Any genus $g$ surface, $\Sigma_{g,n},$ with $n$ boundary components may be given a trinion decomposition: a realization of the surface as a union of $2g-2+n$ trinions glued together along $3g-3+n$ of their boundary circles. Together with the flows of Goldman, Jeffrey and Weitsman use the trinion boundary circles in a decomposition of $\Sigma_{g,n}$ to obtain a Hamiltonian action of a compact torus $(S^1)^{3g-3+n'} $ on an open dense subset of the moduli space of certain gauge equivalence classes of flat $SU(2)-$connections on $\Sigma_{g,n}.$ Jeffrey and Weitsman also provide a complete description of the moment polytopes for these torus actions, and we make use of this description to study the cohomology of associated toric varieties. While we are able to make use of the work of Danilov to obtain the integral (rational) cohomology ring in the smooth (orbifold) case, we show that the aforementioned toric varieties almost always possess singularities worse than those of an orbifold. In these cases we use an algorithm of Bressler and Lunts to recover the intersection cohomology Betti numbers using the combinatorial information provided by the corresponding moment polytopes. The main contribution of this thesis is a computation of the intersection cohomology Betti numbers for the toric varieties associated to trinion decomposed surfaces $\Sigma_{2,0},\Sigma_{2,1},\Sigma_{3,0}, \Sigma_{3,1}, \Sigma_{4,0},$ and $\Sigma_{4,1}.$

We define the phylogenetic model of a trivalent graph as a generalization of a binary symmetric model of a trivalent phylogenetic tree. If the underlining graph is a tree, the model has a parametrization that can be expressed in terms of the tree. The model is always a polarized projective toric variety. Equivalently, it is a projective spectrum of a semigroup ring. We describe explicitly the generators of this projective coordinate ring for graphs with at most one cycle. We prove that models of graphs with the same topological invariants are deformation equivalent and share the same Hilbert function. We also provide an algorithm to compute the Hilbert function, which uses the structure of the graph as a sum of elementary ones. Also, this Hilbert function of phylogenetic model of a graph with g cycles is meaningful for the theory of connections on a Riemann surface of genus g.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 80-84). Available also in a digital version from Dissertation Abstracts.

We consider the pair of a smooth complex projective variety together with an anti-canonical simple normal crossing divisor (we call it "log Calabi- Yau"). Standard examples are toric varieties together with their toric boundaries (we call them "toric pairs"). We provide a numerical criterion for a general log Calabi-Yau to be toric by an inequality between its dimension, Picard number and the number of boundary components. The problem originates in birational geometry and our proof is constructive, motivated by mirror symmetry. / text

Koreguoti kliniškai reikšmingą astigmatizmą būtina, nes visai nekoreguotas ar neteisingoje ašyje iškoreguotas astigmatizmas sukelia ne tik blogą matymą, bet ir astenopiją. Koreguoti kliniškai reikšmingą astigmatizmą būtina, nes visai nekoreguotas ar neteisingoje ašyje iškoreguotas astigmatizmas sukelia ne tik blogą matymą, bet astenopiją. Toriniai kontaktiniai lęšiai (TKL) suteikia galimybę labai tiksliai ištaisyti šią ydą, bet viena būtina sąlyga - jie turi būti stabilioje padėtyje. TKL stabilumas priklauso nuo vokų plyšio, ragenos formos, vokų prispaudimo jėgos, o taip pat nuo refrakcijos ydos dydžio ir astigmatizmo ašies. / N Scientific and Technical Progress century eyesight play an increasingly larger burden. To adjust clinically significant astigmatism is necessary, because completely uncorrected or wrong corrected axis of astigmatism cause not only poor vision, but asthenopia. Toric contact lenses (TKL) allows very precise remedy of this defect, but one necessary condition - they must be in a stable position. TKL stability depends on the eye crack opening, corneal shape, eyelid clamping force, as well as the refractive defect size and astigmatism axis.

In order to discuss topological properties of quaternionic toric 8-manifolds, we introduce the notion of an algebraic morphism in the category of toric spaces. We show that the classification of quaternionic toric 8-manifolds with respect to an algebraic isomorphism is finer than the oriented topological classification. We construct infinite families of quaternionic toric 8-manifolds in the same oriented homeomorphism type but algebraically distinct. To prove that the elements within each family are of the same oriented homeomorphism type, and that we have representatives of all such types of a quaternionic toric 8-manifold, we present and use a method of evaluating the first Pontrjagin class for an arbitrary quaternionic toric 8-manifold.

No description available.

A long-standing problem in cobordism theory has been to find convenient manifolds to represent cobordism classes. For example, in the late 1950's, Hirzebruch asked which complex cobordism classes can be represented by smooth connected algebraic varieties. This question is still open. Progress can be made on this and related problems by studying certain convenient connected algebraic varieties, namely smooth projective toric varieties. The primary focus of this dissertation is to determine which complex cobordism classes can be represented by smooth projective toric varieties. A complete answer is given up to dimension six, and a partial answer is described in dimension eight. In addition, the role of smooth projective toric varieties in the polynomial ring structure of complex cobordism is examined. More specifically, smooth projective toric varieties are constructed as polynomial ring generators in most dimensions, and evidence is presented suggesting that a smooth projective toric variety can be chosen as a polynomial generator in every dimension. Finally, toric varieties with an additional fiber bundle structure are used to study some manifolds in oriented cobordism. In particular, manifolds with certain fiber bundle structures are shown to all be cobordant to zero in the oriented cobordism ring.

Toric geometry studies manifolds M2n acted on effectively by a torus of half their dimension, Tn. Joyce shows that for such a 4-manifold sufficient conditions for a conformal class of metrics on the free part of the action to be self-dual can be given by a pair of linear ODEs and gives criteria for a metric in this class to extend to the degenerate orbits. Joyce and Calderbank-Pedersen use this result to find representatives which are scalar flat K¨ahler and self-dual Einstein respectively. We review some results concerning the topology of toric manifolds and the construction of Joyce metrics. We then extend this construction to give explicit complete scalar-flat K¨ahler and self-dual Einstein metrics on manifolds of infinite topological type, and to find a new family of Joyce metrics on open submanifolds of toric spaces. We then give two applications of these extensions — first, to give a large family of scalar flat K¨ahler perturbations of the Ooguri-Vafa metric, and second to search for a toric scalar flat K¨ahler metric on a neighbourhood of the origin in C2 whose restriction to an annulus on the degenerate hyperboloid {(z1, z2)|z1z2 = 0} is the cusp metric.

Gauge theories is one of the most pervasive and important subject of modern theoretical physics, and there are still many things about them we do not understand. In particular dealing with strongly coupled theories where normal perturbative techniques do not apply is a fundamental open problem. In this thesis, we study a particular class of toy-models that have supersymmetry, which makes them much easier to deal with. We employ the mathematical technique of localization, which for supersymmetric theories lets us evaluate certain path integrals exactly and for any value of the coupling. This is used to study the 5d N=1 theories placed on toric Sasaki-Einstein manifolds and compute their partition functions, finding that they factorize into a product of contributions from each closed Reeb orbit of the manifold. This computation leads us to define two new hierarchies of special functions associated to these manifolds, and we study their properties. Finally, we use the 5d N=1 theories to construct new 4d N=2 theories on a large class of curved backgrounds. These theories have some interesting features, such as supporting both instantons and anti-instantons, and having a position-dependent complexified coupling constant.