The tree of the cusp curve X3 = Y2
in Z3 has a simple structure.
Trees of other varieties are not essentially more complicated.
SFB 878 Groups, Geometry & Actions,
project
Algebraic groups and model theory
Research topics
Main topic: Model theory and arithmetic geometry in henselian valued fields
One project I am working on:
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Setting: The set of solutions (x, y) of a polynomial equation in the p-adic
integers Zp can be represented as a tree:
For each solution (x, y) and each n, the tree has a node at height n
labelled by the pair (last n digits of x, last n digits of y).
The actual solutions correspond to infinite paths in that tree.
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Question: What can these trees look like (after forgetting the labels)?
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(Of course one wants to know this in arbitrary dimension,
for arbitrary varieties or even definable sets, and
if possible in arbitrary Henselian fields.)
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See Trees of definable sets in Zp for details.
Other topics:
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Representation theory (of real reductive groups), symmetric varieties
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Model theory of pseudo-algebraically closed fields
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Combinatorics
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Polyomino achievement games
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Zero-sum problems in finite abelian groups
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Covering codes
CrocoPuzzle