The workshop, which is a follow-up of the school HPRT2023, aims to provide a place for experts in Topological Data Analysis to meet, confront, and present their ideas.
Schedule:
November 29:
13:55 - 14:00 Opening and welcoming
14:00 - 14:45 F. Vaccarino
14:45 - 15:30 S. Settepanella
15:30 - 16:00 Coffee break
16:00 - 16:45 H. Riihimäki
16:45 - 17:30 M. Nurisso
19:30 - Social Dinner
Title ed abstracts:
F. Vaccarino
Title: Diagrams of algebras, persistence, and Hochschild Cohomology
S. Settepanella
Title: Combinatorial Morse theory and minimality of hyperplane arrangements
Abstract: The Combinatorial Morse Theory is a central tool in combinatorial algebraic topology and it is becoming more and more interesting especially for people working on Topological Data Analysis. In this talk we will present an example of a successful application of the combinatorial Morse theory to a known CW complex S in order to get an explicit combinatorial gradient vector field over S and the associated minimal algebraic complex MS. More in detail, S is the Salvetti's compex which gives the homotopy type of the complement to a complexified real arrangement of hyperplanes and the minimal algebraic complex MS computes the local system cohomology of the complement by means of a boundary operator which is effectively computable.
H. Riihimäki
Title: Persistent Hochschild homology of digraphs via connectivity structures, and Morita equivalences of commuting algebras
Abstract: I will begin from observations in simplicial analysis of directed flag complexes in the context of network neuroscience. From here one quickly realises that up-to-homotopy simplicial homology forgets the directionality information inherent in directed graph data. A homological approach of more directed nature comes from Hochschild homology of the associated path algebras, with the caveat that no homological information is present beyond degree 1 in certain cases. To remedy this we can lift to higher connectivity structures, again generating directed graphs, and lift Hochschild homology to higher degrees via these. Moreover, this process can be turned into persistence pipeline through filtrations of digraphs. I will finish by explaining how enhancing this pipeline leads to considering Morita equivalences of category algebras of digraphs, or commuting algebras as recently defined by Green and Schroll. The work presented is a collaboration with Luigi Caputi.
M. Nurisso
Title: Interactions and topological synchronization in the simplicial Kuramoto model
Abstract: Simplicial Kuramoto models have emerged as a diverse and intriguing class of models that capture the dynamics of interacting oscillators placed on the simplices of a simplicial complex. Being formalized with the tools of discrete differential geometry, these models reveal interesting relationships between topology, geometry and dynamics. We leverage their mathematical structure to give a microscopic interpretation of the interaction terms which, we see, include effectively both higher-order and self interactions. This naturally leads us to establish an equivalence between the simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of the underlying simplicial complex being a manifold. Then, we describe the notion of simplicial synchronization, its relation to simplicial homology, and derive bounds on the oscillators’ coupling strength necessary or sufficient for achieving it.
The workshop will take place in the main building of Campus Einaudi, part of the University of Torino, in the classroom H6.
Workshop organisers:
Luigi Caputi (University of Turin)
Takuya Saito (University of Turin)
Simona Settepanella (University of Turin)
For further information, contact one of the organisers.