### Random Networks

Network scientists employ various models to understand real world networks such as the world-wide web. Among the most popular in recent years are those employing the Preferential Attachment paradigm, where new nodes joining the system are more attracted to previous nodes with many connections than to those with few. I have been investigating the infinite limits of such processes.

• *Preferential Attachment Processes Approaching The Rado Multigraph*; submitted. ArXiv Link.

• *A Linear Preferential Attachment Process Approaching The Rado Graph*; Proceedings of the Edinburgh Mathematical Society (May 2020 , pp. 443-455). ArXiv Link.

• *Evolving Shelah-Spencer Graphs*; Mathematical Logic Quarterly (to appear). ArXiv Link.

### Complex Systems

I am interested in evolution and pattern-formation within nonlinear and complex systems. With Andy Lewis-Pye and George Barmpalias, I have been analyzing a fascinating model of racial segregation first proposed by the Nobel-Prize winning economist Thomas Schelling. As an introduction, see my blog-posts (1 & 2) on the subject. Our papers:

• *Digital Morphogenesis via Schelling Segregation*, with Barmpalias and Lewis-Pye, FOCS 2014: 156-165, Arxiv Link

• *Tipping Points in Schelling Segregation*, with Barmpalias and Lewis-Pye, Journal of Statistical Physics, February 2015, Volume 158, Issue 4, pp 806-852, Arxiv Link

• *From Randomness to Order: Unperturbed Schelling Segregation in Two or Three dimensions*, with Barmpalias and Lewis-Pye, Journal of Statistical Physics 164.6 (2016): 1460-1487, Arxiv Link

• *Minority population in the one-dimensional Schelling model of segregation*, with Barmpalias and Lewis-Pye, Journal of Statistical Physics (to appear). Open Access Link

### Mathematical Logic

My background is in Model Theory, a branch of mathematical logic, and its applications to algebra. I’m interested in simple theories, geometric stability theory, the model theory of groups and fields, and the interplay between these.

• *Dimension And Measure In Finite First Order Structures*

PhD thesis, University of Leeds, 2005, (pdf)

• *Asymptotic Classes Of Finite Structures*

**Journal Of Symbolic Logic**, 72, 2, 2007, pp. 418-438; (pdf)

• *A Survey Of Asymptotic Classes and Measurable Structures *(with Dugald Macpherson)

in *Model Theory with Applications to Algebra and Analysis Volume 2* (**Cambridge University Press**, 2008) (pdf)

• *Measurable Groups of Low Dimension* (with Mark Ryten)

**Mathematical Logic Quarterly** 54, No 4, 374-386 (2008)

• *Groups in supersimple and pseudofinite theories* (with Dugald Macpherson, Eric Jaligot, Mark Ryten) **Proceedings of the LMS**, Vol 103, Part 6, 1049-1082. (pdf)

### Lecture Notes

• Slides from my talk *“An amateur’s guide to concrete incompleteness”* (commentary and slides)

• *Galois Maps, Measure, and Generic Automorphisms in Strongly Minimal Sets * (pdf)