51 research outputs found
Whiskered KAM tori of conformally symplectic systems
We investigate the existence of whiskered tori in some dissipative systems,
called \sl conformally symplectic \rm systems, having the property that they
transform the symplectic form into a multiple of itself. We consider a family
of conformally symplectic maps which depend on a drift parameter .
We fix a Diophantine frequency of the torus and we assume to have a drift
and an embedding of the torus , which satisfy approximately the
invariance equation (where
denotes the shift by ). We also assume to have a splitting
of the tangent space at the range of into three bundles. We assume that
the bundles are approximately invariant under and that the
derivative satisfies some "rate conditions".
Under suitable non-degeneracy conditions, we prove that there exists
, and splittings, close to the original ones, invariant
under . The proof provides an efficient algorithm to construct
whiskered tori. Full details of the statements and proofs are given in
[CCdlL18].Comment: 15 pages, 1 figur
Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories
Seiberg-Witten geometry of mass deformed N=2 superconformal ADE quiver gauge
theories in four dimensions is determined. We solve the limit shape equations
derived from the gauge theory and identify the space M of vacua of the theory
with the moduli space of the genus zero holomorphic (quasi)maps to the moduli
space of holomorphic G-bundles on a (possibly degenerate) elliptic curve
defined in terms of the microscopic gauge couplings, for the corresponding
simple ADE Lie group G. The integrable systems underlying, or, rather,
overlooking the special geometry of M are identified. The moduli spaces of
framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1,
the Hitchin systems on curves with punctures, as well as various spin chains
play an important role in our story. We also comment on the higher dimensional
theories. In the companion paper the quantum integrable systems and their
connections to the representation theory of quantum affine algebras will be
discussedComment: 197 page
Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
We present a systematic approach to studying the geometric aspects of Vinberg
theta-representations. The main idea is to use the Borel-Weil construction for
representations of reductive groups as sections of homogeneous bundles on
homogeneous spaces, and then to study degeneracy loci of these vector bundles.
Our main technical tool is to use free resolutions as an "enhanced" version of
degeneracy loci formulas. We illustrate our approach on several examples and
show how they are connected to moduli spaces of Abelian varieties. To make the
article accessible to both algebraists and geometers, we also include
background material on free resolutions and representation theory.Comment: 41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the
occasion of his 65th birthday; v2: fixed some typos and added reference
Mathematical and physical aspects of complex symmetric operators
Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables
Cubic Differentials in the Differential Geometry of Surfaces
We discuss the local differential geometry of convex affine spheres in
\re^3 and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In
each case, there is a natural metric and cubic differential holomorphic with
respect to the induced conformal structure: these data come from the Blaschke
metric and Pick form for the affine spheres and from the induced metric and
second fundamental form for the minimal Lagrangian surfaces. The local
geometry, at least for main cases of interest, induces a natural frame whose
structure equations arise from the affine Toda system for . We also discuss the global theory and applications to
representations of surface groups and to mirror symmetry.Comment: corrected published editio
Hamiltonian spectral flows, the Maslov index, and the stability of standing waves in the nonlinear Schr\"{o}dinger equation
We use the Maslov index to study the spectrum of a class of linear
Hamiltonian differential operators. We provide a lower bound on the number of
positive real eigenvalues, which includes a contribution to the Maslov index
from a non-regular crossing. A close study of the eigenvalue curves, which
represent the evolution of the eigenvalues as the domain is shrunk or expanded,
yields formulas for their concavity at the non-regular crossing in terms of the
corresponding Jordan chains. This, along with homotopy techniques, enables the
computation of the Maslov index at such a crossing. We apply our theory to
study the spectral (in)stability of standing waves in the nonlinear
Schr\"odinger equation on a compact spatial interval. We derive new stability
results in the spirit of the Jones--Grillakis instability theorem and the
Vakhitov--Kolokolov criterion, both originally formulated on the real line. A
fundamental difference upon passing from the real line to the compact interval
is the loss of translational invariance, in which case the zero eigenvalue of
the linearised operator is geometrically simple. Consequently, the stability
results differ depending on the boundary conditions satisfied by the wave. We
compare our lower bound to existing results involving constrained eigenvalue
counts, finding a direct relationship between the correction factors found
therein and the objects of our analysis, including the second-order Maslov
crossing form.Comment: 48 pages, 8 figure
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