The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y, equipped with a Spiny structure. Given a Heegaard splitting of Y = U 0o U Σ U 1 , these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U 0 and U 1 .

https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p03.pdf

Holomorphic disks and topological invariants for closed three-manifolds

Convex Analysis: (pms-28)

In [27], we introduced Floer homology theories HF − (Y,s), HF ∞ (Y,s), HF + (Y,t), � HF(Y,s),and HFred(Y,s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ Spin c (Y ). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with Seiberg-Witten theory. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology.

http://annals.math.princeton.edu/wp-content/uploads/annals-v159-n3-p04.pdf

Holomorphic disks and three-manifold invariants: Properties and applications

Introduction Local behaviour Moduli spaces and transversality Compactness Compactification of moduli spaces Evaluation maps and transversality Gromov-Witten invariants Quantum cohomology Novikov rings and Calabi-Yau manifolds Floer homology Gluing Elliptic regularity Bibliography Indexes.

/pdf/j-holomorphic-curves-and-quantum-cohomology-37huuohzn3.pdf

J-Holomorphic Curves and Quantum Cohomology

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

/pdf/self-dual-instantons-and-holomorphic-curves-1ipcv2eh1u.pdf

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 [30] (see also [24]). Around 1986 Floer generalized this idea and discovered a powerful new approach to infinite dimensional Morse theory now called Floer homology. He used this approach to prove the Arnold conjecture for monotone symplectic manifolds [12] and discovered a new invariant for homology 3-spheres called instanton homology [11]. This invariant can roughly be described as the homology of a chain complex generated by the irreducible representations of the fundamental group of the homology 3-sphere M in the Lie group SU(2). These representations can be thought of as flat connections on the principal bundle M × SU(2) and they appear as the critical points of the Chern-Simons functional on the infinite dimensional configuration space of connections on this bundle modulo gauge equivalence. The gradient flow lines of the Chern-Simons functional are the self-dual Yang-Mills instantons on the 4-manifold M ×R and

/pdf/instanton-homology-and-symplectic-xed-points-52bqxjh2yv.pdf

Instanton homology and symplectic xed points

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

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.

/pdf/assessment-of-the-impact-of-the-planetary-scale-on-the-decay-5c5h2mqxf6.pdf

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

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible SL2(C) representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an R-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.

http://www2.math.umd.edu/~raw/papers/ks.pdf

Stamatis Dostoglou

Papers

Self-dual instantons and holomorphic curves

Instanton homology and symplectic xed points

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

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

Character varieties and harmonic maps to R-trees