Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

Abstract: Every finite simple group can be generated by two elements and furthermore, every nontrivial element is contained in a generating pair. Groups with this property are said to be 3/2-generated. Thompson’s group V, a finitely presented infinite simple group, is one of a small number of examples of infinite noncyclic 3/2-generated groups. I will present a constructive proof of this fact and mention extensions of this theorem to generalizations of V.

Abstract: Constraint satisfaction problems (CSPs) form a class of combinatorial decision problems generalizing graph colorability and Boolean satisfiability. In this expository talk, I will explain how ideas from universal algebra have been instrumental in classifying the computational complexity of CSPs.

Abstract: A sublattice $L$ of the integer lattice $\mathbb{Z}^d$ is called co-cyclic when the quotient $\mathbb{Z}^d/L$ is a cyclic group. Approximately $85\%$ of sublattices of finite index in $\mathbb{Z}^d$ are co-cyclic. This can be proven by either counting solutions to linear congruence equations or using zeta function methods. We show a similar result holds for subgroups of the discrete Heisenberg groups $H_{2d+1}.$

Abstract: BCK-algebras are the algebraic semantics of a non-classical logic. Like for commutative rings, there is a notion of a prime ideal in these algebras, and the set of prime ideals is a topological space called the spectrum. By work of Stone (and later, Priestley), there is a close connection between these spectra and distributive lattices with 0.

In this talk I will discuss some recent results on the interplay between commutative BCK-algebras, their spectra, and distributive lattices.

Abstract: Let $M$ be a finite dimensional $kG$-module for a finite group $G$ over a field $k$ of characteristic $p$. In a recent paper Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M).$ This invariant measures the non-projective proportion of the module $M$. In this talk, we will see some interesting properties of this invariant. We will then determine this invariant for permutation modules of the symmetric group corresponding to two-part partitions and present a combinatorial formula for the same using tools from representation theory and combinatorics.