Persistence: Stability, Computation, and Simplification
Persistent homology, the homology of a filtration, is described up to isomorphism by the persistence diagram (barcode), which encodes the structure of its indecomposable summands. A fundamental theorem states that small changes in the input data lead only to small changes in the persistence barcode. We will study a constructive algebraic approach to this result that admits a high level of generality.
The efficient computation of persistence barcodes has undergone dramatic improvements in recent years. We will discuss recent developments based on cohomology and discrete Morse theory.
A motivation problem for the definition of persistence is to simplify the data in a way that eliminates homological features of small persistence. We will consider different incarnations of this problem and discuss cases in which simplifications may be obtained efficiently, or may not exist at all.
Notes: Lecture 1 - 3.
Computational Geometry and Topology
(I) Delaunay Complexes
The Delaunay triangulation and its dual Voronoi diagram are among the most fundamental concepts in computational geometry. After defining them, we present some of their pertinent properties, and we discuss their generalization to weighted point data.
(II) Discrete Morse Theory
The radius function on the Delaunay triangulation fits a recent generalization of Forman's discrete Morse theory. We describe this algebraic framework and apply it to elucidate important properties of the Delaunay triangulation.
(III) Persistent Homology
Among the many faces of this extension of the classical theory of homology groups is the connection to discrete Morse theory, which it extends by pairing not necessarily incident simplices and derive meaning from this pairing.
Slides: Lecture 1 - 3.
Dynamics Signatures Generated by Regulatory Networks
The results presented in this talk are strongly motivated by attempts to characterize the dynamics of regulatory networks in biological systems. There are a variety of fundamental questions that we attempt to address:
- given a time series of gene expression, e.g. RNA-Seq, can one determine regulatory networks that can generate such a time series;
- given a regulatory network can one determine the robust from of dynamics that it is capable of generating;
- how robust are these networks with respect to expression of particular `dynamic phenotypes’; and
- given a complex set of interactions can one devise a systematic means of choosing minimal models?
There are obvious challenges to answering these questions:
- biological data tends to be extremely noisy and typically imprecisely measured;
- answers to many relevant questions require characterizations of global dynamics;
- since these are multiscale systems the associated mathematical models are derived using heuristic arguments as opposed to first principles;
- the associated parameter spaces are high dimensional and most parameters are either poorly measured or unknown; and
- we need to have confidence in the mathematical answers.
Making use of the abstract theory described in R. Vandervorst’s lecture, we will describe a particular set of computational models and the associated Dynamics Signatures Generated by Regulatory Networks (DSGRN) algorithms and software. This is based on a combinatorial approximation of dynamics that leads to a natural decomposition of parameter space into semi-algebraic sets. We will discuss how this approach leads to efficient computations and mathematically rigorous results that address the above mentioned challenges. We will show how these techniques can be applied to questions related to Malaria and Cancer networks. Finally, we will discuss related open mathematical questions.
Slides: Lecture 1 - 3.
ROBERT VAN DER VORST
Dynamics and Order Theory
In a series of three lectures we discuss the following subjects concerning dynamics and order theory.
- We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some analytical arguments, in order to lay a foundation for the development of algorithms to manipulate these structures computationally.
- The natural lattice structure of attractors can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question of whether all of the algebraic structure of attractors can be captured by these methods.
- The universal construction of Booleanization paves the way to a unified theory of decomposing a dynamical systems in basic sets --- Morse decompositions. The Morse product is an algebraic construction that allows representations of attractor lattices in terms of posets consisting of basic sets --- the Morse sets --- and the partial order combinatorializes dynamics.
The concept discussed in these lectures will be closely related to the theory treated in the lectures by K. Mischaikow.
Slides: Lecture 1 - 3.