Title:  Network Coding and Schubert Varieties Over Finite Fields



Abstract:  Network coding deals with noisy transmission of data over a network, in our case from one sender to several receivers. It turns out that linear vector spaces over a finite field are a good tool for error correction in this setting. In this talk we give a quick introduction to network coding in general and show that many network coding theoretic problems translate into enumerative geometry problems in the Grassmann variety over the finite field. In particular, the Plücker embedding of the Grassmann variety, and Schubert varieties therein, are useful tools for code constructions and decoding algorithms. We show how to set up such an error correcting decoding algorithm and what its advantages and disadvantages are. Moreover, we briefly explain classical Schubert calculus over the complex numbers and show that the classical results do not hold (in general) over finite fields.