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Abstract: I will discuss recent advances in the technique we call "the method  
of interlacing polynomials'' --- a technique that uses polynomials as  
a way to prove existence theorems in linear algebra.  Previous results  
used this method to show the existence of bipartite Ramanujan graphs  
of any size and degree, and subsequent progress was made in the work  
of Cohen in the form of a polynomial time construction.  This talk  
will discuss some recent progress in extending these results to the  
case of biregular, bipartite expanders.  Unlike classical Ramanujan  
graphs, these new constructions can have partitions of different  
sizes, making them more suitable for many applications.

  This is joint work with Aurelien Gribinski.