The mechanisms by which these open sets may be assigned, is arbitrary as long they satisfy the above properties. In practice, the most common way to define open sets is by way of metrics. However, it is important to note that even in this case, the definitions of continuous maps can be much simplified using open sets especially for large dimensions. For example, the classical definition for con...

I develop algebraic-topological theories, algorithms and software for the analysis of nonlinear data and complex systems arising in various scientific contexts. In particular, I employ discrete Morse-theoretic techniques to substantially compress cell complexes built around the input data without modifying their core topological properties. Recently, I have generalized discrete Morse theory its...

The dynamic model of a power system is the combination of the power flow equations and the dynamic description of the generators (the swing equations) resulting in a differential-algebraic equation (DAE). For general DAEs solvability is not guaranteed in general, in the linear case the coefficient matrices have to satisfy a certain regularity condition. We derive a solvability characterization ...

These are colloquium style lecture notes about Hopf rings in algebraic topology. They were designed for use by non-topologists and graduate students but have been found helpful for those who want to start learning about Hopf rings. They are not “up to date,” nor are they intended to be, but instead they are intended to be introductory in nature. Although these are “old” notes, Hopf rings are th...

The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many mathematical invariants have expressions in terms of homotopy groups, and at different times the subject has found itself located in geometric topology, algebra, al...

Notes on homotopy theory, the first part of a trilogy on algebraic topology. Any typos, errors, mistakes, gaffes, etc., are entirely my folly.

The description of deadlocks in concurrent programming is translated to conditions on the topology of the progress graph representing the transaction system.