Siddhartha Gadgil

Extremes of the Indian summer monsoon rainfall, ENSO and equatorial Indian Ocean oscillation

It is well known that anomalies of the Indian Summer Monsoon Rainfall (ISMR) are linked to El Nino and Southern Oscillation (ENSO). We show that large anomalies of the ISMR are also linked to the Equatorial Indian Ocean Oscillation (EQUINOO) between states with enhancement/ suppression of atmospheric convection over the western part of the equatorial Indian Ocean with suppression/enhancement over the eastern part and associated changes in the anomaly of the zonal wind along the equator. EQUINOO is the atmospheric component of the coupled Indian Ocean Dipole mode. There is a strong relation between the large anomalies of ISMR and a composite index which is a linear combination of the indices for ENSO and EQUINOO with all seasons with large deficits ( excess) characterized by small ( large) values of the index.However, the variation of ISMR within one standard deviation is more complex and does not appear to be related to the composite index.

On the geometric simple connectivity of open manifolds

A manifold is said to be geometrically simply connected if it has a proper handle decomposition without 1-handles. By the work of Smale, for compact manifolds of dimension at least 5, this is equivalent to simple connectivity. We prove that there exists an obstruction to an open simply connected n-manifold of dimension n ≥ 5 being geometrically simply connected. In particular, for each n ≥ 4, there exist uncountably many simply connected n-manifolds which are not geometrically simply connected. We also prove that for n ≠ 4, an n-manifold proper homotopy equivalent to a weakly geometrically simply connected polyhedron is geometrically simply connected (for n = 4, it is only end-compressible). We analyze further the case n = 4 and Poenarus conjecture.

The extended Goldman bracket determines intersection numbers for surfaces and orbifolds

In the mid eighties Goldman proved that an embedded closed curve could be isotoped to not intersect a given closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, asked about characterizing simple closed curves algebraically, in terms of the same Lie structure. We show how the Goldman bracket answers these questions for all finite type surfaces. In fact we count self-intersection numbers and mutual intersection numbers for all finite type orientable orbifolds in terms of a new Lie bracket operation, extending Goldmans. The arguments are purely topological, or based on elementary ideas from hyperbolic geometry. These results are intended to be used to recognize hyperbolic and Seifert vertices and the gluing graph in the geometrization of three-manifolds. The recognition is based on the structure of the string topology bracket of three-manifolds.

Homeomorphisms and the homology of non-orientable surfaces

We show that, for a closed non-orientable surfaceF, an automorphism ofH1(F, ℤ) is induced by a homeomorphism ofF if and only if it preserves the (mod 2) intersection pairing. We shall also prove the corresponding result for punctured surfaces.

On the topology of manifolds with positive isotropic curvature

We show that a closed orientable Riemannian n-manifold, n >= 5, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of Sn-1 x S-1.

TOPOLOGICAL GEODESICS AND VIRTUAL RIGIDITY

We introduce the notion of a topological geodesic in a 3-manif- old. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful prop- erties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits us to obtain virtual rigidity results for 3-manifolds. AMS Classication 57M10, 20F67; 57M50 Geodesics in Riemannian manifolds with metrics of negative sectional curva- ture play an essential role in geometry. We show here that, in the case of 3-dimensional manifolds, many crucial properties of geodesics follow from a purely topological characterization in terms of knotting. In particular, we prove two results concerning the virtual rigidity of 3-manifolds following the methods of Gabai (5). We introduce the notion of a topological geodesic in a 3-manifold. We shall prove basic existence and uniqueness results for topological geodesics under suitable hypotheses on the fundamental group. Suppose henceforth that M is a closed 3-manifold with word-hyperbolic (or semi-hyperbolic) 1(M). We refer to the next section for the denition of the semi-hyperbolicity, following Alonso and Bridson. We adopt the convention here that nite groups are not semi-hyperbolic, hence all closed 3-manifolds we consider have innite fundamental group unless the opposite is explicitly stated. In particular, the universal cover f M of M is homeomorphic to R 3 (see

Watson–Crick pairing, the Heisenberg group and Milnor invariants

We study the secondary structure of RNA determined by Watson–Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a lower bound on the number of unpaired bases in an RNA secondary structure. We also show that the Heisenberg invariant can predict allosteric structures for RNA. Namely, if the Heisenberg invariant is large, then there are widely separated local maxima (i.e., allosteric structures) for the number of Watson–Crick pairs found.

A chain complex and quadrilaterals for normal surfaces

We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying that quadrilaterals determine a normal surface up to vertex linking spheres. We also characterise the quadrilateral coordinates that correspond to a normal surface in a (possibly ideal) triangulation.

Real Theta Characteristics And Automorphisms Of A Real Curve

Let X be a geometrically irreductble smooth projective cruve defined over R. of genus at least 2. that admits a nontrivial automorphism, sigma. Assume that X does not have any real points. Let tau be the antiholomorphic involution of the complexification lambda(C) of X. We show that if the action of sigma on the set S(X) of all real theta characteristics of X is trivial. then the order of sigma is even, say 2k and the automorphism tau o (sigma) over cap (lambda) of X-C has a fixed point, where (sigma) over cap is the automorphism of X x C-R defined by sigma We then show that there exists X with a real point and admitting a nontrivial automorphism sigma, such that the action of sigma on S(X) is trivial, while X/ not equal P-R(1) We also give an example of X with no real points and admitting a nontrivial automorphisim sigma such that the automorphism tau o (sigma) over cap (lambda) has a fixed point, the action of sigma on S(X) is trivial, and X/ not equal P-R(1)

Contact structures on elliptic 3-manifolds

We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure if and only if the corresponding group of deck transformations on S 3 (after possibly conjugating by an isometry) preserves the standard contact structure. We also relate universally tight contact structures on 3-manifolds covered by S 3 to the isomorphism SO(4) = (SU(2) ◊ SU(2))/±1. The main tool used is equivariant framings of 3-manifolds. A contact structure on a 3-dimensional manifold M is a smooth, totally non- integrable tangent plane field, i.e., a tangent plane field locally of the form = ker( ) for a 1-form such that ^ d is everywhere non-degenerate. We shall assume that M is oriented. We say is positive if the orientation on M agrees with that induced by the volume form ^ d . Observe that the orientation of ^d does not depend on the sign of , and is thus determined by (even though = ker( ) only locally). A central role in understanding 3-dimensional manifolds has been played by co- dimension one structures - surfaces, foliations and laminations - in these manifolds. Without additional conditions such structures always exist, and are of not much consequence. However, the presence of essential co-dimension one structures - incompressible surfaces, taut foliations and essential laminations, leads to deep