* Preface * Introduction Part I: Compactifications of Riemannian Symmetric Spaces * Review of Classical Compactifications of Symmetric Spaces * Uniform Construction of Compactifications of Symmetric Spaces * Properties of Compactifications of Symmetric Spaces Part II: Smooth Compactifications of Semisimple Symmetric Spaces * Smooth Compactifications of Riemannian Symmetric Spaces G / K * Semisimple Symmetric Spaces G / H * The Real Points of Complex Symmetric Spaces Defined Over R * The DeConcini-Procesi Compactification of a Complex Symmetric Space and its Real Points * The Oshima-Sekiguchi Compactification of G / K and Comparison with G/Hw (R) Part III: Compactifications of Locally Symmetric Spaces * Classical Compactifications of Locally Symmetric Spaces * Uniform Construction of Compactifications of Locally Symmetric Spaces * Properties of Compactifications of Locally Symmetric Spaces * Subgroup Compactifications of o \ G * Metric Properties of Compactifications of Locally Symmetric Spaces o \ X * References * Index

http://dept.math.lsa.umich.edu/~lji/head.pdf

Compactifications of Symmetric and Locally Symmetric Spaces

Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.

Vector Space of Feynman Integrals and Multivariate Intersection Numbers

Middle convolution of Fuchsian systems and the construction of rigid differential systems

http://users.uoa.gr/~rvidunas/halgshort.pdf

Transformations of some Gauss hypergeometric functions

This work has its origins in an attempt to describe systematically the integrable geometries and gauge theories in dimensions one to four related to twistor theory. In each such dimension, there is a nondegenerate integrable geometric structure, governed by a nonlinear integrable differential equation, and each solution of this equation determines a background geometry on which, for any Lie group G, an integrable gauge theory is defined. In four dimensions, the geometry is selfdual conformal geometry and the gauge theory is selfdual Yang{Mills theory, while the lower-dimensional structures are nondegenerate (i.e., nonnull) reductions of this. Any solution of the gauge theory on a k-dimensional geometry, such that the gauge group H acts transitively on an l-manifold, determines a (k+l)-dimensional geometry (k+l≤4) fibering over the k-dimensional geometry with H as a structure group. In the case of an l-dimensional group H acting on itself by the regular representation, all (k +l)-dimensional geometries with symmetry group H are locally obtained in this way. This framework unifies and extends known results about dimensional reductions of selfdual conformal geometry and the selfdual Yang-Mills equation, and provides a rich supply of constructive methods. In one dimension, generalized Nahm equations provide a uniform description of four pole isomonodromic deformation problems, and may be related to the SU(∞) Toda and dKP equations via a hodograph transformation. In two dimensions, the Diff(S1) Hitchin equation is shown to be equivalent to the hyperCR Einstein{Weyl equation, while the SDiff(σ) Hitchin equation leads to a Euclidean analogue of Plebanski's heavenly equations. In three and four dimensions, the constructions of this paper help to organize the huge range of examples of Einstein{Weyl and selfdual spaces in the literature, as well as providing some new ones. The nondegenerate reductions have a long ancestry. More recently, degenerate or null reductions have attracted increased interest. Two of these reductions and their gauge theories (arguably, the two most significant) are also described.

/pdf/integrable-background-geometries-tmtru6vey2.pdf

Integrable Background Geometries

Part 1: The Story of the Configuration Space X (2,4) of Four Points on the Projective Line: Configuration Spaces - The Simplest Case Elliptic Curves Modular Interpretations of X (2,4) Hypergeometric Integrals and Loaded Cycles. Part 2: The Story of the Configuration Space X (2,n) of n Points on the Projective Line: The Configuration Space X (2,5) Modular Interpretation of the Configuration Space X (2,n) The Configuration Space X (3,6) Hypergeometric Functions of Type (3,6) Modular Interpretation of the Configuration Space X (3,6)

正章吉田

Papers

Hypergeometric functions, my love : modular interpretations of configuration spaces

正章 吉田

Papers

Hypergeometric functions, my love : modular interpretations of configuration spaces

正章吉田