# Algebra and Cancer Research

Let me start by acknowledging the existence of this thread: Mathematics and cancer research?

It is well-known that mathematical modeling and computational biology are effective tools in cancer research. When I started college and declared the math major, this was the direction I envisioned myself pursuing. However, I quickly fell in love with algebra, number theory, and "pure" math. So my question is

Are there any ways for an algebraist to contribute to cancer research?

I've recently learned that algebraic geometry has been useful in studying phylogenetic trees in evolutionary biology, so while I cannot even imagine what an affirmative answer to my question might look like, I am hopeful that one exists.

• Developments in Algebraic statistics is what you likely heard about. – user9072 Apr 14 '14 at 12:32
• I visited a talk a while ago, where the speaker described how the dynamics of blood groups worked in a population. This gave rise to a very specific algebraic variety, and the blood group proportions that we see now today, could be explained, using this variety. This is not exactly cancer research, but definitely in the field of medicine. – Per Alexandersson Apr 14 '14 at 12:47
• @PerAlexandersson That sounds interesting. Can you give a reference for further reading? Thanks! – Jeff H Apr 14 '14 at 13:08
• @JeffH: I tried searching for it, but could unfortunately not fond the reference... – Per Alexandersson Apr 14 '14 at 13:14
• Reidun Twarock and others have used group theory to predict the structure of viruses, with serious applications to drug design. – Ben Barber Apr 14 '14 at 13:39

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. Feb 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.

• This is exactly the sort of thing I'm looking for. Thank you! – Jeff H Apr 14 '14 at 13:42

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

• This is excellent! – Jeff H Apr 14 '14 at 14:02

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.

• In case people want to Google this, you should note that Carlsson has a double s. – Neil Strickland Apr 14 '14 at 19:56
• @Strickland - sadly I even know Gunnar. See my dotage comment as a defense :) – meh Apr 14 '14 at 23:30