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.
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.
 A: Contributions of algebraic methods for cancer research are needed i.a. for data analysis and for modeling.
European Association for Cancer Research (EACR):
2 Postdoctoral Positions, Machine Learning, AI, Abstract Algebra, Champalimaud Foundation, Lisbon, Portugal, 2020
"The aim of the positions is to develop theory and methods related to an approach to AI based on Model Theory and Abstract Algebra, read here": Martin-Maroto, F.; De Polavieja, G. G.: Algebraic Machine Learning. 2018
Methods for modeling of biochemical networks and bionetworks (i. a. for cancer) are e.g. neural networks, Petri nets, systems of differential equations, systems of integro-differential equations and the algebraic methods used for that.
Sassone, V.: The Algebraic Structure of Petri Nets. 2000

F. Brouwer, A. F.; Meza, R.; Eisenberg, M. C.: A systematic approach to determining the identifiability of multistage carcinogenesis models. Risk Anal. 37 (2017) (7) 1375–1387
Closed-form solutions could be one means to help deducing new knowledge about unknown interrelations. Closed-form solutions for the above-mentioned methods would be nice.
A: You might like to look at this paper:

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*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.
A: 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
A: 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.  
