Algebra and cancer research

You might like to look at this paper:

• Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Proceedings of the National Academy of Sciences, February 2011.

There is also a lot of other stuff by the same group at http://comptop.stanford.edu

Only a small proportion involves biology, but that might be enough for you.

You might also like to look at the work of Maria-Grazia Ascenzi:

http://ortho.ucla.edu/body.cfm?id=218&ref=39

Her PhD is in mathematics but her current work is in biomedical science. I have heard that she is interested in applying algebraic geometry but I do not know the details.

Here are some examples:

http://arxiv.org/abs/1004.1341 (Algebraic Comparison of Partial Lists in Bioinformatics)

http://arxiv.org/abs/1303.5569 (An algebraic framework to sample the rearrangement histories of a cancer metagenome with double cut and join, duplication and deletion events)

http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0068598 (Searching for Synergies: Matrix Algebraic Approaches for Efficient Pair Screening).

This special issue of the "Bulletin of Mathematical Biology" is devoted to the algebraic methods in mathematical biology: http://link.springer.com/journal/11538/73/4/page/1

For algebraic statistics, I think there are two standard references. The books by Drton, Sturmfels, and Sullivant which can be bought or downloaded as a pdf and 'Algebraic Statistics for Computational Biology' by Sturmfels and Pachter ( the authors- assuming dotage hasn't set in). Phylogenetics has seen some applications from mathematics. There is a book with that name by Semple & Steele (same proviso). Finally no discussion of applications of mathematics in biology is complete (to me) without mentioning the Salmon problem. It was to give explicit equations defining a specific secant variety of some Segre/Veronese embedding of a specific set of projective spaces. It was proposed by Allman who is a professor at the University of Alaska and offered a prize of a self-caught and smoked Copper River salmon. Said salmon was eaten by Shmuel Friedland. Finally, Carlson wrote a survey article on 'topology and data', (in some ams journal) which I think is very good- not really algebra, but worth reading. Someone mentioned a paper of Carlson's on cancer, this is probably a mathematical pre-cursor to that.