Workshop

Certain division algebras arise naturally in noncommutative projective geometry as the ``function skew fields" of noncommutative projective varieties.  Artin conjectured that the division algebras associated in this way to noncommutative surfaces are either finite over their centers, or else fall on a short list of known possibilities.  We discuss the conjecture and other recent work on problems about division rings that are motivated by noncommutative projective geometry.

The Cremona dimension of a finite group G is the minimal dimension of a rationally connected variety which admits a faithful action of G.  In this talk, based on joint work with Giulio Bresciani and Angelo Vistoli, I will discuss new lower bounds on the Cremona dimension of a finite p-group.

The free skew field, introduced by S. Amitsur and studied in depth by P.M. Cohn, is the universal skew field of fractions of the free associative algebra. During the last two decades it came to play an important role  in free noncommutative analysis, the theory of functions of free noncommuting variables represented by matrices of increasing size, that was introduced by J.L. Taylor in his work on noncommutative spectral theory and further developed by D.-V. Voiculescu with a view towards free probability.

In this talk I will introduce the free skew field from free noncommutative analysis perspective. I will discuss the difference-differential calculus and power series expansions (Taylor--Taylor series, named after Brook Taylor of the calculus fame and J.L. Taylor), and their relations to realizations that originated simultaneously in automata theory and in systems and control. Time permitting I will touch on some other results such as the construction of the universal field of fractions of a tensor product of free algebras and the absence of rational identities for free noncommutative random variables (the work of T. Mai, R. Speicher, and S. Yin).

This will be a survey talk, so the content is the work of many authors. My own work on the topic has been done in collaboration with D. Kaliuzhnyi-Verbovtskyi, with I. Klep and Ju. Volcic, and with M. Porat.

While a free associative algebra embeds into various non-isomorphic skew fields, one of them is universal, and called the free skew field. Its fundamental features were developed through a seminal work of P. M. Cohn and others. The aim of this talk is to both survey some of these results and present current developments, with an emphasis on freeness and centralizers in free skew fields. In particular, it is shown that a free skew field contains non-rational subfields, which is based on joint work with Harm Derksen.

Given a polynomial, one can evaluate it at a square matrix via substituting the square matrix in for the unknown. Such preserves various properties of the matrix. For example, if we evaluate at an upper triangular matrix, the output should be upper triangular as well, and if we want to change the coordinates in which the matrix is expressed, we should get the same result if we do that before or after evaluation. Slight refinement and formalization of these properties turns out to give an elementary characterization what is known as the functional calculus- which is usually developed using advanced techniques in real or complex analysis depending on the context. Moreover, such an algebraic formulation of the functional calculus generalization to several variables now known as free noncommutative function theory. We will discuss the basic definitions and move on to various foundational aspects of the theory including the equivalence between stability and analyticity, universal mondromy (which is much stronger than the classical monodromy theorem) and related topics.

In this talk I will discuss some recent results on invariants of group actions on skew fields in joint work with Jurij Volcic: Suppose that G is a finite group acting on a finitely generated skew field M. If L is the set of invariants then L is also a finitely generated skew field. I will describe an algorithm to obtain a set of generators. If M is the free skew field with m generators, and the group action is linear, then L is free with |G|(m-1)+1 generators. However, if the action is non-linear, then the invariant subfield L may not be free.

Michael Wemyss and Will Donovan have developed a method for characterizing commutative rings appearing in algebraic geometry which uses methods from noncommutative ring theory. They introduced contraction algebras, which provide important insight into resolutions of noncommutative singularities. All local contraction algebras were characterised by Michael Wemyss and Gavin Brown by giving generators and relations. Deformations of these algebras give information about Gopakumar–Vafa (GV) invariants, which are very important in algebraic geometry. This leads to purely ring theoretic questions about deformations of local algebras, for example whether or not a given contraction algebra can be deformed to a given semisimple algebra. Some of these problems can be investigated by applying polynomial identities to both algebra A and N when deforming an algebra $N$ into an algebra A.

In this talk, we provide some methods for solving such questions and mention some open questions from algebraic geometry posed by Arend Bayer and Ben Davidson. The talk will be elementary and aimed at people working in noncommutative ring theory.

In this talk, we will present some results concerning the Dixmier property for some families of (Poisson) algebras. One key ingredient to our method is the so-called "relative Dixmier property", which has applications to other topics such as cancellation problem and non-existence of Hopf coactions. 

We will illustrate the richness of the realm of Poisson fields, i.e., fields with Poisson structures.  Even the smallest fields that can support nonzero Poisson structures, namely rational function fields in two variables, carry enormously many non-isomorphic such structures.  We will display a gamut of existence theorems, isomorphism criteria, automorphism groups, and embedding results.  [This is joint work with James Zhang.]