# Mathematics Colloquia and Seminars

The goal of my talk (based on joint work with Vladimir Retakh) is to introduce noncommutative analogs of Catalan numbers which belong to the free Laurent polynomial algebra $$L_n$$ in $$n$$ generators. Our noncommutative Catalan numbers $$C_n$$ admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $$(q,t)$$-versions, another -- to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices $$H_n$$ and introduce two kinds of noncommutative binomial coefficients which are instrumental in computing the inverse of $$H_n$$ and its positive factorizations, and other combinatorial identities involving $$C_n$$.
If time permits, I will explain the relationship of the $$C_n$$ with the noncommutative Laurent Phenomenon, which was previously established for Kontsevich rank 2 recursions and all marked surfaces.