with appropriate boundaryconditions. To do this we present the relevant version of theHodge-Morrey-Friedrichs decomposition theorem for invariant forms interms of the operators $\d_{X_M}$ and $\delta_{X_M}$; the proofinvolves showing that certain boundary value problems are elliptic.We also elucidate the connection between the $X_M$-cohomology groupsand the relative and absolute equivariant cohomology, followingwork of Atiyah and Bott. This connection is then exploited to showthat every harmonic field with appropriate boundary conditions on$N(X_M)$ has a unique corresponding an $X_M$-harmonic field on $M$to it, with corresponding boundary conditions. Finally, we define the interior and boundary portion of $X_M$-cohomology and then we definethe \emph{$X_M$-Poincar\' duality angles} between the interiorsubspaces of $X_M$-harmonic fields on $M$ with appropriate boundaryconditions.Second, In 2008, Belishev and Sharafutdinov [9] showed thatthe Dirichlet-to-Neumann (DN) operator $\Lambda$ inscribes into thelist of objects of algebraic topology by proving that the de Rhamcohomology groups are determined by $\Lambda$.In the second part of this thesis, we investigate to what extent is the equivariant topology of a manifold determined by a variant of the DN map?.Based on the results in the first part above, we define an operator$\Lambda_{X_M}$ on invariant forms on the boundary $\partial M$which we call the $X_M$-DN map and using this we recover the longexact $X_M$-cohomology sequence of the topological pair $(M,\partialM)$ from an isomorphism with the long exact sequence formed from thegeneralized boundary data. Consequently, This shows that for aZariski-open subset of the Lie algebra, $\Lambda_{X_M}$ determinesthe free part of the relative and absolute equivariant cohomologygroups of $M$. In addition, we partially determine the mixed cup product of$X_M$-cohomology groups from $\Lambda_{X_M}$. This shows that $\Lambda_{X_M}$ encodes more information about theequivariant algebraic topology of $M$ than does the operator$\Lambda$ on $\partial M$. Finally, we elucidate the connectionbetween Belishev-Sharafutdinov's boundary data on $\partial N(X_M)$and ours on $\partial M$.Third, based on the first part above, we present the(even/odd) $X_M$-harmonic cohomology which is the cohomology ofcertain subcomplex of the complex $(\Omega^{*}_G,\d_{X_M})$ and weprove that it is isomorphic to the total absolute and relative$X_M$-cohomology groups.

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

The nature of this paper is expository. The purpose is to present the fundamental material concerning actions of infinite discrete groups on the n-sphere and pseudo-Riemannian space forms based on the works of Gehring, Martin and Kulkarni and provide appropriate examples. Actions on the n-sphere split it into ordinary and limit sets. Assuming, additionally, that a group acting on the n-sphere has a certain convergence property, this thesis includes conditions for the existence of a homeomorphism between the limit set and the set of Freudenthal ends, as well as topological and quasiconformal conjugacy between convergence and Mobius groups. Since the certain pseudo-Riemannian space forms are diffeomorphic to non-compact spaces, the work of Hambleton and Pedersen gives conditions for the extension of discrete co-compact group actions on pseudo-Riemannian space forms to actions on the sphere. An example of such an extension is described. / Thesis / Master of Science (MSc)

This thesis concerns the study of the Lefschetz properties of artinian monomial algebras. An artinian algebra is said to satisfy the strong Lefschetz property if multiplication by all powers of a general linear form has maximal rank in every degree. If it holds for the first power it is said to have the weak Lefschetz property (WLP). In the first paper, we study the Lefschetz properties of monomial algebras by studying their minimal free resolutions. In particular, we give an afirmative answer to an specific case of a conjecture by Eisenbud, Huneke and Ulrich for algebras having almost linear resolutions. Since many algebras are expected to have the Lefschetz properties, studying algebras failing the Lefschetz properties is of a great interest. In the second paper, we provide sharp lower bounds for the number of generators of monomial ideals failing the WLP extending a result by Mezzetti and Miró-Roig which provides upper bounds for such ideals. In the second paper, we also study the WLP of ideals generated by forms of a certain degree invariant under an action of a cyclic group. We give a complete classication of such ideals satisfying the WLP in terms of the representation of the group generalizing a result by Mezzetti and Miró-Roig. / <p>QC 20180220</p>

<p>Existence of closed geodesics on compact manifolds was first proved by Lyusternik and Fet in the 1950s using Morse theory, and the corresponding problem for orbifolds was studied by Guruprasad and Haefliger, who proved existence of a closed geodesic of positive length in numerous cases. In this thesis, we develop an alternative approach to the problem of existence of closed geodesics on compact orbifolds by studying the geometry of group actions. We give an independent and elementary proof that recovers and extends the results of Guruprasad and Haefliger for developable orbifolds. We show that every compact orbifold of dimension 2, 3, 5 or 7 admits a closed geodesic of positive length, and we give an inductive argument that reduces the existence problem to the case of a compact developable orbifold of even dimension whose singular locus is zero-dimensional and whose orbifold fundamental group is infinite torsion and of odd exponent. Stronger results are obtained under curvature assumptions. For instance, one can show that infinite torsion groups do not act geometrically on simply connected manifolds of nonpositive or nonnegative curvature, and we apply this to prove existence of closed geodesics for compact orbifolds of nonpositive or nonnegative curvature. In the general case, the problem of existence of closed geodesics on compact orbifolds is seen to be intimately related to the group-theoretic question of finite presentability of infinite torsion groups, and we explore these and other properties of the orbifold fundamental group in the last chapter.</p> / Doctor of Philosophy (PhD)

Dans cette thèse des actions conformes de cohomogénéité un sur l'univers d'Einstein tridimensionel sont classifiées. Notre stratégie est d'établir dans un premier temps quel peut être le groupe de transformations conformes impliqué, à conjugaison près. Nous décrivons aussi la topologie et la nature causale des orbites d'une telle action. / In this thesis, the conformal actions of cohomogeneity one on the three-dimensional Einstein universe are classified. Our strategy in this study is to determine the representation of the acting group in the group of conformal transformations of Einstein universe up to conjugacy. Also, we describe the topology and the causal character of the orbits induced by cohomogeneity one actions in Einstein universe.

Nous étudions des propriétés de rigidité et des propriétés de non-rigidité forte d'actions de groupes sur des espaces Lp non-commutatifs. Récemment, des variantes de la propriété (T) de Kazhdan et de la propriété de point fixe (FH) ont été introduites, appelées respectivement propriété (TB) et propriété (FB), et énoncées en termes de représentations orthogonales sur un espace de Banach B. Nous nous intéressons au cas où B est un espace Lp non-commutatif Lp(M), associé à une algèbre de von Neumann M. Dans un premier temps, nous montrons qu'un groupe possédant la propriété (T) possède la propriété (TLp(M)) pour toute algèbre de von Neumann M. On en déduit que les groupes de rang supérieur ont la propriété (FLp(M)). Nous montrons que pour certaines algèbres, comme par exemple M=B(H), les propriétés (T) et (TLp(M) sont équivalentes. A l'opposé, nous caractérisons les groupes possédant la propriété (Tlp), et montrons que cette classe de groupes est strictement plus grande que celle avec la propriété (T). Dans un second temps, nous introduisons des variantes de la propriété (H) de Haagerup, les propriétés (HLp(M)) et l' a-FLp(M)-menabilité, définies en termes d'actions sur l'espace Lp(M). Nous décrivons les liens entre la propriété (H) et sa variante (HLp(M)) suivant l'algèbre M considérée. Nous montrons que les groupes possédant (H) sont a-FLp(M)-menables pour certaines algèbres M, comme par exemple le facteur II infini hyperfini. / We studied rigidity properties and strong non-rigidity properties for group actions on non-commutative Lp spaces. Recently, variants of Kazhdan's property (T) and fixed-point property (FH) were introduced, respectively called property (TB) and property (FB), and described in terms of orthogonal representations on a Banach space B. We are interested in the case where B is a non-commutative Lp space Lp(M), associated to a von Neumann algebra M. In a first part, we show that if a group has property (T), then it has property (TLp(M)) for any von Neumann algebra M. We deduce that higher rank groups have property (FLp(M)). We show that for some algebras, such as M=B(H), properties (T) and (TLp(M)) are equivalent. By contrast, we characterize groups with property (Tlp), and show that this class of groups is larger than the one with property (T). In a second part, we introduce variants of the Haagerup property (H), namely properties (HLp(M)) and a-FLp(M)-menability, defined in terms of actions on the space Lp(M). We describe relationships between property (H) and its variant (HLp(M)) for different algebras M. We show that groups with property (H) are a-FLp(M)-menable for some algebras M, such as the hyperfinite II infinite factor.

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Tropical geometry is algebraic geometry over the tropical semiring, or min-plus algebra. In this thesis, I discuss the basic geometry of plane tropical curves. By introducing the notion of abstract tropical curves, I am able to pass to a more abstract metric-topological setting. In this setting, I discuss divisors on tropical curves. I begin a study of $G$-invariant divisors and divisor classes.

This  thesis  consists  of two papers  and  a summary.  The  papers  both  deal with  affine algebraic complex  varieties,  and  in particular such  varieties  in dimension  three  that have a non-trivial action  of one of the  one-dimensional  algebraic  groups  Ga   :=  (C, +) and  Gm  :=  (C*, ·).  The methods  used  involve  blowing up  of subvarieties, the correspondances between  Ga - and  Gm - actions  on an affine variety  X with locally nilpotent derivations  and Z-gradings  respectively  on O(X) and passing from a filtered algebra  A to its associated graded  algebra  gr(A). In Paper  I, we study  Russell’s hypersurface  X , i.e. the affine variety  in the affine space A4 given by the equation  x + x2y + z3 + t2 = 0. We reprove by geometric means Makar-Limanov’s result which states  that X is not isomorphic to A3 – a result which was crucial to Koras-Russell’s proof of the linearization conjecture  for Gm -actions on A3. Our method consist in realizing X as an open part  of a blowup M  −→ A3 and to show that each Ga -action on X descends to A3 . This follows from considerations of the graded  algebra  associated to O(X ) with respect  to a certain filtration. In Paper  II, we study  Ga-threefolds X  which have  as their  algebraic  quotient  the  affine plane  A2  = Sp(C[x, y]) and  are a principal  bundle  above the  punctured plane  A2  :=  A2 \ {0}. Equivalently, we study  affine Ga -varieties  Pˆ  that extend  a principal  bundle  P over A2, being P together  with an extra  fiber over the origin in A2. First  the trivial  bundle  is studied,  and some examples of extensions  are given (including  smooth  ones which are not isomorphic  to A2 × A). The  most  basic among  the  non-trivial  principal  bundles  over A2 is SL2 (C)  −→ A2, A  1→  Ae1 where e1  denotes  the first unit  vector,  and we show that any non-trivial  bundle  can be realized as a pullback  of this  bundle  with  respect  to  a morphism  A2  −→ A2. Therefore  the  attention is then  restricted to extensions  of SL2(C)  and  find two families of such extensions  via a study of the  graded  algebras  associated  with  the  coordinate  rings  O(Pˆ)  '→ O(P ) with  respect  to  a filtration  which is defined in terms  of the Ga -actions  on P and Pˆ  respectively.