Summer updates

I worked on creating and implementing a new algorithm for using information theory to make phylogenetic trees from aligned DNA sequences at the Cummings Lab in UMD.

This project motivated many interesting biological questions, but here is a mathematical one that caught my fancy: Given a function \(f: \{0,1\}^n \to \mathbb{R}\), can one find a polynomial time algorithm to compute the global maximum of \(f\)? One might be tempted to say “ah this is just integer programming” but what happens when there is no meaningful extension of \(f\) to \([0,1]^n\)? This problem arises in computing the optimal split in a tree, where we must maximize the function \(\vert A \vert H(A) + \vert A^c \vert H(A^c)\) over all possible partitions \(A \sqcup A^c = X\) of a finite set \(X\). Here \(H(A)\) is the (empirical) entropy of \(A\). Email me if you have suggestions and find out more about this project here.