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MATHEMATICS COLLOQUIUM
“Geometry and
Combinatorics via Finite Fields”
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Date: |
Download-files: |
Time: |
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Wednesday 07. Dec 2016 |
Video-Recording for any system with MP4-support
- Video.mp4 (ca.421
Mb) |
15:15 – 16:15 |
Karen Smith,
Abstract :
Many natural
problems in mathematics can be framed in terms of solutions to polynomial
equations. These
solution sets, called algebraic varieties, have many interesting geometric
properties which
have been studied for centuries for
their own intrinsic beauty, as well as
used to solve
seemingly unrelated problems in many different contexts. In this talk, I will
explain how
certain geometric properties of
varieties can be understood by shifting
focus
to the algebraic
features of their corresponding coordinate rings. We will see that by
``reducing modulo
p" we can introduce effective new tools for understanding the
singularities
of a complex
variety. As a recent application, we discuss an application to an important
new type of
algebra arising in combinatorial representation theory called a cluster algebra
due to Fomin and
Zelevinsky. Specifically, we prove that
certain cluster algebras always
have rational
singularities over the real or complex
numbers, by studying the corresponding
cluster algebras
over finite fields.