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 am part of Lancaster's Discrete Mathematics and Geometric Rigidity group. Details about our group are available on our webpage. Typical problems in geometric rigidity theory 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.

**Submitted papers:**

1. Rigidity of frameworks on expanding spheres, with Bernd Schulze, Shin-Ichi Tanigawa and Walter Whiteley, 22 pages, http://arxiv.org/abs/1501.01391

2. Inductive constructions for combinatorial local and global rigidity, with Elissa Ross, 27 pages. Invited chapter for Handbook of Geometric Constraint Systems Principles, edited by Jessica Sidman, Meera Sitharam and Audrey St John.

3. Rigid cylindrical frameworks with two coincident points, with Bill Jackson and Viktoria Kaszanitzky, 21 pages, https://arxiv.org/abs/1607.02039

4. Point-hyperplane frameworks, slider joints, and rigidity preserving transformations, with Yaser Eftekhari, Bill Jackson, Bernd Schulze, Shin-Ichi Tanigawa and Walter Whiteley, 33 pages, https://www.arxiv.org/abs/1703.06844

5. Global rigidity of generic frameworks on the cylinder, with Bill Jackson, 31 pages, https://arxiv.org/abs/1610.07755

6. Tensegrity, with Robert Connelly, 21 pages. Invited chapter for Handbook of Geometric Constraint Systems Principles, edited by Jessica Sidman, Meera Sitharam and Audrey St John.

7. Double-distance frameworks and mixed sparsity graphs, with Stephen Power, 29 pages, https://arxiv.org/abs/1709.06349.

8. Change of metrics in rigidity theory, with Walter Whiteley, 31 pages. Invited chapter for Handbook of Geometric Constraint Systems Principles, edited by Jessica Sidman, Meera Sitharam and Audrey St John.

Information on my papers is also available at google scholar.

**Collaborators:**

Robert Connelly (Cornell), Yaser Eftekhari (York), Nick Gill (South Wales), Neil Gillespie (Bristol), Bill Jackson (Queen Mary), Viktoria Kaszanitzky (Budapest), 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), Adnan Sljoka (Kyoto), Shin-ichi Tanigawa (Tokyo), Louis Theran (St Andrews), Mike Thorpe (Arizona), Walter Whiteley (York).

**Event organisation:**

- 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.

This year (2017-18) I am helping to organise the Algebra, Combinatorics and Measure seminar. We also have regular informal seminars in geometric rigidity with mostly internal speakers.

**2017/2018**

Students on the course should consult the moodle page, https://modules.lancaster.ac.uk/course/view.php?id=22118

Students on the course should consult the moodle page, https://modules.lancaster.ac.uk/course/view.php?id=22168

**2016/2017**

**2015/2016**

**2014/2015**

- MATH103 Matrix Methods
- MATH143 Differential Equations