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Anthony Nixon supervises 6 postgraduate research students. If these students have produced research profiles, these are listed below:

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Dr Anthony Nixon SFHEA

Reader

Fylde College

LA1 4YF

Lancaster

PhD supervision

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.

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 of ladders on the line, with Bryan Chen, Robert Connelly, Steven Gortler and Louis Theran, https://arxiv.org/abs/2207.08763.

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

3. Rigidity of symmetric frameworks on the cylinder, with Bernd Schulze and Joseph Wall, https://arxiv.org/abs/2210.06060.

4. 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

5. 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

6. Rigidity of symmetric linearly constrained frameworks in the plane, with Bernd Schulze and Joseph Wall.

7. Rigid frameworks with dilation constraints, with Sean Dewar and Andrew Sainsbury.

8. Rigidity of nearly planar classes of graphs, with Sean Dewar, Georg Grasegger, Eleftherios Kastis and Brigitte Servatius.

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), Mahdi Sadjadi (Arizona), Andrew Sainsbury (Lancaster), Bernd Schulze (Lancaster), Jason Semeraro (Loughborough), Brigitte Servatius (WPI), Meera Sitharam (Florida)Adnan Sljoka (Kyoto), Shin-ichi Tanigawa (Tokyo), Louis Theran (St Andrews), Mike Thorpe (Arizona), Joseph Wall (Lancaster), Walter Whiteley (York).

 

 

Event organisation - upcoming:

Event organisation - past:

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:

Ronan Dance (2023-)

Rebecca Monks (2023-)

Daniel Hodgson (2021-)

Jack Trainer (2021-)

Andrew Sainsbury (2020-, part-time)

Joseph Wall (2019-)

John Hewetson (2018-2022)

 

Current Teaching

2023/2024

  • MATH326 Graph Theory

2022/2023

  • MATH326 Graph Theory

2021/2022

  • MATH326 Graph Theory

2020/2021

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 2023 - present: Reader in Pure mathematics, Lancaster University.
  • August 2021 - July 2023: 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

  • PDR Group lead, since 2022.
  • 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), 2018-2022.
  • 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.

 

 

Research Grants

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.

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