Transformation Groups Seminar

Speaker: Li Cai (Xi'an Jiaotong-Liverpool University)

"On Graph Products of Simplicial Groups"

Time: 09:00

Room: Zoom

In this talk we give a homotopical interpretation of a result of Panov and Veryovkin on the commutator subgroup of a graph product of discrete groups, using the language of simplicial groups. It turns out that the commutator subgroup of a graph product of simplicial groups gives a model of the loop spaces of certain polyhedral products. The Pontryagin algebra of these loop spaces will also be discussed. We will also introduce some questions that require further work.

Geometry and Topology

Speaker: Taylor Brysiewicz (Western)

"Algebraic Matroids, Monodromy, and the Heron Variety"

Time: 15:30

Room: MC 107

Heron's formula gives the area of a triangle in terms of the lengths of its sides. More generally, the volume of any simplex is determined by its edge-lengths via a Cayley-Menger determinant.

In work-in-progress with Seth Asante and Michelle Hatzel, we ask

*Which other sets of volumes of faces of an n-simplex, when known, determine the remaining unknown face-volumes?*.

An answer to this question is encoded in the algebraic matroid of the Heron variety. Moreover, we ask

*When are these unknown volumes recoverable via formulae in the known volumes?*

We answer these questions for n<5 by combining techniques in computational group theory, computer algebra, field theory, and numerical algebraic geometry. Of particular focus is the problem of recovering the 10 edge lengths of a 4-simplex from its 10 triangular face areas, a problem motivated by applications in theoretical physics.

Ph.D. Public Lecture

Speaker: Jacob Imre (Western)

"Asymptotic Approximations of Lambert W and Related Functions"

Time: 10:00

Room: MC 108

In the realm of multivalued functions, certain specimens run the risk of being elementary or complex to a fault. The Lambert W function serves as a middle ground in a way, being non-representable by elementary functions yet admitting several properties which have allowed for copious research. W utilizes the inverse of the elementary function xe^x , resulting in a multivalued function with non-elementary connections between its branches. W_k(z), the solution to the equation z = W_k(z)e^W_k(z) for a “branch number” k ∈ Z, has both asymptotic and Taylor series for its various branches.

This lecture will primarily focus on the asymptotic series approximations of any branch of the lambert W function. The series previously used to approximate the "principal" branch of the Lambert W can be generalized to any branch. Similarly, the presentation will focus on improvements to these series and why they're possible. Also included is a section on a family of functions closely related to Lambert W, and how their asymptotic approximations can be derived from W's.

Colloquium

Speaker: Richard Szabo (Edinburgh)

"Random partitions, instantons and enumerative geometry"

Time: 15:30

Room: MC 107

Counting partitions in diverse dimensions is a long-standing problem in enumerative combinatorics. It also plays a prominent role in the physics of instanton counting and in algebraic geometry through the computation of Donaldson-Thomas invariants. In this talk I will give an overview of these counting problems, and discuss how recent developments in the computation of instanton/Donaldson-Thomas partition functions clarify some open problems in the enumeration of higher-dimensional partitions.

Algebra Seminar

Speaker: Aaron Landesman (MIT)

"The distribution of Selmer groups and ranks of abelian varieties in quadratic twist families over function fields"

Time: 14:30

Room: MC 108

The minimalist conjecture predicts that, in quadratic twist families of abelian varieties, half have rank 0 and half have rank 1. This fits into the larger picture of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics, which predict the distribution of Selmer groups of these abelian varieties. In joint work with Jordan Ellenberg, we prove a version of these heuristics: over function fields over the finite field $\mathbb{F_q}$, we show that the above heuristics are correct to within an error term in $q$, which goes to 0 as $q$ grows. The main inputs are a new homological stability theorem in topology for a generalized version of Hurwitz spaces and an expression of average sizes of Selmer groups in terms of the number of rational points on these Hurwitz spaces over finite fields.

Graduate Seminar

Speaker: Elaine Murphy (Western)

" The Mathematical Structure of Point Mutations"

Time: 15:30

Room: MC 107

Mutation is the engine of evolution. By considering only single point mutations (SNPs) on DNA sequences, we see a natural group theoretic model of mutations acting on the set of nucleotides. In this talk, we will investigate the implications of this structure for synonymous mutations (mutations that do not change the encoded amino acids) and how this affects the notion of distance between two genetic sequences.