Project homepage: Solving Algebraic Equations
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Project description
The overall goal is to develop tools and techniques for the construction
and the geometric understanding of the solution variety of polynomial
equations in real and complex affine space, in power series rings and in p-adic
spaces.
This is our notion of "solving". Moreover, we want to use
these techniques, together with already existing tools, in order
to provide devices for representing algebraic varieties
to an audience inside and outside mathematics.
The project has four specific themes:
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Synthetic geometry:
Conceptualize mathematically the geometry of algebraic varieties as they are
perceived by the human eye and stored by the brain. Develop tools to construct
varieties with prescribed geometric properties.
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Arc spaces:
Study solutions of algebraic equations in power series spaces and in the field
of p-adic numbers using methods of global analysis
(e.g., the Rank Theorem for analytic maps between power series spaces).
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Fibrations:
Describe surfaces as families of moving curves (as in cartesian products, ruled
surfaces, surfaces of revolution). Give criteria and constructions
for natural fibrations.
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Visualization:
Produce authentic pictures of real surfaces exhibiting special
geometric configurations. Compose these pictures to a movie which
explains main geometric concepts and phenomena of algebraic varieties.
Examples of visualizations
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figure 1
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figure 2
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figure 3
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figure 4
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figure 5
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figure 1:
The cartesian product of a plane cusp and node embedded
isomorphically into R^3.
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figure 2:
Example of singularity which is not mikado at the common
intersection point, of equation y*z*(x^2+y-z) = 0.
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figure 3:
Family of four lines with varying cross ratio of equation x*y*(x-y)*(x-zy).
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figure 4:
Degeneration of node to a cusp. The equation is x^2=z^3-y^2*z^2.
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figure 5:
The generation of a surface by a fibration with curves.
Cartesian product of plane cusp with itself embedded into R^3.
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