Thanks, I’m glad people are still finding this blog useful, though I haven’t touched it in years. I’m no longer in mathematics so I don’t have any deep insights on rank 2 bundles on elliptic curves, but the main thing is that it’s a family of planes for each point on the curve. If there’s some vector space they all fit into (there should be for some finite N, though no clue how to bound it) then that means its a map from the elliptic curve to the Grassmannian. I vaguely recall something about vector bundles over abelian group schemes decomposing into direct sums of line bundles, but I can’t verify it. Good luck!

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It is the reduction to a pencil that is the nontrivial part of the proof. The rest is just Sard’s theorem. ]]>

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which can be easily solved to find

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where is an unknown constant. Now we know that the electric field will cause a steady state drift, thus, we can assume that the velocity is of the form,

1 INTRODUCTION

The differences between these types of materials can be understood from solid state theory. [ ]

Conductors

SemiconductorsIn each of these we make assumptions about the materials and describe them.

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This trial solution can be plugged into our original equation of motion to determine,

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Adding these equations we find,

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Because of our original drift solution, we know that,

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