A seminar series.
Matthew Krauel
(Sacramento State)
Classification theories surrounding lattices and other algebraic structures
Given a definition of an algebraic structure, a common question is "how many" of these structures are there? That is, we'd like to classify all the algebraic structures satisfying the definition. For example, if you've taken group theory, you may know how many groups of order four are there (up to isomorphism). In this talk we discuss some important classifications of algebraic structures, and in particular, those to do with rank 24 lattices. We will then discuss recent research surrounding some classifications of structures known as vertex operator algebras. This talk will be experimental inasmuch as it will be leaning heavily on looking at webpages and research articles while attempting to put together an overview of some large research areas. Most (if not all) technical aspects will be suppressed. Therefore, while having taken group theory will be useful, it will not be necessary for the talk.
Joshua Wiscons
(Sacramento State)
Representations of Sym(n) of minimal dimension
A representation of a group G typically refers to a homomorphism from G into a matrix group, thus giving a way to concretely realize G in terms of matrices and allowing for additional (linear algebraic) tools to analyze it.
In this talk, we discuss representations of the symmetric group Sym(n) (i.e. all permutations of n identical items), focusing on determining the smallest dimension d for which Sym(n) can be faithfully represented as d x d matrices. This particular result goes back to Leonard Dickson in 1907, and here we also present a new, more general context where the same result is obtained, though many questions remain. Moreover, this talk serves to introduce a series of talks slated for Spring 2021 around similar topics and with a variety of open questions.
The talk will begin with a brief introduction to representations; prior exposure to groups is all that is required.