# Topology, Geometry and Laplacians of Simplicial Complexes

Lead Research Organisation:
Durham University

Department Name: Mathematical Sciences

### Abstract

Simplicial complexes are natural abstract mathematical objects which play a prominent role in several fields of mathematics. They appear as triangulations of surfaces or more general higher dimensional spaces, which are useful in topology for the computation of invariants like the Euler characteristic. Other important examples of simplicial complexes, in connection with geometric group theory, are buildings. They were first introduced by Jacques Tits in his work on Klein's Erlangen program and provide a very successful geometric approach to group theory.

Being combinatorial objects, simplicial complexes can serve as simplified models of smooth geometric spaces. Their combinatorial nature allows explicit computations of their fundamental groups. Fundamental groups are a basic algebraic tool to describe the equivalence of closed paths under continuous deformations. In this project, we aim to obtain a better understanding of the fundamental groups of simplicial complexes and their properties. Another attractive property of simplicial complexes is that they can be endowed with additional geometric structures and become gateways to a much richer world than the classical surfaces and their generalisation to higher dimensions (manifolds). In this project, we aim is to generalize known geometric concepts of curvature into this richer world. Another consequence of the simplicity and versatility of simplicial complexes is their wide use as geometric representations in the fields of industrial design and medical imaging. The better understanding of their mathematical properties will lead to improved processing algorithms that can be used in these applications.

Being combinatorial objects, simplicial complexes can serve as simplified models of smooth geometric spaces. Their combinatorial nature allows explicit computations of their fundamental groups. Fundamental groups are a basic algebraic tool to describe the equivalence of closed paths under continuous deformations. In this project, we aim to obtain a better understanding of the fundamental groups of simplicial complexes and their properties. Another attractive property of simplicial complexes is that they can be endowed with additional geometric structures and become gateways to a much richer world than the classical surfaces and their generalisation to higher dimensions (manifolds). In this project, we aim is to generalize known geometric concepts of curvature into this richer world. Another consequence of the simplicity and versatility of simplicial complexes is their wide use as geometric representations in the fields of industrial design and medical imaging. The better understanding of their mathematical properties will lead to improved processing algorithms that can be used in these applications.

### Planned Impact

Simplicial complexes are the most commonly used geometric representations for modelling 3D objects. Even when other representations are used (e.g. splines), they are usually converted to simplicial complexes for visualisation, or for running numerical simulations (e.g. Finite Element Methods).

Areas in the commercial private sector where simplicial complexes are widely used for geometric modelling include animation, computer games and industrial design (Computer Aided Design/Manufacturing).

In the public sector, two typical applications of geometric modelling involving simplicial complexes are the digitisation of cultural heritage and the analysis of medical scans. Museums create digital 3D models of their artefacts for analysis and preservation and for making them accessible to a wider audience though the web. Doctors process 3D scan data from medical devices such as CT, MRI and ultrasound, creating simplicial complex representations which can be automatically analysed.

The theoretical results of the project will lead to a better understanding of the fundamental properties of simplicial complexes and could have a long term impact in ways that we cannot predict at this stage. The applied results can have immediate practical implications, by developing methods for measuring and improving the quality of the models.

Areas in the commercial private sector where simplicial complexes are widely used for geometric modelling include animation, computer games and industrial design (Computer Aided Design/Manufacturing).

In the public sector, two typical applications of geometric modelling involving simplicial complexes are the digitisation of cultural heritage and the analysis of medical scans. Museums create digital 3D models of their artefacts for analysis and preservation and for making them accessible to a wider audience though the web. Doctors process 3D scan data from medical devices such as CT, MRI and ultrasound, creating simplicial complex representations which can be automatically analysed.

The theoretical results of the project will lead to a better understanding of the fundamental properties of simplicial complexes and could have a long term impact in ways that we cannot predict at this stage. The applied results can have immediate practical implications, by developing methods for measuring and improving the quality of the models.

### Publications

Atay F
(2020)

*Cheeger constants, structural balance, and spectral clustering analysis for signed graphs*in Discrete Mathematics
Atay F. M.
(2014)

*Cheeger constants, structural balance, and spectral clustering analysis for signed graphs*in Preprint Server arXiv.org, MPI MiS Preprint 11/2014
Barker N
(2016)

*The power conjugacy problem in Higman-Thompson groups*in International Journal of Algebra and Computation
Barker N
(2015)

*An Infinite Family of 2-Groups with Mixed Beauville Structures*in International Mathematics Research Notices
Barker N.
(2015)

*The power conjugacy problem in Higman-Thompson groups*in Preprint Server arXiv.org
Bauer F
(2017)

*Modern Approaches to Discrete Curvature*
Bonnefont M
(2020)

*Magnetic-Sparseness and Schrödinger Operators on Graphs*in Annales Henri Poincaré
Bromberg L
(2015)

*Navigating in the Cayley graph of $$SL_2(\mathbb {F}_p)$$ S L 2 ( F p ) and applications to hashing*in Semigroup Forum
Cushing D
(2017)

*The Graph Curvature Calculator and the curvatures of cubic graphs*
Cushing D
(2019)

*The Graph Curvature Calculator and the Curvatures of Cubic Graphs*in Experimental MathematicsDescription | Our research efforts led us to obtain deeper insights into interesting connections between our project with other fields. Some of them are in the applied field like (i) new kinds of spectral clusterings based on rgaphs with generalised signings, (ii) Hash functions and navigation problems in computational complexity, (iii) developing a notion of entropy applied to simplicial complexes, (iv) point set analysis based on energy-like spherical functions. Other more theoretical fields comprise (i) spectra of Markov operators via approximations, (ii) connections between buildings and algebraic surfaces, (iii) interplay between discrete spaces and smooth spaces like surfaces and more general Riemannian manifolds, (iv) discrete Bakry-Emery type curvature notions based on generalized Laplacians like magnetic Laplacians and Connection Laplacians, (v) generalizations of matching polynomials for polygonal complexes. We also supervised a student implementing efficient algorithms for the construction of simplicial complexes from point sets, a student implementing a calculational tool for concrete representations of specific groups acting simply transitively on Euclidean buildings, and an early career programmer developing software tools for star-graph and link investigations of polygonal complexes and and developing the software for a discrete curvature calculator of finite graphs. |

Exploitation Route | We made our research results available on the preprint server arxiv.org and submitted it to high impact journals. We also participated in various international conferences to present our result. During a joint 1 month research visit in summer 2015 at the Max Planck Institut in Bonn of 3 of the researchers in this grant, each of us presented certain results in a seminar talk to other researchers of this institute. |

Sectors | Digital/Communication/Information Technologies (including Software),Education,Culture, Heritage, Museums and Collections,Other |

URL | http://www.maths.dur.ac.uk/~dma0np/epsrc2013/TGLSC.html |

Description | Max Planck Institute Bonn Collaboration (Liu, Peyerimhoff, Vdovina) |

Amount | € 9,000 (EUR) |

Organisation | Max Planck Society |

Department | Max Planck Institute for Mathematics |

Sector | Academic/University |

Country | Germany |

Start | 06/2015 |

End | 08/2015 |

Description | Max Planck Institute Bonn Collaboration (Vdovina) |

Amount | € 3,000 (EUR) |

Organisation | Max Planck Society |

Department | Max Planck Institute for Mathematics |

Sector | Academic/University |

Country | Germany |

Start | 06/2016 |

End | 08/2016 |

Description | University of California, Berkely, Visiting Scholar |

Amount | $3,000 (USD) |

Organisation | University of California |

Sector | Academic/University |

Country | United States |

Start | 09/2016 |

End | 12/2016 |

Description | Arithmetic lattices in trees and buildings |

Organisation | European University Viadrina Frankfurt (Oder) |

Department | Institute of Mathematics |

Country | Germany |

Sector | Academic/University |

PI Contribution | Working on a joint research project; Alina Vdovina was also external PhD examiner there. |

Collaborator Contribution | Working on a joint research project. |

Impact | Research papers in the following journals: Monatshefte fuer Mathematik 2015 (published, see publications), Mathematical Proceedings of Cambridge Philosophical Society (accepted) |

Start Year | 2009 |

Description | Hyperbolic buildings project |

Organisation | Aalto University |

Department | Department of Mathematics |

Country | Finland |

Sector | Academic/University |

PI Contribution | Alina Vdovina is collaborating with Riikka Kangaslampi in various computational problems related to hyperbolic buildings. |

Collaborator Contribution | Riikka Kangaslampi created software (see, e.g., http://www.maths.dur.ac.uk/~dma0np/epsrc2013/Software.html) for the joint research projects. |

Impact | Joint publications: R. Kangaslampi and A. Vdovina, Hyperbolic triangular buildings without periodic planes of genus two, arxiv:1409.1401, October 2014, To appear in Experimental Mathematics. |

Description | Spectral methods for clustering and spatial representations of graph vertices |

Organisation | Technical University of Munich |

Country | Germany |

Sector | Academic/University |

PI Contribution | Our collaboration has resulted in a jointrecent publication (Liu/Lange/Peyerimhoff/Post) and is about to lead to another article on spectral representation of graphs via eigenfunctions. Carsten Lange visited us at Durham as a Grey Fellow in Epiphany 2014 (see https://www.dur.ac.uk/mathematical.sciences/distinguishedfellows/greyfellows/) and several times afterwards within the EPSRC grant to continue in our oint research projects. |

Collaborator Contribution | Norbert Peyerimhoff visited Dr Lange at the TU Munich for collaboration and gave a talk in their joint SFB Colloquium with the TU Berlin (14.07.2015). This meeting was also very useful to make progress in our research project. |

Impact | Joint publication C. Lange, Sh. Liu, N. Peyerimhoff, O. Post, Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians, arXiv:1502.06299, Calculus of Variations and Partial Differential Equations 54 (2015), pp. 4165-4196. |

Start Year | 2013 |

Title | Matlab computations of complex constructions |

Description | Given point set inputs, it computes several complexes such as Vietoris/Rips, Cech and alpha-shapes. The variables for these constructions are user defined parameters. The Cech complex construction is based on work of two of the CI's (Dantchev and Ivrissimtzis), while the other constructions are based on standard state of the art algorithms. A GUI (graphical user interface) has been implemented for ease of use and documentation has been written. |

Type Of Technology | Software |

Year Produced | 2014 |

Open Source License? | Yes |

Impact | At the moment, the software is not publicly available, but it is used by members of the project for research purposes. |

Title | Python program for calculation of stargraphs and links |

Description | The software allows to calculate stargraphs and links of polygonal complexes given via labellings of finitely many polygons. |

Type Of Technology | Software |

Year Produced | 2015 |

Open Source License? | Yes |

Impact | The software is useful to find polyhedral complexes with certain properties or to prove that they do not exist. This is part of a larger project overseen by Alina Vdovina. |

URL | http://www.maths.dur.ac.uk/~dma0np/epsrc2013/software/david_cushing_1/linkcalculator.html |

Title | Python programs for matrix computations of groups |

Description | It is a very useful tool to carry out explicit calculations in matrix representations of particular groups acting simply transitively on specific Euclidean buildings. |

Type Of Technology | Software |

Year Produced | 2015 |

Open Source License? | Yes |

Impact | The software allowed us to verify certain identities which are important in connection with Beauville structures and may ultimatively be useful to find a proof of our "finite width conjecture". But this is a long-term aim. |

URL | http://www.maths.dur.ac.uk/~dma0np/epsrc2013/software/francesca_bianchi/matrixcalc.html |

Description | Seminar GGA (Groups, Geometry and Analysis) for grant members and other staff of maths and computer science departments with occasional international guests |

Form Of Engagement Activity | A formal working group, expert panel or dialogue |

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Other audiences |

Results and Impact | So far, there were 29 talks in our GGA seminar. Sometimes, we had speakers from other academic institutions and at other times from our own working group. The audience was also far beyond just members of our working group. The talks are open ended, have informal character and are intended to stimulate discussions and potential collaboration. You can find a list of all talks in the GGA seminar so at the webpage http://www.maths.dur.ac.uk/~dma0np/epsrc2013/TGLSC.html Several collaborations (also with academics who are not members of the EPSRC grant) were initiated by these meetings. |

Year(s) Of Engagement Activity | 2013,2014,2015,2016 |

URL | http://www.maths.dur.ac.uk/~dma0np/epsrc2013/TGLSC.html |

Description | Workshop "Geometry and Computation on Groups and Complexes", held from 6-10 June 2016 |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | The activity brought together high profile international speakers as well as researchers at an earlier stage of their career to present their results in June 6-10, 2016. We set up a webpage about the conference. and we have currently 26 speakers and around 30 registrations to attend the workshop. |

Year(s) Of Engagement Activity | 2016 |

URL | http://www.maths.dur.ac.uk/~dma0np/epsrc2013/workshop/gcompgcomp.html |