# Analysis Seminar: "Free Analysis and Tensor Products of Skew Fields: or, the Modern Prometheus"

*Abstract: Recent advances in Free Analysis have yielded remarkable analogs of classical theorems in analysis, algebra and geometry. In particular, the Free Inverse Function Theorem has a stronger conclusion than its classical counterpart while the Free Jacobian Conjecture is true and its proof is quite tractable.*

*Recall that a free polynomial mapping in g freely noncommuting variables sends g-tuples of matrices (of the same size) to g-tuples of matrices (of the same size). The Free Grothendieck Theorem asserts that an endomorphism of the free algebra is an isomorphism if and only if its induced polynomial mapping is injective on all g-tuples of matrices if and only if its free Jacobian matrix is invertible over the tensor product of the free algebra.*

*In this talk, we discuss the free rational version of the above theorem: if we have an endomorphism of the free skew field, then what injectivity conditions on its induced free rational mapping imply the endomorphism is an isomorphism and how does this relate to invertibility over the tensor product of the free skew field?*

*Our goal is to reduce certain implications to a conjecture about the invertible elements of the tensor product of the free skew field: the only invertible elements are the rank one tensors.*

*Host: Brett Wick*