## The Finite R-Transform

#### Nikhil SrivastavaUniversity of California, Berkeley (UC Berkeley)

We will define a convolution operation on degree n real-rooted polynomials mimicking the free convolution of measures in free probability, and show how the R-transform can be used to control the extreme roots of this finite free convolution. We will briefly describe how these bounds imply the existence of optimal bipartite expander graphs of all sizes and degrees.

Joint work with A. Marcus and D. Spielman

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