72 research outputs found
Diagonal Subschemes and Vector Bundles
We study when a smooth variety , embedded diagonally in its Cartesian
square, is the zero scheme of a section of a vector bundle of rank on
. We call this the diagonal property (D). It was known that it holds
for all flag manifolds .
We consider mainly the cases of proper smooth varieties, and the analogous
problems for smooth manifolds (the topological case).
Our main new observation in the case of proper varieties is a relation
between (D) and cohomologically trivial line bundles on , obtained by a
variation of Serre's classic argument relating rank 2 vector bundles and
codimension 2 subschemes, combined with Serre duality. Based on this, we have
several detailed results on surfaces, and some results in higher dimensions.
For smooth affine varieties, we observe that for an affine algebraic group
over an algebraically closed field, the diagonal is in fact a complete
intersection; thus (D) holds, using the trivial bundle. We conjecture the
existence of smooth affine complex varieties for which (D) fails; this leads to
an interesting question on projective modules.
The arguments in the topological case have a different flavour, with
arguments from homotopy theory, topological K-theory, index theory etc. There
are 3 variants of the diagonal problem, depending on the type of vector bundle
we want (arbitrary, oriented or complex). We obtain a homotopy theoretic
reformulation of the diagonal property as an extension problem for a certain
homotopy class of maps. We also have detailed results in several cases:
spheres, odd dimensional complex projective quadric hypersurfaces, and
manifolds of even dimension with an almost complex structure.Comment: 33 page
EduCycle: Bicycle-based Low Power Lighting Demonstration
This report describes EduCycle, a system for demonstrating the reduced energy requirements of modern lighting in a hands-on manner to elementary school students. EduCycle consists of a bicycle trainer connecting a bicycle to a generator and a bank of incandescent, compact fluorescent (CFL), and light emitting diode (LED) light bulbs. The EduCycle system facilitates experiential learning by allowing the user to physically experience the amount of power required by each type of lighting. EduCycle has been designed from the ground-up to be portable, durable and safe, and was beta tested on 3rd grade elementary school students. Surveys collected from the beta test indicate that the students achieved the intended learning outcomes of the project
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