378 research outputs found
Algebro-Geometric Solutions of the Boussinesq Hierarchy
We continue a recently developed systematic approach to the Bousinesq (Bsq)
hierarchy and its algebro-geometric solutions. Our formalism includes a
recursive construction of Lax pairs and establishes associated
Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations
for analogs of Dirichlet and Neumann divisors. The principal aim of this paper
is a detailed theta function representation of all algebro-geometric
quasi-periodic solutions and related quantities of the Bsq hierarchy.Comment: LaTeX, 48 page
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Convex and Algebraic Geometry
The subjects of convex and algebraic geometry meet primarily in the theory of toric varieties. Toric geometry is the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Recent developments in toric geometry that were discussed during the workshop include applications to mirror symmetry, motivic integration and hypergeometric systems of PDE’s, as well as deformations of (unions of) toric varieties and relations to tropical geometry
Yang-Mills theory and Tamagawa numbers
Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills
functional to calculate the Betti numbers of moduli spaces of vector bundles
over a Riemann surface, rederiving inductive formulae obtained from an
arithmetic approach which involved the Tamagawa number of SL_n. This article
surveys this link between Yang-Mills theory and Tamagawa numbers, and explains
how methods used over the last three decades to study the singular cohomology
of moduli spaces of bundles on a smooth complex projective curve can be adapted
to the setting of A^1-homotopy theory to study the motivic cohomology of these
moduli spaces.Comment: Accepted for publication in the Bulletin of the London Mathematical
Societ
Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies
Combining algebro-geometric methods and factorization techniques for finite
difference expressions we provide a complete and self-contained treatment of
all real-valued quasi-periodic finite-gap solutions of both the Toda and
Kac-van Moerbeke hierarchies. In order to obtain our principal new result, the
algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we
employ particular commutation methods in connection with Miura-type
transformations which enable us to transfer whole classes of solutions (such as
finite-gap solutions) from the Toda hierarchy to its modified counterpart, the
Kac-van Moerbeke hierarchy, and vice versa.Comment: LaTeX, to appear in Memoirs of the Amer. Math. So
Extended Rate, more GFUN
We present a software package that guesses formulae for sequences of, for
example, rational numbers or rational functions, given the first few terms. We
implement an algorithm due to Bernhard Beckermann and George Labahn, together
with some enhancements to render our package efficient. Thus we extend and
complement Christian Krattenthaler's program Rate, the parts concerned with
guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of
Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's
qGeneratingFunctions.m.Comment: 26 page
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