The problem Consider two simplicial complexes X and Y as pictured below. Our goal is to understand an algorithm which reveals how a map of complexes induces a map on (co)homology. This post is laid out in the following two parts: We will start by reviewing an algorithm others have shared for computing the basis … Continue reading An Elementary Algorithm for Persistent Cohomology

## Sheafification Via Hypercovers (Under Construction)

The Goal / An Informal Introduction Loosely speaking, the definition for sheaf many of us first learn is that it is a presheaf, $latex F$, which satisfies some glueing condition: for each open subset $latex U$ of our fixed topological space $latex X$, and open covering $latex \{U_i \}$, sections of $latex F(U)$ are precisely … Continue reading Sheafification Via Hypercovers (Under Construction)

## Computing the Direct Limit

The motivation for this post is to go through some details necessary in the formal definition of Cech Cohomology. In that definition, we come across the notion of the direct limit of a directed system. This post will serve to: Explicitly write out some of the details needed to see why the direct limit of a directed … Continue reading Computing the Direct Limit

## Why can we add to both sides of an equation?

(Originally written for my Intro to Homological Algebra Summer Seminar Group at SJC) One of the first things you learn in Algebra, is how to manipulate equations. Given the equation: $latex 2x + 3 = 4,$ we quickly learned how to subtract 3 from both sides of the equation. In this post, I would like to go … Continue reading Why can we add to both sides of an equation?

## Quotients of Groups: Where 2+3 = 1.

(Meant for my Intro to Homological Algebra Summer Seminar Group at SJC) Last time we met we discussed groups, where we can recall that A group, $latex G$, is a set with a binary operation $latex \mu: G \times G \to G$ which is associative, has an identity, and has inverse elements. Example: $latex \mathbb{Z} … Continue reading Quotients of Groups: Where 2+3 = 1.

## An Introduction to Linear Regression

(A Note for my SJC MAT 151 class - Summer 2018) What we would have talked about on Friday, had we met for class, was Linear Regression. Consider this Body Fat Data Set I found which compares patients' body fat measurements to other measurements on their bodies. The first sheet in the excel spreadsheet is simply all … Continue reading An Introduction to Linear Regression