During the Fall 2010 semester, the graduate algebra seminar is organized by Josh Wiscons and Jason Hill. The seminar is scheduled for Tuesday at 4:00PM in MATH220.

## Schedule

### Matthew Moore (CU) - Natural Dualities II: Conditions for Duality

Tuesday, November 16, 4:00PM in MATH220
Working from the definitions given last time, we determine necessary and sufficient conditions for a topological space to yield a duality. With the correct assumptions, these conditions become finitary and will lead us to the main theorem: If the lattice of congruences of an algebra is distributive and it has a near unanimity term, then it is dualizable. The converse to this theorem will be discussed, and we will use the near unanimity duality theorem to construct the Priestley duality for distributive lattices.

A handout containing definitions and pertinent examples will be distributed, so
attendance to the first part of this talk is not a prerequisite.

### Joshua Sanders (CU) - DPSC, Part 1

Tuesday, November 2, 4:00PM in MATH220
Abstract: In Part 1 of my two-part talk, I will define/discuss most of the concepts needed to talk about definable principle sub-congruences (DPSC), such as algebras, varieties, congruences, DPC, etc, taking examples from varieties of groups.

### Matthew Moore (CU) - Natural Dualities I: Motivation, Definition, and Basic Results

Tuesday, October 18, 4:00PM in MATH220
Abstract: A natural duality is a adjoint-like property for a pair of representable contravariant functors. For a finite algebra A and algebraic variety V=HSP(A) and given certain conditions on A, we construct a categorical natural duality between V and the topological variety T=HS_cP_+(A'). This general duality yields many interesting dualities as well as the classic Stone, Priestly, and Pontragin dualities, which we will examine as motivation for the definition. Time permitting, we will examine the (finitary) algebraic conditions necessary for a duality to exist, as opposed to the obvious categorical conditions needed.

### Jason B. Hill (CU) - Permutation Group Algorithms, Part 3

Tuesday, October 12, 4:00PM in MATH220
Abstract: There are two topics for today. First, we'll consider tree-pruning and base rewriting as methods for improving subgroup search problems that are, in general, not known to be in polynomial-time. (E.g., set stabilizers, centralizers and normalizers.) Then, we'll finish with a survey of known polynomial-time reductions of such searches based on polynomial-time decidable group structures.

### Jason B. Hill (CU) - Permutation Group Algorithms, Part 2

Tuesday, October 5, 4:00PM in MATH220
Abstract: We'll finish discussing polynomial time algorithms with the construction of stabilizer chains and strong generating sets via the Schreier-Sims algorithm. We'll then utilize our stabilizer chains to introduce the notion of backtrack, where we'll find ourselves exceeding polynomial time in the search for elements and subgroups that satisfy certain properties.

Here is a PDF of slides from this talk.

The following GAP code only implements a basic version of backtrack on the full domain (not only the base). The main purpose is just to help one understand the process of backtrack. Please e-mail me if you have questions about its use.

GAP Backtrack Functions

### Jason B. Hill (CU) - Permutation Group Algorithms, Part 1

Tuesday, September 28, 4:00PM in MATH220
Abstract: Permutation group algorithms comprise one of the most developed areas of computational group theory. During the next couple of weeks, we'll be examining computation in permutation groups. We'll begin this week by restricting our discussion to the theory and implementation of algorithms in polynomial time. Implementations in GAP and Sage will be discussed.

Here is a link to a PDF of notes on CGT by Alexander Hulpke.

Here is a PDF of slides from this talk.

The following GAP code allows one to construct orbits, transversals, Schreier vectors, Schreier generators, stabilizer chains, strong generating sets and bases. It uses the most basic deterministic version of Schreier-Sims. (Of course, GAP does all of these things internally. The purpose of this code is simply to help one learn the algorithms themselves.) Please e-mail me if you have questions about how to use the file or functions.

GAP Schreier-Sims Functions