Speakers:

Kitty Meeks, University of Glasgow 

Sihem Mesnager, University of Paris VIII 

Emily Roff, University of Edinburgh

Anna Seigal, Oxford University

Date:

December 12, 2019

Location:

 Oxford Mathematical Institute. 

Local organiser:

 Heather Harrington, University of Oxford

Registration :

If you wish to attend this conference, please register by sending an e-mail to Heather Harrington. If you wish to attend the meeting in Oxford and require financial support with the travel and accommodation costs, please also email Dimitra Kosta (Scottish based students) or Emilie Dufresne (rest of UK).

Schedule:

• 10:00-11:00 tea and coffee, welcome

• 11:00-12:00 Kitty Meeks (University of Glasgow)  Computational complexity in multi-layer structures

• 12:00-13:00 Lunch

• 13:00-14:00 Sihem Mesnager (University of Paris VIII)  Linear codes from functions over finite fields

• 14:00-15:00 Emily Roff (University of Edinburgh)  Magnitude and diversity

• 15:00-15:30 tea and coffee

• 15:30-16:30: Anna Seigal (Oxford)  Tensors under congruence action

Titles and Abstracts 

Kitty Meeks: Computational complexity in multi-layer structures

Real-world systems often involve qualitatively different types of relationships between different objects: for example, in order to understand a social network, we can consider both online and face-to-face contact.  Many computational questions we might want to answer about such systems are intractable unless we have some additional information about the structure.  The different layers typically have different structural properties (for example, face-to-face contact will be much more influenced by geography than online contact) and this talk will address the following question: when can we make use of algorithmically useful structure in the individual layers to solve problems efficiently on the whole system? 

Sihem Mesnager: Linear codes from functions over finite fields  

Linear codes have diverse applications in secret sharing schemes, authentication codes, communication data storage devices, consumer electronics, association schemes, strongly regular graphs and secure two-party computation. In this talk, we focus on linear codes from cryptographic functions and polynomials over finite fields.

We shall discuss two particular classes of linear codes: minimal linear codes (which have significant applications in secret sharing schemes and secure two-party computation) and locally recoverable codes (which currently form one of the rapidly developing topics in coding theory because of their applications in distributed and cloud storage systems).

Emily Roff : Magnitude and diversity

Magnitude is a numerical invariant of enriched categories, whose specialisation in familiar settings gives rise to size-like invariants such as cardinality and Euler characteristic. Any metric space may be regarded as an enriched category, and when interpreted for metric spaces magnitude turns out to encode classical geometric information including volume, dimension, and curvature. Applied to point clouds, it captures qualitative features such as `effective number of points' and `effective dimension'.

In this talk, I will tell the story of magnitude and an associated homology theory, with reference to work by Leinster, Shulman, Otter and others. I will also introduce a family of distance-sensitive measures of the spread, or entropy, of a probability distribution on a compact metric space. These measures are referred to as diversities, because their definition in a finite setting (in work by Cobbold and Leinster) was first motivated by problems in theoretical ecology. I will describe recent joint work with Tom Leinster, in which, extending results due to Leinster and Meckes, we prove a maximisation theorem for diversity on compact spaces. The theorem yields a numerical invariant - the maximum diversity of a metric space - which is closely related to magnitude. Studying diversity-maximising measures on a space as its metric is scaled, we also obtain something like a canonical `uniform’ measure on any compact metric space.

Anna Seigal : Tensors under congruence action

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. This talk is based on joint work with Max Pfeffer and Bernd Sturmfels.

Sponsors:

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