Presented By: Department of Mathematics
Student Algebraic Geometry
Knot Theory and Problems on Affine Space
Affine n-space A^n is one of the most basic objects in algebraic geometry. Unfortunately, there are many basic things we don't know about its geometry and its symmetries: for example,
(1) Is every embedding of A^k into A^n linearizable?
(2) What is the automorphism group of A^n?
(3) (The Jacobian Conjecture) Is every endomorphism of A^n with invertible Jacobian an isomorphism?
The Abhyankar-Moh theorem provides answers to some of these and other related questions in dimension n=2. What's surprising is that even though the statement is purely algebraic, the shortest known proof utilizes knot theory, specifically that of torus knots. We will explain how torus knots naturally appear in algebraic geometry, and present Rudolph's knot-theoretic proof of the Abhyankar-Moh theorem. Speaker(s): Takumi Murayama (UM)
(1) Is every embedding of A^k into A^n linearizable?
(2) What is the automorphism group of A^n?
(3) (The Jacobian Conjecture) Is every endomorphism of A^n with invertible Jacobian an isomorphism?
The Abhyankar-Moh theorem provides answers to some of these and other related questions in dimension n=2. What's surprising is that even though the statement is purely algebraic, the shortest known proof utilizes knot theory, specifically that of torus knots. We will explain how torus knots naturally appear in algebraic geometry, and present Rudolph's knot-theoretic proof of the Abhyankar-Moh theorem. Speaker(s): Takumi Murayama (UM)
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