Professor Anthony Nixon SFHEA
Professor in MathematicsResearch 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. Rigidity of symmetric frameworks on the cylinder, with Bernd Schulze and Joseph Wall, https://arxiv.org/abs/2210.06060.
2. 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.
3. 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.
4. Rigidity of symmetric linearly constrained frameworks in the plane, with Bernd Schulze and Joseph Wall, https://arxiv.org/abs/2312.12158.
5. Rigid frameworks with dilation constraints, with Sean Dewar and Andrew Sainsbury, https://arxiv.org/abs/2402.14093.
6. Rigidity of nearly planar classes of graphs, with Sean Dewar, Georg Grasegger, Eleftherios Kastis and Brigitte Servatius, https://arxiv.org/abs/2402.17499.
7. Angular constraints on planar frameworks, with Sean Dewar, Georg Grasegger, Zvi Rosen, William Sims, Meera Sitharam and David Urizar, https://arxiv.org/abs/2403.16145.
8. Single-cell 3D genome reconstruction in the haploid setting using rigidity theory, with Sean Dewar, Georg Grasegger, Kaie Kubjas and Fatemeh Mohammadi, https://arxiv.org/abs/2407.10700.
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), Zvi Rosen (Florida Atlantic), Mahdi Sadjadi (Arizona), Andrew Sainsbury (Lancaster), Bernd Schulze (Lancaster), Jason Semeraro (Loughborough), Brigitte Servatius (WPI), William Sims (Florida), Meera Sitharam (Florida), Adnan Sljoka (Kyoto), Shin-ichi Tanigawa (Tokyo), Louis Theran (St Andrews), Mike Thorpe (Arizona), David Urizar (Florida Atlantic), Joseph Wall (Lancaster), Walter Whiteley (York).
Event organisation - upcoming:
- Semester program on Geometry of materials, packings and rigid frameworks, ICERM, Brown, January-May 2025, https://icerm.brown.edu/programs/sp-s25/.
- Heilbronn focused research group, September 2024, Lancaster University.
Event organisation - past:
- Rigidity in action workshop, part of a RICAM special semester, April 8-12, 2024, https://www.ricam.oeaw.ac.at/specsem/specsem2024/
- Focus program on Geometric constraint systems, Fields Institute for mathematical research, Toronto, July-August 2023, http://www.fields.utoronto.ca/activities/23-24/constraint.
- 19th Cologne-Twente workshop on graphs and combinatorial optimization, program committee member, June 14-16 2023, Garmisch-Partenkirchen, Germany, see https://ctw2023.comtessa.org/.
- Graph rigidity and applications, April 17-21 2023, Lancaster University, https://www.lancaster.ac.uk/maths/graph-rigidity-and-applications-2023/.
- 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.
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:
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
- Graduate course on combinatorial and geometric rigidity, taught at the Fields Institute, videos of the lectures available, http://www.fields.utoronto.ca/activities/20-21/constraint-CRDG
- MATH326 Graph Theory
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.
- 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.
External Roles
- Associate editor, Journal of Algebraic Combinatorics, since 2024.
- Member of the British Combinatorial Committee, since 2020.
PhD Supervision Interests
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.
Structural stability: combinatorics, geometry and topology
01/05/2024 → 31/08/2024
Research
Abstract Rigidity for natural stability problems
01/12/2023 → 30/11/2026
Research
A New Approach to the Graph Rigidity Problem in Arbitrary Dimension
01/08/2022 → 31/08/2023
Research
29th British Combinatorial Conference - Invited Speakers
01/07/2022 → 31/07/2022
Research
29th British Combinatorial Conference - Reception
01/07/2022 → 31/07/2022
Research
29th British Combinatorial Conference - Support
01/07/2022 → 31/07/2022
Research
2British Combinatorial Conference - mini-symposia (BCC-MS)
01/07/2022 → 31/07/2022
Research
Topological Graph Rigidity and Maximum Likelihood Thresholds
28/03/2022 → 08/04/2022
Research
Heilbronn fellowship
01/07/2020 → 31/08/2021
Research
Globally Rigid Linearly Constrained Frameworks
11/01/2020 → 10/04/2020
Research
Discrete structures: algebra, combinatorics and geometry
10/01/2020 → 09/02/2020
Research
Linearly constrained frameworks
01/11/2018 → 31/01/2019
Research
Graph rigidity and applications
01/01/1900 → …
Research
18th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
Participation in workshop, seminar, course
Circle packing and geometric rigidity
Participation in conference -Mixed Audience
Hakan Guler
Hosting an academic visitor
Discrete Structures
Participation in workshop, seminar, course
American Institute of Mathematics: Rigidity and flexibility of microstructures
Participation in workshop, seminar, course
Galway Mathematics Seminar
Invited talk
Bristol combinatorics seminar
Invited talk
British combinatorial conference
Participation in conference -Mixed Audience
Loughborough applied algebraic geometry meeting
Invited talk
SIAM conference on applied algebraic geometry
Participation in conference -Mixed Audience
DIMACS workshop on optimization in distance geometry
Invited talk
Geometric Constraint Systems: rigidity, flexibility and applications
Participation in conference -Mixed Audience
Japanese-Hungarian symposium on discrete mathematics
Participation in conference -Mixed Audience
British Mathematical Colloquium
Participation in conference -Mixed Audience
Liverpool selected topics in mathematics seminar
Invited talk
Rigidity and flexibility of geometric structures
Participation in conference -Mixed Audience
Circle packing and geometric rigidity
Participation in workshop, seminar, course
Stability of flat structures
Participation in workshop, seminar, course
Bond-node structures: rigidity, combinatorics and chemistry
Participation in conference -Mixed Audience
Egervary research group on combinatorial optimization seminar
Invited talk
Sheffield Automatic Control and Systems Engineering Seminar
Invited talk
Cornell conference on rigidity
Participation in conference -Mixed Audience
Bond-node structures
Participation in conference -Mixed Audience
Japanese-Hungarian symposium on discrete mathematics
Participation in conference -Mixed Audience
St Andrews centre for interdisciplinary research in computer algebra seminar
Invited talk
Queen Mary Combinatorics Study Group Seminar
Invited talk
WPI Colloquium Talk
Invited talk
Cornell Discrete Geometry and Combinatorics Seminar
Invited talk
Unusual Configuration Spaces
Participation in conference -Mixed Audience
Distance Geometry
Participation in conference -Mixed Audience
Workshop on geometric rigidity
Participation in conference -Mixed Audience
Geometric rigidity theory and applications
Participation in conference -Mixed Audience
Discrete mathematics and game theory seminar
Invited talk
Workshop on rigidity, submodularity and discrete convexity
Participation in workshop, seminar, course
Heilbronn Annual Conference
Participation in conference -Mixed Audience
Plymouth Mathematics Seminar
Invited talk
Global rigidity
Participation in conference -Mixed Audience
Advances in combinatorial and geometric rigidity
Participation in conference -Mixed Audience
Geometric Rigidity Workshop
Participation in workshop, seminar, course
Configuration spaces of linkages
Participation in workshop, seminar, course
Combinatorics, Geometric Rigidity
Combinatorics, Energy Lancaster, Geometric Rigidity
Statistical Artificial Intelligence, STOR-i Centre for Doctoral Training
Combinatorics, Geometric Rigidity
- Combinatorics
- Geometric Rigidity