In the three main sections of this thesis (chapters II, III, and IV; chapter I consists of definitions) we explore three methods of studying Alexander polynomials of links which are alternatives to Fox' free differential calculus. In chapter II we work directly with a presentation of the link group and show how to obtain a presentation for the Alexander invariant. From this we deduce that the order ideal of the Alexander invariant is principal for links of two or three components (the case of one component is well known) but nonprincipal in general for links of four or more components. In any event we show that only one determinant is needed to obtain the Alexander polynomial. In chapter III we use surgery techniques to characterize Alexander invariants of links of two components in terms of their presentation matrices. We then use this to show that the Torres conditions characterize link polynomials when the linking number of the two components is zero or both components are unknotted and the linking number is two. Chapter IV uses Seifert surfaces to prove a generalization of a theorem of Kidwell which relates the individual degrees of the Alexander polynomial to the linking complexity, to present an algorithm for calculating the Alexander polynomial of a two-bridge link from a two-bridge diagram and to prove a conjecture of Kidwell in the special case of two-bridge links. These results are then used to generate link polynomials from allowable pairs (a concept introduced in chapter III) and these results in turn are used to produce methods of generating allowable pairs. / Science, Faculty of / Mathematics, Department of / Graduate

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In this paper we examine the effectiveness of reducing the second order radial equation, of the hyperspherical coordinate solution to the two-electron Schrodinger equation, into a set of coupled first order linear equations as suggested by Klar. All results have been obtained in a completely nonadiabatic formalism thereby ensuring accuracy. We arrive at the conclusion that our application of the reduction process is in some way inconsistent and suggest a possible resolution to this anomaly.

Toric varieties are a class of geometric objects with a combinatorial structure encoded in polytopes. P1 x P2 is a well known variety and its polytope is the triangular prism. Studying the symmetries of the triangular prism and its truncations can lead to symmetries of the variety. Many of these symmetries permute the elements of the cohomology ring nontrivially and induce nontrivial relations. We discuss some toric symmetries of P1 x P2, and describe the geometry of the polytope of the corresponding blowups, and analyze the induced action on the cohomology ring. We exhaustively compute the toric symmetries of P1 x P2.

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Le développement de nouvelles stratégies d’immunothérapie représente de nos jours un enjeu majeur de santé publique. Dans ce contexte, les lymphocytes T Natural Killer invariant (iNKT) exercent de puissantes activités immuno-modulatrices. Ces cellules ont la particularité de reconnaitre par l’intermédiaire de leur récepteur T (TCR), des (glyco) lipides, présentés par la molécule CD1d exprimée par les cellules présentatrices d’antigènes (APC), notamment les cellules dendritiques (DC). Initialement découvert à partir d’une éponge marine pour ses activités anti-métastatiques, l’alpha-galactosylcéramide (a-GalCer ou KRN) induit rapidement une sécrétion massive, par les cellules iNKT, de cytokines immunomodulatrices telles l’IFN-g, conduisant à la transactivation de nombreuses cellules immunitaires, notamment les cellules Natural Killer (NK), les DC ou encore les lymphocytes Tgd. Cette propriété unique permet aux cellules iNKT de contrôler, chez la souris, le développement des réponses immunes, notamment la réponse anti-tumorale. Face à ces résultats encourageants, des essais cliniques chez l’homme ont été réalisés mais les résultats se sont avérés décevants. Actuellement, il existe deux stratégies pour cibler un antigène à une population cellulaire particulière : 1) le ciblage passif, basé sur la taille des particules ou leur composition et 2) le ciblage actif, basé sur les fortes interactions anticorps/antigène ou ligand/récepteur. Nous avons testé les deux types de ciblages et avons utilisé le même polymère pour constituer nos vecteurs : le PLGA ou poly(lactic coglycolic acid). Dans une première étude (ciblage passif), nous avons comparé l’efficacité de l’encapsulation de l’a-GalCer dans des particules de tailles différentes: des nano (NP) et microparticules (MP). L’a-GalCer vectorisé dans les NP et les MP est endocyté par les DC (voie des clathrines) et active les cellules iNKT in vitro et in vivo mais ne peuvent empêcher leur anergie. Ces résultats décevants nous ont conduits à opter pour le ciblage actif des DC. Dans un premier temps, étant donné les études controversées sur le rôle des DC dans l’anergie des cellules iNKT, nous avons revisité l’implication de ces dernières dans la primo-activation/anergie des cellules iNKT. Nous confirmons ainsi le rôle primordial de DC dans la primo-activation des cellules iNKT mais surtout, nous montrons qu’elles n’induisent pas leur anergie. Représentant une population hétérogène, nous montrons que parmi les DC, les DC CD8a+ sont de puissantes activatrices des cellules iNKT. Nos résultats nous ont ainsi menés à délivrer spécifiquement l’a-GalCer aux DC CD8a+. Pour cela, nous avons greffé sur les NP de PLGA un anticorps anti-DEC205, récepteur lectinique fortement exprimé par les DC CD8a+. In vitro et in vivo, les NP/DEC205/a-GalCer induisent une plus forte activation des cellules iNKT comparativement à l’a-GalCer libre. De même, la co-délivrance d’a-GalCer et d’ovalbumine (OVA) au sein des DC CD8a+ améliore les propriétés adjuvantes de l’a-GalCer en induisant des réponses humorale et cellulaire (lymphocytes T CD8+) spécifiques de l’OVA plus importantes comparativement à la délivrance des deux composés sous leur forme libre. Finalement, de façon intéressante, nous montrons que suite à une primo-activation par les NP/DEC205/a-GalCer, les cellules iNKT sont capables de répondre de nouveau à une seconde stimulation, traduisant l’absence d’anergie des cellules iNKT. En conclusion, nos résultats indiquent que la délivrance spécifique de l’a-GalCer aux DC CD8a+ amplifie la primo-activation des cellules iNKT tout en évitant la mise en place du phénomène d’anergie et ouvrent de nombreuses perspectives dans le cadre de thérapies anti-tumorales et anti-infectieuses. / Nowdays, the development of new immunotherapy strategies represent a major issuein public health. In this context, the invariant Natural Killer T lymphocytes (iNKT) have strongimmunomodulatory properties. This cell population recognizes (glycol)lipid presented by theCD1d molecule expressed by antigen presenting cell (APC) as dendritic cells (DC). Initiallyfound in a marine sponge for its anti-metastatic activities, alpha-galactosylceramide (-GalCer or KRN) induces a massive cytokine production (IFN-, IL-4, IL-17) by iNKT cells.This cytokine burst lead to downstream activation of numerous immune cells like naturalkiller cell (NK), DC or CD8+ T cells. Through this property, iNKT cells regulate numerousimmune responses, including anti-tumoral response. Based on encouraging results in themouse model, -GalCer has been used in anti-tumour therapy in human. Although the drugwas well tolerated, no or moderate clinical responses were observed in patients repeatedlyinoculated with -GalCer. As observed in the mouse system, one potential explanation forthis disappointing observation may lie in the induction of a long-term anergy of human iNKTcells, thus preventing cytokine release upon a recall stimulation. Although controversial,various studies suggest that this phenomonen should be due to a lack of delivery of -GalCer into dendritic cells (DC) and so its presentation by non adequate antigen presentingcell (APC) as B cells. The objective of our work was to optimize -GalCer activity by avoidingiNKT anergy using vectorisation approach. Actually, there are 2 strategies using nanotechnologies to target an antigen to a specific cell population: 1) passive targeting based on the size of the particles, their composition and their surface charge and 2) active targeting based on the strong interactions between an antibody and its antigen or a ligand and its receptor. We have tested the two strategies and therefore we use the same composition of our particles: PLGA or poly(lactic co glycolic acid). This biodegradable and biocompatible molecule is already used in therapy.In a first study (passive targeting), we compared the efficiency of -GalCer encapsulation inparticles with different size: nano (NP) and microparticle (MP). Vectorised -GalCer in NPand MP rapidly activates iNKT cells in vitro. Both type of particles are uptake by DC via aclathrin dependent mechanism. In in vivo approaches, NP/-GalCer and MP/-GalCeractivate iNKT cells but unfortunately could not prevent iNKT cell anergy. These disappointingresults led us to use an active targeting. In first time, because of controversial role of DC iniNKT anergy, we have revisited the role of DC in iNKT primo-activation and anergy. Weconfirm the primordial role of DC in iNKT primo-activation but especially we show that DC donot induce iNKT anergy. DC are heterogeneous and we show that among DC, CD8a+ DCsubpopulation are potent iNKT cells activation. Our results led us to deliver specifically a-GalCer to CD8+ DC. For this, anti-DEC205 antibodies were covalently linked to the surfaceof PLGA NP, DEC205 being highly expressed by CD8+ DC. In vitro and in vivo,NP/DEC205/-GalCer induce a stronger iNKT cell activation relative to free -GalCer (or NPIgG/-GalCer). Moreover, -GalCer and ovalbumin co-delivery in CD8+ DC improve -GalCer adjuvanticity leading to more important humoral and cellular responses. Interestingly,after a primo-activation by NP/DEC205/-GalCer, iNKT cells are able to respond to secondstimulation thus avoiding iNKT cell anergy.In conclusion, our results indicate that specific -GalCer delivery to CD8+ DC improve iNKT cells primo-activation and avoid anergy phenomenon. These findings open several perspectives in anti-tumoral and anti-infectious therapies.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 163-165. / Ph.D.

Chan, Kai Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 106-109). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.7 / Chapter I --- Background Scenery --- p.10 / Chapter 1 --- Seiberg-Witten Invariants --- p.11 / Chapter 1.1 --- Preliminaries --- p.11 / Chapter 1.2 --- Construction of Seiberg-Witten Invariants --- p.17 / Chapter 1.2.1 --- Seiberg-Witten Equations and the Moduli Space --- p.17 / Chapter 1.2.2 --- Seiberg-Witten Invariants --- p.19 / Chapter 1.2.3 --- Remarks --- p.20 / Chapter 1.2.4 --- Seiberg-Witten Invariants for b+2= 1 --- p.22 / Chapter 1.3 --- Important Results of Seiberg-Witten Invariants --- p.23 / Chapter 1.3.1 --- Manifolds Admit Positive Scalar Metrics --- p.23 / Chapter 1.3.2 --- Connected Sums --- p.25 / Chapter 1.3.3 --- Kahler Surfaces --- p.27 / Chapter 1.3.4 --- Symplectic Manifolds --- p.30 / Chapter 2 --- Intersection Forms --- p.32 / Chapter 2.1 --- Intersection Forms of 4-manifolds --- p.32 / Chapter 2.2 --- Classification Theorem --- p.34 / Chapter 2.3 --- Review: Van Kampen's Theorem --- p.35 / Chapter 3 --- Kirby Calculus --- p.37 / Chapter 3.1 --- Review on Handle Decompositions --- p.37 / Chapter 3.1.1 --- Constructions --- p.38 / Chapter 3.1.2 --- Handle Slides and Cancellations --- p.42 / Chapter 3.1.3 --- Calculation of Homology Groups --- p.44 / Chapter 3.2 --- Kirby Diagrams --- p.45 / Chapter 3.2.1 --- Constructions --- p.45 / Chapter 3.2.2 --- Handle Slides and Cancellations --- p.50 / Chapter 3.2.3 --- Dotted Notation for 1-handles --- p.56 / Chapter 3.3 --- 3-Manifolcis: As Boundaries of 4-Manifolds --- p.60 / Chapter 3.3.1 --- Introduction --- p.60 / Chapter 3.3.2 --- Lens spaces --- p.62 / Chapter 3.4 --- Linear Plumbing --- p.63 / Chapter 3.5 --- Rational Blowdown --- p.65 / Chapter II --- Examples of Exotic Structures --- p.71 / Chapter 4 --- mCP2#kCP2 --- p.72 / Chapter 4.1 --- Introduction --- p.72 / Chapter 4.2 --- Example: CP2#7CP2 --- p.73 / Chapter 4.3 --- Progress of Researches --- p.85 / Chapter 5 --- Gluing Results in Seiberg-Witten Theory --- p.89 / Chapter 5.1 --- Revisit of Seiberg-Witten Invariants --- p.89 / Chapter 5.2 --- Fiber Sums and its Generalization --- p.91 / Chapter 5.3 --- Logarithmic transformations and its Generalization --- p.93 / Chapter 5.4 --- Knot Theory and Alexander Polynomials --- p.96 / Chapter 5.5 --- Main Theorem --- p.102 / Bibliography --- p.106

1In this dissertation we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. To impose the symmetry at the quantum level, on both the connection and its conjugate momentum, the method used necessarily has similiarities to the Gupta-Bleuler method of quantizing the electromagnetic field. Lastly, a strategy for embedding states of loop quantum cosmology into this new homogeneous isotropic sector, and using this embedding to compare the physics, is presented. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection