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

Abstract: In collaboration with Robert Bieri and Daniel Studenmund, we have been studying tessellations of Euclidean spaces which arise from families of hyperplanes. A rich class of examples come from a finite type root system and associated finite Weyl group, W, whose affine extension acts on the tessellation. We have also seen examples which do not come from a root system and Weyl group, so we want to understand exactly what geometric properties of the hyperplane families are needed for our project. Our goal has been to define and study piecewise isometry groups acting on such tessellations. In this talk I will discuss the details of some Weyl and some non-Weyl tessellations.

Abstract: The conjugacy problem for a finitely generated group $G$ asks for an algorithm which, on input a pair of words u and v, declares whether or not they represent conjugate elements of $G$. The conjugator length function $CL$ is its most direct quantification: $CL(n)$ is the minimal $N$ such that if $u$ and $v$ represent conjugate elements of $G$ and the sum of their lengths is at most $n$, then there is a word $w$ of length at most $N$ such that $uw=wv$ in $G$. I will talk about why this function is interesting and how it can behave, and I will highlight some open questions. En route I will talk about results variously with Martin Bridson, Conan Gillis, and Andrew Sale, as well as recent advances by Conan Gillis and Francis Wagner.

Abstract: If $\mathfrak{P}$ is a property pertaining to subgroups of a $p$-group $G$, and if each subgroup with property $\mathfrak{P}$ contains $Z(G)$, then a group $G$ whose subgroups are dense with respect to property $\mathfrak{P}$ must satisfy the following criteria:

$|Z(G)|= p$ and every subgroup $H$ of order at least $p^2$ contains $Z(G)$.

I will discuss our progress in obtaining a classification of all such $p$-groups. This is joint work with Mark Lewis and Tae Young Lee.

Abstract: It is a well-known fact that if more than 3/4 of the elements of a finite group are involutions then the group is abelian. Berkovich proved that if more than 4/15 are involutions then the group must be solvable. Motivated by these results, Deaconescu asked the following question: If at least half of the elements are of the same order, $k$, does the group have to be solvable? In this talk, we prove this when $k = p^a$ for primes $p$ except when $p = 2,3$ and $a > 1$, and give counterexamples for larger powers of 2 and 3 except $k = 4$, and also for several other types of composite numbers. We also show that when $k > 4$, it is always possible to find a non-solvable group such that at least 3/19 of its elements have order $k$. This is a joint work with Ryan McCulloch.

Abstract: It is well known that for a finite group G and a proper subgroup A of G, it is impossible to cover G with the conjugates of A. Thus, instead of the conjugates of A, we take the conjugates of the coset Ax in G and check if the union of $(Ax)^g$ covers G-{1} for g in G. Moreover, if $(Ax)^g$ covers G for all Ax in Cos(G:A), we say that (G,A) is CCI. We are aiming to classify all such pairs. It has been proven by Baumeister-Kaplan-Levy that this can be reduced to the case where A is maximal in G, and so that the action of G on Cos(G:A) is primitive, here Cos(G:A) stands for the set of right cosets of A in G. And they showed that (G,A) is CCI if G is 2-transitive. By O'Nan-Scott Theorem and CFSG (classification of finite simple groups), we see that G is either an affine group or almost simple. In the paper by Baumeister-Kaplan-Levy, it is shown that affine CCI groups are 2-transitive. Thus, it remains to consider the almost simple groups. By employing the knowledge of buildings, representation theory, and Aschbacher-Dynkin theorem, we prove that, apart from finitely many small cases, the CCI almost simple groups are 2-transitive.

Abstract: A lattice L is a subgroup of $\mathbb{R}^n$ isomorphic to $\mathbb{Z}^n$. Finding a vector in L of the shortest length has many applications in number theory, cryptography and optimisation. While finding a vector with the shortest length is an NP hard problem, the LLL algorithm finds a “short enough” vector in Polynomial time. In this talk, we will go over the LLL algorithm and demonstrate one of its applications, finding a Diophantine approximation for a finite set of rational numbers.

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