# Combinatorics

## Group Members

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## Projects

LMS Research School on Rigidity, Flexibility and Applications

18/07/2022 → 22/07/2022

Research

29th British Combinatorial Conference - Invited Speakers

01/07/2022 → 31/07/2022

Research

29th British Combinatorial Conference - Support

01/07/2022 → 31/07/2022

Research

2British Combinatorial Conference - mini-symposia (BCC-MS)

01/07/2022 → 31/07/2022

Research

Positive Fuss-Catalan Combinatorics and Representation Theory

05/07/2021 → 29/08/2021

Research

Research in Pairs - Scheme 4 - Support of collaborative research with Professor Walter Whiteley at York University, Toronto

01/02/2021 → 31/05/2021

Research

The Alliance Hubert Curien Programme 2021: Noncummuatative Probabilty, Matrix Analysis and Quantum Groups (NP, MA & Q Groups)

01/01/2021 → 31/12/2022

Research

Permutation Patterns as Noncommutative Central Limits

01/10/2020 → 30/09/2021

Research

Mathematical Theory of Symbiosis

01/08/2020 → 31/07/2021

Research

Support of Collaborative Research with Prof E. Steingrimmson at Hrísey, Iceland

01/08/2020 → 31/07/2021

Research

Heilbronn fellowship

01/07/2020 → 31/08/2021

Research

Globally Rigid Linearly Constrained Frameworks

11/01/2020 → 10/04/2020

Research

Discrete structures: algebra, combinatorics and geometry

10/01/2020 → 09/02/2020

Research

IWOTA Lancaster 2020: International Workshop on Operator Theory and its Applications to be held at Lancaster from 17th to 21st August 2020 (Postponed until 16-20 August 2021)

01/09/2019 → 31/08/2020

Research

LIMITS: Limits of Structures in Algebra and Combinatorics

01/02/2019 → 31/01/2024

Research

Linearly constrained frameworks

01/11/2018 → 31/01/2019

Research

Infinite bond-node frameworks

03/07/2017 → 02/07/2019

Research

LMS Scheme 4 - 41434

01/09/2015 → 30/09/2015

Other

Comibinatorial Rigidity, Symmetric Geometric Constraint Systems and Applications

01/06/2015 → 31/08/2016

Research

EPSRC - Interpretation functors and infinite-dimensional representations of finite-dimensional algebras

09/09/2013 → 08/09/2016

Research

Crystal Frameworks, Operator Theory and Combinatorics

01/09/2012 → 31/08/2014

Research

## Research Activity

Combinatorics is concerned with arrangements of discrete objects according to constraints and the study of discrete structures such as graphs. Large graphs underpin many aspects of data science and can be used to model networks.

Lancaster’s research interests in combinatorics are diverse and closely connected to each of the other pure mathematics research themes in the department.

Additive combinatorics explores how to count arithmetic configurations within unstructured sets of integers - those sets for which we have only combinatorial information, such as their density. Techniques in this field have been developed to tease out the latent structure present in such sets, separating it from random 'noise'. Resolving these combinatorial problems can then feed back into more classical problems of number theory, such as detecting arithmetic structures in the primes.

As combinatorial structures provide convenient means of indexing words in noncommuting variables, many problems in probability and analysis can be approached using methods of diagrammatic calculus. Such methods provide a fundamental bridge between combinatorics and algebra, on the one hand, and probability and analysis, on the other. A well-known example is the moment method, through which the Catalan numbers, ubiquitous in combinatorics, are interpreted as the moments of the semicircle law, the analogue of the Gaussian random variable in random matrix theory. Research in our combinatorics group is closely connected to classical and noncommutative probability, including random matrix theory, and mathematical physics.

Measurable combinatorics concerns extensions of combinatorial theorems in the context of Borel graphs with applications including classical geometrical questions. There is also interest in sparse and dense graph limits where, with a suitable metric imposed on the set of finite graphs, limits of convergent graph sequences are studied.

Matroids are a mathematical structure that extends the notion of linear independence of a set of vectors. They have numerous important applications, for example, in Operational Research and Combinatorial Optimisation. Our research in matroid theory concentrates on real world occurrences of the 2 fundamental motivating examples of matroids: row matroids of matrices and matroids defined on graphs.

Combinatorial aspects of geometric rigidity theory (and its applications in fields such as engineering and robotics) such as recursive constructions of classes of (coloured) sparse graphs. This topic underlies the flexibility and rigidity properties of geometric structures, such as finite and infinite bar-joint frameworks in finite dimensional spaces. Further geometric aspects of combinatorics such as polyhedral scene analysis, equidissection and triangulation problems, and the rich interactions between discrete geometry, combinatorics and symmetry.

Many aspects of mathematics share the same underlying classifications in terms of combinatorics such as Coxeter combinatorics and Fuss-Catalan combinatorics. These underlying combinatorial similarities often indicate deeper connections, e.g. the ubiquitous "finite type" classifications in terms of Dynkin diagrams. In representation theory of algebras, combinatorics is used extensively to get concrete understanding of abstract objects. For example, certain algebras can be associated to ribbon graphs, which in turn can be embedded in surfaces, and directed graphs may be used to encode the multiplication in noncommutative algebras.