I would be interested in discussing PhD opportunities with a student interested in graph theory, matroid theory, discrete geometry, algebraic geometry, algebraic statistics or matrix/tensor product completions. Specifically I work in combinatorial rigidity which combines ideas from combinatorics, algebra and geometry to study problems related to each of the above topics. Unifying these topics is the study of geometric graphs and their configuration spaces. As well as the above theoretical topics, I am interested in applications of these topics, for example to biophysical materials and control of robotic formations.

• Combinatorial rigidity
• Graphs and matroids
• Discrete and computational geometry
• Algebraic and geometric techniques in discrete mathematics

I lead Lancaster's Combinatorics research theme and I'm also part of our Geometric Rigidity research theme. 

I'm typically interested in combinatorial problems in geometric rigidity theory. These involve determining the nature of the solutions to systems of equations arising from geometric constraint systems. I am particularly interested in the generic behaviour and in understanding this behaviour in purely combinatorial terms.

At a basic level we consider the rigidity or flexibility of structures defined by geometric constraints (fixed length, angle, direction, etc.) on a set of rigid objects (points, lines, etc.). The fundamental example being that of bar-joint frameworks which are geometric realisations of graphs with edges represented by stiff bars and vertices by revolute joints.

To study such frameworks, rigidity uses a range of techniques from analysis, algebra, combinatorics and geometry. In particular the combinatorial side uses ideas from structural graph theory, combinatorial optimization and matroid theory, while the geometric side uses diverse ideas from projective geometry, matrix analysis, real (semi-)algebraic geometry and semi-definite programming, among others. 

From January to April 2021 I taught a graduate course on combinatorial and geometric rigidity at the Fields Institute. The course information (including links to the lecture recordings) is here, http://www.fields.utoronto.ca/activities/20-21/constraint-CRDG. I also have typed (very rough) lecture notes that I can share on request.

Submitted papers:

1. Rigidity of symmetric frameworks on the cylinder, with Bernd Schulze and Joseph Wall, https://arxiv.org/abs/2210.06060.

2. Identifiability of points and rigidity of hypergraphs with algebraic constraints, with James Cruickshank, Fatemeh Mohammadi and Shin-Ichi Tanigawa, https://arxiv.org/abs/2305.18990.

3. On the uniqueness of collections of pennies and marbles, with Sean Dewar, Georg Grasegger, Kaie Kubjas and Fatemeh Mohammadi, https://arxiv.org/abs/2307.03525.

4. Rigidity of nearly planar classes of graphs, with Sean Dewar, Georg Grasegger, Eleftherios Kastis and Brigitte Servatius, https://arxiv.org/abs/2402.17499.

5. Angular constraints on planar frameworks, with Sean Dewar, Georg Grasegger, Zvi Rosen, William Sims, Meera Sitharam and David Urizar, https://arxiv.org/abs/2403.16145.

6. Single-cell 3D genome reconstruction in the haploid setting using rigidity theory, with Sean Dewar, Georg Grasegger, Kaie Kubjas and Fatemeh Mohammadi, https://arxiv.org/abs/2407.10700.

7. Spanning disks in triangulations of surfaces, with Katie Clinch, Sean Dewar, Niloufar Fuladi, Maximilian Gorsky, Tony Huynh, Eleftherios Kastis and Brigitte Servatius, https://arxiv.org/abs/2410.04450.

8. Realizations of frameworks, with Zvi Rosen and Jessica Sidman.

9. k-fold circuits in matroids, with Bill Jackson and Ben Smith, https://arxiv.org/abs/2412.14782.

10. Stable cuts, NAC-colourings and flexible realisations of graphs, with Katie Clinch, Daniel Garamvolgyi, John Haslegrave, Tony Huynh and Jan Legersky,https://arxiv.org/abs/2412.16018.

11. Rigid three-dimensional realisations of maximal planar graphs on parallel planes and lines, with Viktoria Kaszanitzky.

12. A tropical approach to rigidity: counting realisations of frameworks, with Oliver Clarke, Sean Dewar, Daniel Green Tripp, James Maxwell, Yue Ren and Ben Smith, https://arxiv.org/abs/2502.10255.

Information on my papers is also available at my google scholar page.

Collaborators:

Daniel Bernstein (Tulane), Bryan Chen, Oliver Clarke (Durham), Katie Clinch (New South Wales), Robert Connelly (Cornell), James Cruickshank (NUI Galway), Sean Dewar (Bristol), Yaser Eftekhari (York), Niloufar Fuladi, Daniel Garamvolgyi (Eotvos Lorand), Nick Gill (Open U), Neil Gillespie (Riverlane), Maximilian Gorsky, Steven Gortler (Harvard), Georg Grasegger (RICAM, Linz), Daniel Green Tripp (Bristol), Hakan Guler (Kastomonu), John Haslegrave (Lancaster), John Hewetson (Lancaster), Tony Huynh, Bill Jackson (Queen Mary), Eleftherios Kastis (Lancaster), Viktoria Kaszanitzky (Budapest), Derek Kitson (Mary Immaculate College), Kaie Kubjas (Aalto), Jan Legersky (Prague), James Maxwell (Bristol), Tom McCourt (Queensland), Fatemeh Mohammadi (Leuven), Harshit Motwani (Leuven), John Owen (Siemens), Stephen Power (Lancaster), Sean Prendiville (Lancaster), Yue Ren (Durham), Elissa Ross (Metafold), Zvi Rosen (Florida Atlantic), Mahdi Sadjadi (Arizona), Andrew Sainsbury (Lancaster), Bernd Schulze (Lancaster), Jason Semeraro (Loughborough), Brigitte Servatius (WPI), Jessica Sidman (Amherst), William Sims (Florida), Meera Sitharam (Florida), Adnan Sljoka (Kyoto), Ben Smith (Lancaster), Shin-ichi Tanigawa (Tokyo), Louis Theran (St Andrews), Mike Thorpe (Arizona), David Urizar (Florida Atlantic), Joseph Wall (Lancaster), Walter Whiteley (York).

Event organisation - upcoming:

• Mathematics, AI and Data Science for Materials Innovation, a DSI and MARS workshop, Lancaster Univeristy, June 9-13 2025.
• Semester program on Geometry of materials, packings and rigid frameworks, ICERM, Brown, January-May 2025, https://icerm.brown.edu/programs/sp-s25/.

Event organisation - past:

• Heilbronn focused research group, Discrete Structures, September 2024, Lancaster University.
• Rigidity in action workshop, part of a RICAM special semester, April 8-12, 2024, https://www.ricam.oeaw.ac.at/specsem/specsem2024/
• Focus program on Geometric constraint systems, Fields Institute for mathematical research, Toronto, July-August 2023, http://www.fields.utoronto.ca/activities/23-24/constraint.
• 19th Cologne-Twente workshop on graphs and combinatorial optimization, program committee member, June 14-16 2023, Garmisch-Partenkirchen, Germany, see https://ctw2023.comtessa.org/.
• Graph rigidity and applications, April 17-21 2023, Lancaster University, https://www.lancaster.ac.uk/maths/graph-rigidity-and-applications-2023/.
• 29th British Combinatorial Conference, July 11-15 2022, Lancaster University, https://www.lancaster.ac.uk/maths/bcc2022/
• Thematic program on Geometric constraint systems, framework rigidity, and distance geometry, Fields Institute for mathematical research, Toronto, January - June 2021, http://www.fields.utoronto.ca/activities/20-21/constraint
• 18th Cologne-Twente workshop on graphs and combinatorial optimization, program committee member, September 14-16 2020, Ischia, Italy, see http://ctw2020.iasi.cnr.it/
• Graph rigidity and control of robotic formations, Research in Groups, ICMS (Edinburgh), July-August 2020.
• Circle packings and geometric rigidity, ICERM, July 6-10 2020, see https://icerm.brown.edu/topical_workshops/tw-20-cpgr/.
• Heilbronn focused research group, Discrete Structures, January 2020, Lancaster University, see https://www.lancaster.ac.uk/maths/discrete-structures-2020/.
• Rigidity and flexibility of microstructures, American Institute of Mathematics, November 4-8 2019, see https://aimath.org/workshops/upcoming/flexmicro/
• SIAM applied algebra and geometry (minisymposium - Algebraic geometry and combinatorics of jammed structures), Bern, July 9-13 2019, see https://mathsites.unibe.ch/siamag19/
• Geometric constraint systems: rigidity, flexibility and applications, Lancaster, June 11-14 2019, see https://www.lancaster.ac.uk/maths/geometric-constraint-systems-2019/
• British Mathematical Colloquium (combinatorics workshop), Lancaster, April 8-11 2019, see https://www.lancaster.ac.uk/maths/bmc2019/
• Rigidity and flexibility of geometric structures, Erwin Schrodinger Institute for mathematics and physics, Vienna, September 24-28 2018, see https://www.esi.ac.at/activities/events/2018/rigidity-and-flexibility-of-geometric-structures
• Circle packings and geometric rigidity, Collaborate@ICERM, ICERM (Brown), August 2018.
• Stability of flat structures, Research in Groups, ICMS (Edinburgh), July 2018.
• Bond-node structures: rigidity, combinatorics and chemistry (Lancaster) June 2018, see http://www.lancaster.ac.uk/maths/bond-node-structures-2018/.
• Bond-node structures: rigidity, combinatorics and materials science (Lancaster) June 2017, see http://www.lancaster.ac.uk/maths/bond-node-structures/.
• Geometric Rigidity workshop (Lancaster) June 2016, see http://www.lancaster.ac.uk/maths/news-and-events/events/?view=fulltext&month=02&day=25&year=2016&id=d.en.237588&timestamp=1465131600&.
• Global Rigidity workshop (BIRS, Canada) July 2015, see http://www.birs.ca/events/2015/2-day-workshops/15w2199.
• Geometric and Topological Graph Theory (Bristol) April 2013, see http://www.maths.bris.ac.uk/~maakn/GTGT2013.

My group:

Postdocs:

Zeyuan He (2024 - 2025, ICERM)

Ben Smith (2023 - 2026, EPSRC)

John Hewetson (2022-2023, EPSRC)

Daniel Bernstein (2021, Fields Institute)

Sean Dewar (2021, Fields Institute)

Georg Grasegger (2021, Fields Institute)

Alexander Heaton (2021, Fields Institute)

Eleftherios Kastis (2021, Fields Institute)

PhD students:

Rebecca Monks (2023-)

Daniel Hodgson (2021-)

Jack Trainer (2021-)

Andrew Sainsbury (2020-, part-time)

Joseph Wall (2019-2024)

John Hewetson (2018-2022)

2023/2024

• MATH326 Graph Theory

2022/2023

• MATH326 Graph Theory

2021/2022

• MATH326 Graph Theory

2020/2021

• Graduate course on combinatorial and geometric rigidity, taught at the Fields Institute, videos of the lectures available, http://www.fields.utoronto.ca/activities/20-21/constraint-CRDG
• MATH326 Graph Theory

2018/2019

• MATH326 Graph Theory

2018/2019

• MATH326 Graph Theory
• MATH491 (pure math) and MATH492 (statistics) Dissertation Coordinator

2017/2018

• MATH105 Linear Algebra
• MATH326 Graph Theory

2016/2017

• MATH105 Linear Algebra

2015/2016

• MATH103 Matrix Methods

2014/2015

• MATH103 Matrix Methods
• MATH143 Differential Equations

• August 2024 - present: Professor in Mathematical Sciences, Lancaster University
• August 2023 - July 2024: Reader in Pure mathematics, Lancaster University.
• August 2021 - July 2023: Senior Lecturer in Pure mathematics, Lancaster University.
• July 2015 - July 2021: Lecturer in Pure Mathematics, Lancaster University.
• September 2014 - June 2015: Temporary Lecturer in Pure Mathematics, Lancaster University.
• January 2014 - August 2014: Postdoctoral fellow at York University, Canada.
• January 2012 - December 2013: Postdoctoral fellow at the University of Bristol.
• July 2011 - December 2011: Postdoctoral fellow at the Fields Institute, University of Toronto, for the special semester on Discrete Geometry and Applications.
• Ph.D. Mathematics, Lancaster University, November 2011.

• PDR Group lead, since 2022.
• Theme lead, Combinatorics, since 2021.
• Director of Postgraduate Research, since 2021.
• Deputy Director of Natural Science (Part I Director of Studies), 2018-2022.
• Year 4 Director of Studies (Mathematics and Statistics), 2018-19.
• Member of the LMS scientific committee for the British Mathematical Colloquium, 2016-2020.
• Natural science coordinator (within Mathematics and Statistics), 2014-2019.
• Department representative for British Combinatorial Bulletin, since 2014.

2023-2026, EPSRC grant, Abstract rigidity for natural stability problems, £428,712.

2022-2023, EPSRC grant, The graph rigidity problem in arbitrary dimension, £45,952.

2020-2021, Heilbronn Institute for Mathematical Research, fellowship, £42,135.

See the 'projects tab' for a number of further small grants.

• Associate editor, Journal of Algebraic Combinatorics, since 2024.
• Member of the British Combinatorial Committee, since 2020.