The fact that the inner automorphism group of a group is isomorphic to the quotient of the group by its center leads to the following observation: the inner automorphism group is abelian if and only if the group itself is nilpotent of class at most 2. Loops are ``nonassociative groups’’, and as with most results in group theory, this observation does not extend to loops exactly as stated. The question as to how far the result does extend is of interest in loop theory because it reveals the interplay between loops and the permutation groups that act upon them.
Work on this by various people over several years has led to what is now known as the AIM Conjecture (AIM = Abelian Inner Mappings), which I will state precisely in the talk. The conjecture remains open, but is known to hold in many important classes of loops. What makes all this of interest outside the seemingly esoteric world of nonassociative mathematics has been the approach to the problem. The conjecture can be formulated purely equationally and so it can be quite naturally attacked using automated deduction tools such as Prover9. This is where all of the recent success in establishing the conjecture in special cases has come from.
In this talk, after a quick introduction to loops, I will talk about how automated deduction has been used in loop theory over the last couple of decades, and current work on the AIM Conjecture. No background in either loops or automated deduction is needed.
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The finite subgroups of the special unitary group $S\phantom{\rule{0}{0ex}}U(2)$ can be classified into two infinite families and three exceptional subgroups. The McKay correspondence uses representation theory to put these subgroups in bijection with simply laced affine Dynkin diagrams. For a finite subgroup $G$ of $S\phantom{\rule{0}{0ex}}U(2)$, the centralizer algebras of $G$ are the $\u2102\phantom{\rule{0}{0ex}}G$-module endomorphisms of the $k$-fold tensor power of the defining representation, and the dimension of the $k$th centralizer algebra of $G$ can be found by counting closed walks on the Dynkin diagram associated to $G$. I’ll discuss how the combinatorics of each Dynkin diagram controls the structure of the corresponding family of centralizer algebras; in particular I will describe a basis for each algebra which can be derived uniformly from closed walks, and give a presentation for each algebra. Of interest may be that Temperley-Lieb algebras $T\phantom{\rule{0}{0ex}}L(2)$ are the centralizers of $S\phantom{\rule{0}{0ex}}U(2)$, and subalgebras of each other centralizer. This is joint work with Tom Halverson.
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The first square is never a mine... Explosion!!!
The Minesweeper Consistency Problem (MCP) poses the following question: given a partially completed Minesweeper board, is there a way to place mines that satisfies the board? We will delve into the MCP and discuss what it means to be NP-complete along with a standard approach for showing a problem is NP-complete. Some familiarity with the game Minesweeper would be useful (you can play it for free online). No other prerequisite knowledge needed; an overview of the relevant topics in computational complexity will be given.
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See abstract at http://euclid.colorado.edu/~jnc/Fels-abstract.pdf