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Dr Anthony Nixon

Senior Lecturer in Pure Mathematics

Fylde College

LA1 4YF

Lancaster

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 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:

 

 

Current Teaching

2021/2022

  • MATH326 Graph Theory

2020/2021

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

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

 

Career Details

  • August 2021 - present: 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.

Professional Role

  • Theme lead, Combinatorics, since 2021.
  • Director of Postgraduate Research, since 2021.
  • Member of the British Combinatorial Committee, since 2020.
  • Deputy Director of Natural Science (Part I Director of Studies), since 2018.
  • 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.

 

 

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