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Dr Anthony Nixon

Lecturer in Pure Mathematics

Fylde College

LA1 4YF

Lancaster

Tel: +44 1524 594695

PhD supervision

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.

Research Interests

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

 

 

Current Research

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. A constructive characterisation of circuits in the simple (2,1)-sparse matroid, with Thomas McCourt, 25 pages, http://arxiv.org/abs/1604.05226

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

5. 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

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

7. 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.

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

 

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).

 

 

Conference/workshop organisation:

 

 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.

 

Current Teaching

2017/2018

  • MATH105 Linear Algebra

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

  • MATH326 Graph Theory

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

 

2016/2017

  • MATH105 Linear Algebra

 

2015/2016

  • MATH103 Matrix Methods

 

2014/2015

  • MATH103 Matrix Methods
  • MATH143 Differential Equations

 

Career Details

  • July 2015 - present: 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.
  • 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.

Professional Role

  • Natural Science Coordinator
  • Department representative for British Combinatorial Bulletin

 

 

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