Analysis Seminar: "A Free Analytic approach to Invertibility in the Tensor Product of Skew Fields"

Speaker: Meric Augat, Washington University in Saint Louis

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.
It is a classical result that if $A$ and $B$ are commutative domains, then their tensor product is a commutative domain whose invertible elements are exactly the pure tensors: $a\otimes b$, where $a$ and $b$ are units. While extending this to the noncommutative setting is false in general, we wish to study the problem for the tensor product of two free skew fields. Recall that the free skew field can be realized as the set of all noncommutative rational functions that can be evaluated on at least one tuple of matrices.
In this talk we will discuss the domain of the tensor product of rational functions and how this relates to a conjecture on invertibility. Moreover, we will use a few magical algebraic results coupled with Complex Analysis to prove that a rank 2 tensor of rational functions is not invertible over the tensor product of free skew fields.

Host: Brett Wick