Abstracts for Topics in Topology

The homotopy principle (h-principle) is a general observation that differential relations in differential geometry can often be solved by means of homotopy theory. For example, in the case of immersions this implies the Smale paradox asserting that the standard embedding of the sphere $S^2$ into $R^3$ can be turned inside out through a family of immersions. According to a general theorem of Gromov, the h-principle holds true in a very general setting. Often solutions of a differential relation R form a monoid under operation defined by taking the disjoint union.

Subsurfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.

I will give a precise formulation of T-duality for Ramond-Ramond fields. This leads to a canonical isomorphism between the "geometrically invariant" subgroups of the twisted differential K-theory of certain principal torus bundles, which combines topological T-duality with the Buscher rules found in physics.

Elliptic cohomology is at the heart of many recent developments in topology. What led to its discovery was Ochanine's observation in the 1980s that there are many more multiplicative genera for spin fiber bundles than for oriented fiber bundles, one for each elliptic curve with a marked point of order 2. Given that multiplicative genera for spin manifolds have led to such unexpectedly rich developments, it seems reasonable to investigate multiplicative genera for other types of fiber bundles, for example O<8> fiber bundles.