#### Submission Date

7-24-2017

#### Document Type

Paper- Restricted to Campus Access

#### Department

Mathematics

#### Second Department

Computer Science

#### Faculty Mentor

Nicholas A. Scoville

#### Project Description

In topology, simplicial complexes can be reduced to smaller complexes of the same simple homotopy type through a series of elementary collapses. It is a common goal to try to find the smallest complex that is of the same simple homotopy type as another. We use the concept of strong collapses in which at least 2 faces are removed, to speed up the process of reducing the complexes. In our heuristic, we first check to see if we can run a strong collapse. If not, we attempt to perform a random elementary collapse if there are free pairs available. If there are no free pairs available, we then remove a random top face from the complex and try again. For the last two options, we keep track of how many top faces and elementary collapses were required in a vector. We obtain this strong discrete Morse v*ector* by increasing the *n*^{th} index in the vector each time an *n*-dimensional face is removed by a method other than a strong collapse. Since the heuristic relies on random selections, the vector may be different if run multiple times. The average vector obtained gives some indication of the complexity of the complex.

#### Recommended Citation

Peralta, Nikolai X., "Random Strong Discrete Morse Theory" (2017). *Mathematics Summer Fellows*. 8.

https://digitalcommons.ursinus.edu/math_sum/8

## Comments

Presented during the 19th Annual Summer Fellows Symposium, July 21, 2017 at Ursinus College.