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                                          MATHEMATICS COLLOQUIUM

                            “Geometry and Combinatorics via Finite Fields”





 Wednesday 07. Dec 2016

    Video-Recording for any system with MP4-support


       -   Video.mp4  (ca.421 Mb)


 15:15 – 16:15



                                             Karen Smith, University of Michigan


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.


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