I would be interested in discussing PhD opportunities with a student interested in combinatorics, geometry or both. In combinatorics I am interested in graph theory, matroid theory and combinatorial rigidity. In geometry I am interested in discrete and computational geometry, sphere packing and concrete aspects of differential and algebraic geometry. Unifying these topics is the study of geometric graphs and their configuration spaces.
As well as the above purely theoretical topics, I am interested in applications of these topics 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. Universal rigidity on the line, point order, with Bryan Chen, Robert Connelly and Louis Theran, https://arxiv.org/abs/2104.01892.

2. Maximum likelihood thresholds via graph rigidity, with Daniel Bernstein, Sean Dewar, Steven Gortler, Meera Sitharam and Louis Theran, https://arxiv.org/abs/2108.02185.

3. Coincident-point rigidity in normed planes, with Sean Dewar and John Hewetson, https://arxiv.org/abs/2112.10480.

4. Quotient graphs of symmetrically rigid frameworks, with Sean Dewar, Georg Grasegger and Eleftherios Kastis, https://arxiv.org/abs/2202.09165.

5. Uniquely realisable graphs in analytic normed planes, with Sean Dewar and John Hewetson, https://arxiv.org/abs/2206.07426.

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

**Collaborators:**

Daniel Bernstein (Tulane), Bryan Chen, Katie Clinch, Robert Connelly (Cornell), James Cruickshank (NUI Galway), Sean Dewar (RICAM, Linz), Yaser Eftekhari (York), Nick Gill (South Wales), Neil Gillespie (Bristol), 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), Tom McCourt (Queensland), John Owen (Siemens), Stephen Power (Lancaster), Elissa Ross (MESH consultants), Mahdi Sadjadi (Arizona), Bernd Schulze (Lancaster), Jason Semeraro (Leicester), Brigitte Servatius (WPI), Meera Sitharam (Florida), Adnan Sljoka (Kyoto), Shin-ichi Tanigawa (Tokyo), Louis Theran (St Andrews), Mike Thorpe (Arizona), Walter Whiteley (York).

**Event organisation:**

- Focus program on Geometric constraint systems, Fields Institute for mathematical research, Toronto, July-August 2023, http://www.fields.utoronto.ca/activities/23-24/constraint.
- 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.

**2021/2022**

**2020/2021**

Students on the course should consult the moodle page, https://modules.lancaster.ac.uk/course/view.php?id=31433

**2018/2019**

**2018/2019**

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

**2017/2018**

- MATH105 Linear Algebra
- MATH326 Graph Theory

**2016/2017**

**2015/2016**

**2014/2015**

- MATH103 Matrix Methods
- MATH143 Differential Equations