Abstract Morse function is called strong if all its critical points have different values. Given such a $f$ and field $\mathbb{F}$ Barannikov constructed pairing of some the $f$, which now also known as barcode. With every pair (a.k.a. bar in barcode), we naturally associate (up to sign) an element $\mathbb{F} \setminus \{ 0 \}$; call it Bruhat number. The paper devoted study these numbers. We ...

In this paper, we study merge trees induced by a discrete Morse function on tree. Given function, provide method to constructing an tree and define new notion of equivalence functions based the We then relate matching number certain invariant Finally, count that can be star graph characterize

For a class of functions (called minimal Radó functions) that arise naturally in surface theory, we bound the number interior critical points (counting multiplicity) terms boundary data and Euler characteristic domain function.

Forman’s discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.