Abstract: We will use multi-sorted first-order logic to show that, if A is a finite algebra with strongly solvable radical tau and A has a term operation which is sensitive to changes within tau at more than a small number of variables, then A cannot belong to any finitely decidable variety.

Abstract: I will survey some of my recent results about computable Artinian rings and Euclidean domains, while relating these results to classical (i.e. noncomputable) ring theory. In particular I will show that annihilator ideals play a central role in the theory of Artinian rings, and then construct transfinite Euclidean domains of all cardinalities, answering a question of P. Samuel, Nagata, and others.

Abstract: Symmetry breaking in combinatorics involves coloring the elements of a structure so that there are no nontrivial automorphisms of the structure which respect the coloring.  We say that such a coloring distinguishes the structure.

We apply computability theory to this notion and show that there is a computable, finite-valence, pointed graph which is distinguished by a 2-coloring but not by any computable 2-coloring.

We also show that if a computable, finite-branching tree has a distinguishing 2-coloring, then it must have a 0”-computable distinguishing 2-coloring.

Abstract: I conceive the talk to be like a ramble through general algebra and related fields of mathematics such as logic, model theory, graph theory and computational complexity. It will be in part a highly personal and biased offering of some of my reflections on personalities, good results and false starts collected over a long career in research. But also, I will strive to convey a convincing picture of a youthful field currently enjoying a vigorous state of healthy development.

Abstract: For each Turing machine T, we construct an algebra A'(T) such that the variety generated by A'(T) has definable principal subcongruences if and only if T halts, thus proving that the property of having definable principal subcongruences is undecidable. Using this, we present another proof that A. Tarski’s finite basis problem is undecidable.

Abstract: While studying the complexity of the Constraint Satisfaction Problem, Libor Barto and Marcin Kozik discovered the idea of absorption. If B is an absorbing subalegebra of the algebra A, then many kinds of connectivity properties of A are also true for B. This is very useful for proofs by induction, and absorption has since played a role in several other universal algebraic situations.

After giving a taste of how absorption works, we would like to talk about our current project (which is a joint work with Libor Barto): How to decide, given an algebra A with finitely many basic operations and some B subalgebra of A, if B absorbs A.