# Dr Anthony Nixon

Lecturer in Pure Mathematics## 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 symmetric frameworks in normed spaces, with Derek Kitson and Bernd Schulze, 36 pages, https://arxiv.org/abs/1808.04484.

2. Global rigidity of 2-dimensional linearly constrained frameworks, with Hakan Guler and Bill Jackson, 33 pages, https://arxiv.org/abs/1906.10926.

3. Assur decompositions of direction-length frameworks, 12 pages.

4. Flexible circuits in the d-dimensonal rigidity matroid, with Georg Grasegger, Hakan Guler and Bill Jackson, 15 pages, https://arxiv.org/abs/2003.06648.

Information on my papers is also available at my google scholar page.

**Collaborators:**

Katie Clinch (Tokyo), Robert Connelly (Cornell), James Cruickshank (NUI Galway), Yaser Eftekhari (York), Nick Gill (South Wales), Neil Gillespie (Bristol), Georg Grasegger (RICAM, Linz), Hakan Guler (Kastomonu), Bill Jackson (Queen Mary), Viktoria Kaszanitzky (Budapest), Derek Kitson (Lancaster), 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:**

- 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
- 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/
- 18th Cologne-Twente workshop on graphs and combinatorial optimization, program committee member, June 15-17 2020, Ischia, Italy, see http://ctw2020.iasi.cnr.it/
- 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.

## Current Teaching

**2019/2020**

- MATH326 Graph Theory

Students on the course should consult the moodle page, https://modules.lancaster.ac.uk/course/view.php?id=27463

**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

- 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, 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

- 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, 2016-2019.
- Natural science coordinator (within Mathematics and Statistics), 2014-2019.
- Department representative for British Combinatorial Bulletin, since 2014.

## PhD Supervision Interests

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.

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

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

Loughborough applied algebraic geometry meeting

Invited talk

SIAM conference on applied algebraic geometry

Participation in conference

DIMACS workshop on optimization in distance geometry

Invited talk

Geometric Constraint Systems: rigidity, flexibility and applications

Participation in conference

Japanese-Hungarian symposium on discrete mathematics

Participation in conference

British Mathematical Colloquium

Participation in conference

Liverpool selected topics in mathematics seminar

Invited talk

Rigidity and flexibility of geometric structures

Participation in conference

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

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

Bond-node structures

Participation in conference

Japanese-Hungarian symposium on discrete mathematics

Participation in conference

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

Distance Geometry

Participation in conference

Workshop on geometric rigidity

Participation in conference

Geometric rigidity theory and applications

Participation in conference

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

Plymouth Mathematics Seminar

Invited talk

Global rigidity

Participation in conference

Advances in combinatorial and geometric rigidity

Participation in conference

Geometric Rigidity Workshop

Participation in workshop, seminar, course

Configuration spaces of linkages

Participation in workshop, seminar, course

Discrete Mathematics and Geometric Rigidity

- Algebra and Geometry
- Discrete Mathematics and Geometric Rigidity