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.

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 symmetric linearly constrained frameworks in the plane, with Bernd Schulze and Joseph Wall, https://arxiv.org/abs/2312.12158.

5. Rigid frameworks with dilation constraints, with Sean Dewar and Andrew Sainsbury, https://arxiv.org/abs/2402.14093.

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

7. 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.

8. 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.

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

**Collaborators:**

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

**Event organisation - upcoming:**

- Semester program on Geometry of materials, packings and rigid frameworks, ICERM, Brown, January-May 2025, https://icerm.brown.edu/programs/sp-s25/.
- Heilbronn focused research group, September 2024, Lancaster University.

**Event organisation - past:**

- 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×tamp=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:*

Ben Smith (2023 - 2026)

John Hewetson (2022-2023)

Daniel Bernstein (2021)

Sean Dewar (2021)

Georg Grasegger (2021)

Alexander Heaton (2021)

Eleftherios Kastis (2021)

*PhD students:*

Rebecca Monks (2023-)

Daniel Hodgson (2021-)

Jack Trainer (2021-)

Andrew Sainsbury (2020-, part-time)

Joseph Wall (2019-)

John Hewetson (2018-2022)

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.