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Mathematics Colloquium


Date:  Tuesday, April 22, 2014

Title:  Characteristic p tricks in algebra, geometry and combinatorics.

Abstract:  Consider a commutative ring of prime characteristic p, such as a polynomial ring over a finite field of p elements. For any two elements x and y, we have(x+y)p = xp +yp because the binomial coefficients are all divisible by p for 0 < i < p. This simple algebraic fact is remarkably powerful, often leading to deep theorems even about algebras over Q or C with surprisingly easy proofs. An early example is the Hochster Roberts theorem on the Cohen-Macaulayness of rings of invariants. In algebraic geometry, tricks involving p-th powers have led to strong vanishing theorems for line bundles on certain projective varieties. More recently, the Frobenius (or p-th power) map has been used to clarify the structure of certain cluster algebras (a topic especially popular in our own department). In this talk, we hope to introduce the magic of the p-th power map to non-experts, with examples drawn from commutative algebra, algebraic geometry, and combinatorics.

Speaker:  Karen Smith
Institution:  University of Michigan


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