The talk will survey results in free algebraic geometry and free analysis over the last few years. Hopefully some might eventually be useful in free probability.
Real Algebraic Geometry concerns polynomial inequalities and equalities. Say p >0 where q>0, then there is a algebraic relationship (called Positivstellensatz) between p and q which is equivalent to this happening. As to equalities, for a polynomial p in noncommutative variables there are 3 natural notions of zero. Hence one seeks 3 different Nullstellensatze and recently one of these (possibly the most useful one) was emerged.
The talk will present this, how it combines with Positivstellensatz to produce applications and how it fits with a variety of other results. As to techniques: a powerful one used here, also used in free probability, is "linearization".
Most of the work is joint with Meric Augat, Eric Evert, Igor Klep, Scott McCullough, Jurij Volcic.
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