Topological and Statistical Physics methods for data analysis

Tuesday, October 25, 2016

9:30 a.m.

ISI Red Room

Mario Rasetti, Pierluigi Contucci, Francesco Vaccarino


Aggregation models on hypergraphs.
Mario Rasetti (ISI Foundation)

TDFT is essentially a gauge field theory. As such it requires that a gauge group G is defined. The general formal approach requires that G = G p ∧ G MC , where G MC is the ‘mapping class group’ of the data space D, i.e. the group of all homeomorphisms of D that leave its topology (global, not only local) invariant, whereas G p is the group associated to the ‘process algebra’, namely to the set of possible oper ations that we can perform on the elements of D. A general scheme to construct all the irreducible representations of G MC in [su(p, q) ⨂(2g+1) ] q has been constructed. We exemplify it for 2-D surfaces, where the basic group is su(1,1). The analogous thing for G p requires focusing on the specific application at hand, because it depends on the nature of the data in D. We are constructing a general scheme resorting to ‘path algebras’ and their ‘quiver representations’ that drastically reduces the ‘specific’ part of the construction.

Topological Data Field Theory
Pierluigi Contucci (UNIBO)

Following a newly introduced approach by Rasetti and Merelli we investigate the possibility to extract topological information about the space where interacting systems are modeled. From the statistical datum of their observable quantities, like the correlation functions, we show how to reconstruct the activities of their constitutive parts which embed the topological information. The procedure is implemented on a class of polymer models with hard-core interactions. We show that the model fulfills a set of iterative relations for the partition function that generalize those introduced by Heilmann and Lieb for the monomer-dimer case. After translating those relations into structural identities for the correlation functions we use them to test the precision and the robustness of the inverse problem. Finally the possible presence of a further interaction of peer-to-peer type is considered and a criterion to discover it is identified. (Joint work with D.Alberici, E.Mingione and M.Molari)

Coffee Break

Data Field Theory of BF type via Hochschild Cohomology
Francesco Vaccarino (POLITO & ISI Foundation)

We attempt to build on the top of reasonable data ( e.g. weighted networks) so-called abstract BF-theories (a kind of topological quantum field theories), we will extend the usual approach of multidimensional persistent (co-)homology by considering diagrams of incidence algebras of finite posets. By using results o f Gerstenhaber et al. we will show what is the link between diagram cohomology, Hochschild cohomology, and multidimensional persistence, thus connecting seemingly uncorrelated research areas as topological data analysis and topological quantum field theory.