#### Submission Date

7-23-2015

#### Document Type

Paper

#### Department

Mathematics

#### Faculty Mentor

Nicholas Scoville

#### Project Description

The goal of this project is to construct a discrete Morse function which induces both a unique gradient vector field and homological sequence on a given tree. After reviewing the basics of discrete Morse theory, we will show that the two standard notions of equivalence of discrete Morse functions, Forman and homological equivalence, are independent of one another. We then show through a constructive algorithm the existence of a discrete Morse function on a tree inducing a desired gradient vector field and homological sequence. After proving that our algorithm is correct, we give an example to illustrate its use.

#### Recommended Citation

Rand, Ian B., "Discrete Morse Functions, Vector Fields, and Homological Sequences on Trees" (2015). *Mathematics Summer Fellows.* Paper 2.

http://digitalcommons.ursinus.edu/math_sum/2

## Comments

Presented during the 17th Annual Summer Fellows Symposium, July 24, 2015 at Ursinus College.