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Arithmetic Geometry, Number Theory, and Computation
This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research. Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.
The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations
This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centres on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coicides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. a number of new examples illustrate the effectiveness of this approach.
Elliptic Curves, Hilbert Modular Forms and Galois Deformations
The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations ...
On the Existence of Feller Semigroups with Boundary Conditions
This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Ventcel' (Wentzell) boundary conditions. This paper considers the non-transversal case and solves from the viewpoint of functional analysis the problem of construction of Feller semigroups for elliptic Waldenfels operators. Intuitively, our result may be stated as follows: One can construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it "dies" at which time it reaches the set where the absorption phenomenon occurs.
A Generalization of Riemann Mappings and Geometric Structures on a Space of Domains in C$^n$
Similar in philosophy to the study of moduli spaces in algebraic geometry, the central theme of this book is that spaces of (pseudoconvex) domains should admit geometrical structures that reflect the complex geometry of the underlying domains in a natural way. With its unusual geometric perspective of some topics in several complex variables, this book appeals to those who view much of mathematics in broadly geometrical terms.
Imbeddings of Three-Manifold Groups
This paper deals with the two broad questions of how 3-manifold groups imbed in one another and how such imbeddings relate to any corresponding [lowercase Greek]Pi1-injective maps. In particular, we are interested in 1) determining which 3-manifold groups are no cohopfian, that is, which 3-manifold groups imbed properly in themselves, 2) determining the knot subgroups of a knot group, and 3) determining when surgery on a knot [italic]K yields a lens (or "lens-like") space and the relationship of such a surgery to the knot-subgroup structure of [lowercase Greek]Pi1([italic]S3 - [italic]K). Our work requires the formulation of a deformation theorem for [lowercase Greek]Pi1-injective maps between certain kinds of Haken manifolds and the development of some algebraic tools.
Projective Modules over Lie Algebras of Cartan Type
This paper investigates the question of linkage and block theory for Lie algebras of Cartan type. The second part of the paper deals mainly with block structure and projective modules of Lies algebras of types W and K.
Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras
We present a new proof of the identities needed to exhibit an explicit [bold]Z-basis for the universal enveloping algebra associated to an affine Lie algebra. We then use the explicit [bold]Z-bases to extend Borcherds' description, via vertex operator representations, of a [bold]Z-form of the enveloping algebras for the simply-laced affine Lie algebras to the enveloping algebras associated to the unequal root length affine Lie algebras.
Neumann Systems for the Algebraic AKNS Problem
This work is concerned with an algebraically completely integrable Hamiltonian system whose solutions may be used to describe the finite gap solutions of the AKNS spectral problem, a first order two-by-two matrix linear system. Trace formulas, constraints, Lax paris, and constants of motion are obtained using Krichever's algebraic inverse spectral transform. Computations are carried out explicityly over the class of spectral problems with square matrix coefficients.
