## Algebra Seminar-Alun Stoke/Dan Barake - Classifying the Image of $\phi_p$ by Parameters of the Moduli Space for its $\phi$-congruence Subgroup/A primer on vertex operator algebra theory

### Description

Speaker: Alun Stokes (McMaster University)

Title: Classifying the Image of $\phi_p$ by Parameters of the Moduli Space for its $\phi$-congruence Subgroup

Abstract: The congruence subgroup problem over the modular group, $\modgp = \PSLn[2]{\bbZ}$, is quite mysterious. Most subgroups are non-congruence, but we know of almost no clear descriptions of meaningful families. As such, the search for methods by which to do so is of much interest. Following in the ideas of a recently published work of Babei, Fiori, and Franc's on the generalisation of congruence subgroups to $\phi$-congruence subgroups of the modular group, we investigate the case where $\phi$ takes codomain $\SLn[3]{\bbZ}$. However, instead of addressing provability of lifting to $p$-adics, we elect to consider the images generated by each set of parameters in a moduli representation of $\SLn[3]{\Fq[p]}$. We particularly interest ourselves with when this map surjects --- important when discussing the profinite completion.

With fairly standard character theory, we proved a family of cases that always surject, plus some other uniqueness. Determining why a parameter set may not surject in general is a much more difficult issue, and since our proof covered only one family, we must consider surjective maps separate from our straight-forward case, especially because other sets surject. We optimised abstract syntax trees with evolutionary methods with parsimony, to produce algebraic conditions that indicate surjectivity or otherwise, just in terms of algebraic/boolean operators, but no domain knowledge. We extended this to explicitly describe all such relationships delineating image orders, and identify groups. Currently, this has allowed us to produce several proofs that exactly verify the guess of our models.

Speaker: Dan Barake (McMaster University)

Title: A primer on vertex operator algebra theory

Abstract: Vertex operator algebras (VOAs) are modern and fundamental algebraic structures intimately related to various areas in both mathematics and physics. In this talk we will present these objects alongside two concrete examples, and proceed to introduce results on the theory of coordinate transformations which is instrumental in establishing the connection between VOAs and modular forms.

Location: HH/410
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