In this thesis we construct an analogue in tropical geometry for a class of Schubert varieties from classical geometry. In particular, we look at the collection of tropical lines contained in the fan tropical plane. We call these tropical spaces "tropical Schubert prevarieties", and develop them after creating a tropical analogue for flag varieties that we call the "flag Dressian". Having constructed this tropical analogue of Schubert varieties we then determine that the 2-skeleton of these tropical Schubert prevarieties is realizable. In fact, as long as the lift of the fan tropical plane is in general position, only the 2-skeleton of the tropical Schubert prevariety is realizable. / text

This thesis describes a geometric approach to integrable systems. In the first part we describe the geometry of Drinfeld--Sokolov integrable hierarchies including the corresponding tau-functions. Motivated by a relation between Drinfeld--Sokolov hierarchies and certain physical partition functions, we define a dispersionless limit of Drinfeld--Sokolov systems. We introduce a class of solutions which we call string solutions and prove that the tau-functions of string solutions satisfy Virasoro constraints generalizing those familiar from two-dimensional quantum gravity. In the second part we explain how procedures of Hamiltonian and quasi-Hamiltonian reductions in symplectic geometry arise naturally in the context of shifted symplectic structures. All constructions that appear in quasi-Hamiltonian reduction have a natural interpretation in terms of the classical Chern-Simons theory that we explain. As an application, we construct a prequantization of character stacks purely locally. / text

This thesis begins a study of principal series categories in geometric representation theory using the Beilinson-Drinfeld theory of chiral algebras. We study Whittaker objects in the unramified principal series category. This provides an alternative approach to the Arkhipov-Bezrukavnikov theory of Iwahori-Whittaker sheaves that exploits the geometry of the Feigin-Frenkel semi-infinite flag manifold. / Mathematics

No description available.

Les deux premières parties de cette thèse s'inscrivent dans le contexte des analogies entre l'irrégularité pour les connexions méromorphes et la ramification sauvage des faisceaux l-adiques. On y développe l'analogue pour les connexions méromorphes de la construction d'Abbes et Saito, tout d'abord dans le cas d'un trait, puis en dimension supérieure. En première partie, on prouve une formule explicite reliant les invariants produits par la construction d'Abbes et Saito appliquée à un module différentiel M aux parties les plus polaires des formes différentielles intervenant dans la décomposition de Levelt-Turrittin de M. Dans la seconde, on généralise en dimension supérieure l'observation issue de la première partie que sur un corps algébriquement clos, les modules produits par la construction d'Abbes et Saito sont des sommes finies de modules exponentiels associés à des formes linéaires. Dans la dernière partie de cette thèse, on montre que le lieu des points stables d'une connexion méromorphe M le long d'un diviseur lisse est un sous-ensemble de l'intersection des lieux où les faisceaux d'irrégularité de M et End M sont des systèmes locaux. Enfin, on discute d'une stratégie d'attaque de l'inclusion réciproque, et on démontre à l'aide d'un critère d'André pour les points stables que si elle est vraie en dimension 2, alors elle est vraie en toute dimension.

Ce travail comporte deux parties dont la première concerne l'ensemble asymptotique d'une application polynomiale $F : \C^n \to \C^n$. Dans les années 90s, Z. Jelonek a montré que cet ensemble est une variété algébrique complexe singulière de dimension (complexe) $n-1$. Nous donnons une méthode, appelée méthode façon, pour strati fier cet ensemble. Nous obtenons une strati cation de Thom-Mather. En utilisant les façons, nous donnons un algorithme pour expliciter l'ensemble asymptotique d'une application quadratique dominante en trois variables. Nous obtenons aussi une liste des ensembles asymptotiques dans ce cas. La deuxième partie concerne l'ensemble $V_F$ , appelé l'ensemble des Valette. L'année 2010, Anna et Guillaume Valette ont construit une pseudo-variété réelle $V_F \in R^{2n+ p}$, où $p > 0$, associée à une application polynomiale $F : C^n \to C^n$. Dans le cas $n= 2$, ils ont prouvéque si $F$ est une application polynomiale de déterminant jacobien partout non nul, alors $F$ n'est pas propre si et seulement si l'homologie (ou l'homologie d'intersection) de $V_F$ n'est pas triviale en dimension 2. Nous donnons une généralisation de ce résultat, dans le cas d'une application polynomiale $F= (F_1, \ldots, F_n : \C^n to \C^n$ de jacobien partout non nul. Nous donnons aussi une méthode pour stratifi er l'ensemble $V_F$ . Comme applications, nous avons les strati cations de l'ensemble $K_{\infty}F$ des valeurs critiques asymptotiques de $F$, de l'ensemble $B(F)$ des points bifurcation de F.

The motivating question behind this body of research is Smale’s 17th problem: Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm? While certain aspects and viewpoints of this problem have been solved, the analog over the real numbers largely remains open. This is an important question with applications in celestial mechanics, kinematics, polynomial optimization, and many others. Let A = {α_1, . . . , α_n+k } ⊂ Zn. The A-discriminant variety is, among other things, a tool that can be used to categorize polynomials based on the topology of their real solution set. This fact has made it useful in solving aspects and special cases of Smale’s 17th problem. In this thesis, we take a closer look at the structure of the A-discriminant with an eye toward furthering progress on analogs of Smale’s 17th problem. We examine a mostly ignored form called the Horn uniformization. This represents the discriminant in a compact form. We study properties of the Horn uniformization to find structural properties that can be used to better understand the A-discriminant variety. In particular, we use a little known theorem of Kapranov limiting the normals of the A-discriminant amoeba. We give new O(n^2) bounds on the number of components in the complement of the real A-discriminant when k = 3, where previous bounds had been O(n^6) or even exponential before that. We introduce new tools that can be used in discovering various types of extremal A-discriminants as well as examples found with these tools: a family of A-discriminant varieties with the maximal number of cusps and a family that appears to asymptotically admit the maximal number of chambers. Finally we give sage code that efficiently plots the A-discriminant amoeba for k = 3. Then we switch to a non-Archimedean point of view. Here we also give O(n^2) bounds for the number of connected components in the complement of the non- Archimedean A-discriminant amoeba when k = 3, but we also get a bound of O(n^(2(k−1)(k−2)) )when k > 3. As in the real case, we also give a family exhibiting O(n^2) connected components asymptotically. Finally we give code that efficiently plots the p-adic A-discriminant amoeba for all k ≥ 3. These results help us understand the structure of the A-discriminant to a degree, as yet, unknown. This can ultimately help in solving Smale’s 17th problem as it gives a better understanding of how complicated the solution set can be.

For pairing-based cryptographic protocols to be both efficient and secure, the underlying genus 2 curves defined over finite fields used must satisfy pairing-friendly conditions, and have small rho-value, which are not likely to be satisfied with random curves. In this thesis, we study two specific families of genus 2 curves defined over finite fields whose Jacobians do not split over the ground fields into a product of elliptic curves, but geometrically split over an extension of the ground field of prescribed degree n=3, 4, or 6. These curves were also studied extensively recently by Kawazoe and Takahashi in 2008, and by Freeman and Satoh in 2009 in their searches of pairing-friendly curves. We present a new method for constructing and identifying suitable curves in these two families which satisfy the pairing-friendly conditions and have rho-values around 4. The computational results of the rho-values obtained in this thesis are consistent with those found by Freeman and Satoh in 2009. An extension of our new method has led to a cryptographic example of a pairing-friendly curve in one of the two families which has rho-value 2.969, and it is the lowest rho-value ever recorded for curves of this type. Our method is different from the method proposed by Freeman and Satoh, since we can prescribe the minimal degree n =3,4 or 6 extension of the ground fields which the Jacobians of the curves split over. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-11-08 18:57:59.988

In this thesis, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that, a surface admitting a smooth fibration as above is elliptic and we employ results on the moduli of polarized elliptic surfaces, to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes H(1,X(d),n) of morphisms of degree n from elliptic curves to the modular curve X(d), d&amp / #8804 / 3. Ultimately, we show that the moduli spaces, considered, are fiber spaces over the affine line A&sup1 / with fibers determined by the components of H (1,X(d),n).

Thesis (M.Sc.) - Carleton University, 2008. / Includes bibliographical references (p. 212-213). Also available in electronic format on the Internet.