abstracts combinatorial-2024

The first step in computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics using delay-coordinate embedding; the second is the construction of an appropriate simplicial complex from the results. The first of these steps involves a number of free parameters, though, and the computation of homology for a large number of simplices can be expensive. I will discuss some approaches for computing the homology efficiently and effectively, in the face of these challenges, using a range of examples from the classic Lorenz system through musical instruments and honeybee colonies to demonstrate the associated ideas.

Connection matrices, a generalization of Morse boundary operators from the classical Morse theory, capture the connection made by the flow among critical structures like attractors, repellers, and orbits in a vector field. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Recently, in the context of a combinatorial framework, an efficient persistence-like algorithm to compute connection matrices has been proposed in [DLMS24]. We show that, actually, the classical persistence algorithm in topological data analysis with exhaustive reduction indeed retrieves the connection matrix thus simplifying the algorithm of [DLMS24] even further. We also present an approach for updating a connection matrix under changes of a minimal Morse decomposition of the vector field in play. We conclude the talk by proposing a concept of persistence in the vector field derived from its connection matrix and presenting some preliminary experimental results.

[DLMS24] Computing Connection Matrices via Persistence-Like Reductions. Tamal K. Dey, Michal Lipinski, Marian Mrozek, and Ryan Slechta. SIAM J. Appl. Dyn. Syst., Vol. 23, No. 1, pages 81-97, 2024.

We present a combinatorial topological approach to analyze the dynamics of a family of parametrized ODEs. The method discretizes the state space and applies algebraic topological tools to derive dynamical information. Specifically, given a parameter, we propose analytical bounds to realize the combinatorial structure within the phase space, obtaining transversality results. Motivated by gene regulatory networks, we will explore examples of ramp systems and analyze the outcomes.

We present a combinatorial topological method to compute the dynamics of a parameterized family of ODEs. A discretization of the state space of the systems is used to construct a combinatorial representation from which recurrent versus non-recurrent dynamics is extracted. Algebraic topology is then used to validate and characterize the dynamics of the system. We will discuss the combinatorial description and the algebraic topological computations and will present applications to systems of ODEs arising from gene regulatory networks.

We describe a close connection between two seemingly disparate types of gene regulatory network models: ordinary differential equations and monotone Boolean maps. We show that switching (Glass) models connect the world of continuous dynamics of ODEs and discrete dynamics of Boolean maps by describing connection between collection of all parameterizations of switching systems and collections of monotone Boolean maps compatible with the network structure. This connection allows combin