# Work Packages

- WP1: Topology of data
- WP2: Probabilistic Methods
- WP3: Formal Methods
- WP4: Bio-oriented methods
- WP5: Dissemination and Joint Collaborative Tasks
- WP6: Management

## WP1: Topology of data

In this WP we will develop computational algebraic topology approaches to address problems of feature detection and shape recognition in high-dimensional data sets.

#### O1.1 Elementary coarse-grained features of data sets

The principal motivation is that it is expedient to deal with data sets: 1) replacing a given set of data points with a family of simplicial complexes, indexed by a proximity parameter. Such operation converts the data set into a global topological object; 2) dealing with the resulting topological complexes by the tools of algebraic topology – specifically, via a theory of persistent homology tailored to parameterized families; 3) encoding the persistent homology of a data set in the form of a parameterized version of Betti numbers.

#### O1.2 Gromov’s spaces for simplicial geometry

Gromov space of bounded geometries has been shown to be a fruitful tool to construct a discretized version of quantum gravity in the framework of simplicial general relativity to construct invariant measures and probability distributions for quantum states.

This, in particular, has been done by computing entropy estimates by counting minimal geodesic ball coverings with given Euler characteristic in a given representation of the fundamental group with fixed analytical or Reidemeister torsion.

Our aim is to evaluate to which extent it is possible to extend this approach in a metric-free perspective by weakening the strong smoothness and regularity assumptions made in the gravity context, in order to be able to describe the wild nature and heterogeneity of simplicial complexes (filtered or not) built out of data, relying on the results of O1.1.

In particular, we will establish various entropy estimates, extending the approach based on the analysis of the asymptotic distribution of combinatorially inequivalent triangulated n-manifolds, to random simplicial complexes, also by means of combinatorial Laplacians’ spectra.

#### O1.3 Coverings: simplicial complexes vs. Riemannian manifolds

There are known deep results on the bounding of the number of distinct geodesic ball coverings of n-dimensional Riemannian manifolds of bounded geometry for fixed Euler characteristic and torsion in a given representation of the fundamental group. Much in the same way as in O1.2, we will investigate to which extent it is possible to extend and possibly adapt the above results for a general simplicial setting, in order to provide a coherent path toward a meaningful definition of the analogue of the thermodynamic limit of statistical mechanics in that context.

#### O1.4 Gibbs families

Finally, in view of constructing a well defined statistical mechanics – more precisely a statistical field theory – we will define a generalization of Gibbs families to the case where the substrate is not only a graph but a simplicial complex. This will include the case when the substrate underlying the Gibbs field is itself random.

## WP2: Probabilistic Methods

#### O2.1 Basic features of probabilistic approaches

The main goal of this WP is to develop probabilistic/information methods can be exploited for describing complex systems starting from different point of views.

#### O2.2 Resilient Information and topology constrains

Moving from a microscopic-like description we will devise an approach based on the information dissipation through a network. In this case the topology enters through the links of the network.

#### O2.3 Information processing during critical transitions and relevant variables

Moving from a macroscopic-like description we will devise an approach based on informational geodesic flows on curved statistical manifolds. In this case the topology enters in the curvature of statistical manifolds and in the information-constrained dynamics. Both approaches will take advantage of the results from WP1 and will be applied to relevant systems, compared, and validated through the tools provided in WP3.

#### O2.4 Application of probabilistic methods

The two approaches complement each other in a multilevel vision and will advance both the description and the understanding of complex systems.

## WP3: Formal Methods

#### O3.1 Basic features of process algebraic approaches

The aim of this work package is to develop a formal methods expressive enough for modelling and analysing multi-level complex systems as composed of a structural (macro) level describing the global properties and the functional requirements of the system; and a behavioural (micro) level accounting for the functional behaviour of and the interactions among the many different agents of systems. These methods will rely on well-established formal techniques in concurrency theory like process algebras including behavioral equivalences and formal verification, and will exploit the knowledge and properties of the topology of data (WP1) and topology of statistical dynamics (WP2) to be suitable for modeling and analysing multi-level systems.

#### O3.2 Characterization of S[B]-systems

In the proposed approach, denoted S[B]-systems, a complex system consists of a pair S and B, where S represents the global properties (dynamic laws) of the environmental structure and B represents the network of interacting agents in the environment S. The macro (structural) level S describes the functional requirements of the system and the micro (behavioural) level B accounts for the dynamics of the system. We will define suitable mappings, from probabilistic and topological methods to S[B]-systems. We will also exploit the compositionality property of S[B] for analysing the robustness of such methods.

#### O3.3 Formal analysis and run-time verification

Finally, we will characterize behaviour equivalences and pre-orders for analysing different models and answer the questions “what is the ‘influence’ of one element on the behaviour of the system as a whole?”, “what is the ‘influence’ of the spatial distribution of interacting elements on the system’s global properties?” for probabilistic systems.

#### O3.4 S[B] methods for probabilistic and bio-oriented systems

Application, verification and validation of S[B] methods on biological data.

## WP4: Bio-oriented methods

#### O4.1 γ-structures and irreducible shadows

The main goal of this WP is the development of methods oriented to biological systems and which account for topological constraints. It will benefit from results of WP2 and WP3 besides tools of WP1. In the context of RNA folding we will first study the polynomial of irreducible shadows and try to establish algebraic properties of γ-structures.

#### O4.2 Molecular information network and Analogues of Waterman’s recursion

Then, we will look for a bijective proof for a higher genera analogue of Waterman’s recursion.

#### O4.3 Shapes over a topological surface and Topology of molecular information network

Furthermore, we will explore the connection between shapes as cell-decompositions and complex structures on a topological surface. A symbolic combinatorics of the generating function of γ-structures will be pursued. Finally, the derived results will be applied to implement a folding algorithm of RNA pseudoknot structures.

#### O4.4 Topological folding algorithm

In the context of defining the emergence of new properties in complex systems described as networks, and in analogy with the topological theory of thermodynamic phase transitions, we aim at finding new topological tools to characterize phase transition like phenomena similar to those appearing in random graphs after the Erdös-Rényi theorem. This problem is posed by the astonishing degree of auto-organization of the network of molecular reactions whithin living cells and, in particular, by epigenetic regulatory networks.

## WP5: Dissemination and Joint Collaborative Tasks

#### O5.1 Supporting the coordination and dissemination of results

Project coordination support and dissemination of results. Creation of an excellent website for internal coordination of the project and external dissemination, including the publication of periodic newsletter, a video library of lectures and presentations Protection and dissemination of intellectual property rights. The consortium will monitor industrial potential of research results delivered by the project and create mechanisms for sharing intellectual property rights arising from collaborations.

#### O5.2 Elaboration of a plan for joint and collaborative tasks

Joint Publications. Collaboration with other projects in the DYM-CS programme to promote the Proactive Initiative: web portal, journals, reports, presentations, joint press releases, joint books, newsletters, press kit, training material for students. Contribute to special issues dedicated to Topology Driven Methods in Complex Systems, in appropriate journal with IF > 2, such as FGCS, …

Joint Events † Organise and participate in joint events for the Proactive Initiative: participation to the FET conference, contribution to: dedicated workshops & working groups, including with industry, consultation meetings, summer schools and clustering meetings (at least once a year). Organize a series of Workshops on Topology Driven Methods in Complex Systems as a satellite event of international conferences. Participate to the call for conference, workshop and summer schools at Institute Mittag-Leffer for the Summer 2013.

#### O5.3 Definition of Strategy and Roadmapping

Contributing to the strategy and roadmapping activities of the Proactive Initiative: contribution to the initiative’s impact assessment of results for research take-up and exploitation, contribution to white papers for new research topics and future workprogrammes. Organize the Grand Finale Art-Mathematics-Science TOPDRIM dissemination events.

#### O5.4 International Co-Operation and Mobility

International Co-Operation and Mobility activities for the Proactive Initiative: contribution to relevant national and international activities (ex. joint workshops, calls, etc… for example with NSF-US, Japan, China, India, Brazil and South America, Russia, Africa, …) Facilitating mobility † In addition, optional joint activities may include: contribution to short-term visiting research students and post-docs in partner laboratories, training, demonstrations .