Stamatis Dostoglou

Self-dual instantons and holomorphic curves

A gradient flow of a Morse function on a compact Riemannian manifold is said to be of Morse-Smale type if the stable and unstable manifolds of any two critical points intersect transversally. For such a Morse-Smale gradient flow there is a chain complex generated by the critical points and graded by the Morse index. The boundary operator has as its (x, y)-entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the Morse indices is 1. The homology of this chain complex agrees with the homology of the underlying manifold M and this can be used to prove the Morse inequalities (cf. [33], [26]). Around 1986, Floer generalized this idea to infinite-dimensional variational problems in which every critical point has infinite Morse index but the moduli spaces of connecting orbits form finite-dimensional manifolds for every pair of critical points. The dimensions of these spaces give rise to a relative Morse index and the boundary operator is defined by counting connecting

Assessment of the impact of the planetary scale on the decay of blocking and the use of phase diagrams and enstrophy as a diagnostic

It was shown that abrupt changes in the large-scale structure of atmospheric flows may lead to the rapid decay of blocking. Analysis of phase diagrams made it possible to identify when sharp changes occurred in the dynamics of the system. The connection of these changes to the decay of blocking was estimated for three blocking events in the Southern Hemisphere. In addition to phase diagrams, enstrophy was used as a diagnostic tool for the analysis of blocking events. From the results of this analysis, four scenarios for the decay mechanisms were determined: (i) decay with a lack of synoptic-scale support, (ii) decay with an active role for synoptic processes, and (iii–iv) either of these mechanisms in the interaction with an abrupt change in the character of the planetary-scale flow.

On the Morgan-Shalen compactification of the SL(2,ℂ) character varieties of surface groups

A gauge theoretic description of the Morgan-Shalen compactification of the

Cauchy-Riemann Operators, Self—Duality, and the Spectral Flow

\SL

Stable distributions and the term structure of interest rates

character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the Hitchin map.

Families of SU(2) representations for mapping cylinders of periodic monodromy

Atiyah, Patodi and Singer [3] observed that the Fredholm index of the operator

ON HYDRODYNAMIC EQUATIONS AT THE LIMIT OF INFINITELY MANY MOLECULES

{D_A}\, = \,\frac{d}{{dt}}\, + \,A\left( t \right)

Instanton homology and symplectic xed points

with invertible limits \( {A^ \pm }\, = \,\mathop {\lim }\limits_{t \to \pm \infty } \,A\left( t \right) \) is given by the spectral flow of the self-adjoint operator family A(t) (the number of eigenvalues crossing 0 counted with signs). Such operators appear in infinite dimensional analogues of Morse theory as the linearisation of the gradient flow equation. The Fredholm index is the dimension of the space of gradient flow lines connecting two critical points. It can be thought of as the relative Morse index in cases where the absolute Morse index (the number of negative eigenvalues of the Hessian) is infinite.