17th meeting,
12th Dec. 2023, University of York

Speakers:

Izzy Friedlander, Durham University

Gillian Grindstaff, University of Oxford

Jack Southgate, University of St Andrews

Date:

December 12th, 2023

Location:

Dusa McDuff Room (G/N/135),  on the first floor of Department of Mathematics, University of York.

Local organiser:

Emilie Dufresne, University of York

Funding:

We have some funding to support the travel costs of PhD students/early career researchers who wish to attend the meeting. Please send an email e-mail 📬 to Emilie Dufresne (emilie.dufresne@york.ac.uk).

Registration :

If you wish to attend the meeting, please register by sending an e-mail 📬 to Emilie Dufresne (emilie.dufresne@york.ac.uk).

Schedule:

13:30-14:15 Izzy Friedlander, The MacWilliams Identity for Krawtchouk Association Schemes

14:20-15:05 Jack Southgate, Studying Volume Rigidity of Simplicial Complexes

15:10-15:40 Tea and biscuits ☕ 🍪

15:40-16:25 Gillian Grindstaff, Expanding statistics in phylogenetic tree space

Izzy Friedlander, Durham University

Title: The MacWilliams Identity for Krawtchouk Association Schemes.

Abstract: The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of its dual is the MacWilliams Identity, first developed for the Hamming association scheme. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via eigenvalues of the association scheme. The functional transformation form can, in particular, be used to derive important moment identities for the weight distribution of codes. In this talk, we focus initially on extending the functional transformation to codes based on skew-symmetric and Hermitian matrices. A generalised b-algebra and new fundamental homogeneous polynomials are then identified and proven to generate the eigenvalues of a specific subclass of association schemes, Krawtchouk association schemes. Based on the new set of MacWilliams Identities, we derive several moments of the weight distribution for all of these codes.

Gillian Grindstaff, University of Oxford

Title: Expanding statistics in phylogenetic tree space

Abstract: For a fixed set of n leaves, the moduli space of weighted phylogenetic trees is a fan in the n-pointed metric cone. As introduced in 2001 by Billera, Holmes, and Vogtmann, the BHV space of phylogenetic trees endows this moduli space with a piecewise Euclidean, CAT(0), geodesic metric. This has been used to define a growing number of statistics on point clouds of phylogenetic trees, including those obtained from different data sets, different gene sequence alignments, or different inference methods. However, the combinatorial complexity of BHV space, which can be most easily represented as a highly singular cube complex, impedes traditional optimization and Euclidean statistics: the number of cubes grows exponentially in the number of leaves. Accordingly, many important geometric objects in this space are also difficult to compute, as they are similarly large and combinatorially complex. In this talk, I’ll discuss specialized regions of tree space and their subspace embeddings, including affine hyperplanes, partial leaf sets, and balls of fixed radius in BHV tree space. Characterizing and computing these spaces can allow us to extend geometric statistics to areas such as supertree contruction, compatibility testing, and phylosymbiosis.

Jack Southgate, University of St Andrews

Title: Studying Volume Rigidity of Simplicial Complexes

Abstract: Rigidity theory is concerned with when and how we may deform sets of points in some space, subject to (usually geometric) constraints, such as being the vertices of a graph in  Euclidean space whose edge-lengths are fixed, or being a subset of entries of a PSD matrix. Arrays of tools have been developed to study rigidity that are often specific to the setting being considered. In this talk we introduce the setting of, and the tools used to study the volume rigidity of simplicial complexes - noting how these relate to the theories of rigidity in different, more well-understood, settings.

Sponsors:

We are grateful for the financial support from the Glasgow Mathematical Journal Learning and Research Support Fund, from the Edinburgh Mathematical Society and the London Mathematical Society.