What types of mathematics are useful for studying genetics? Of course, statistics is, and I'm guessing set theory and group theory, but anything else? What about ordinary and partial differential equations?
Genetics is a very wide field of knowledge and many different areas of mathematics are being applied to genetics. Here is a probably not exhaustive list of the fields of mathematics that I have encountered in genetics (you will note that the elements of the list are not mutually exclusive)
- diffusion equations
- differential equations
- markov processes
- birth-death process
- probability theory (a lot of it)
- Bayesian probabilities
- Math related to MCMC processes as well as HMC processes
- Many fields of theoretical computer science are being applied and I guess theoretical computer science is a field of mathematics.
- Graph theory
- numerical integration
- Analysis of dynamic systems
- set theory
- Yes of course
- group theory
- Probably not so much or maybe I jsut don't really know what it is
- ordinary and partial differential equations
- I think pretty much any field of science applying mathematics will use differential equations.
But of course, not all geneticists are good mathematicians. Each geneticist specialize in a particular subfield. Some of them do a lot of mathematical modelling, others don't do any. Statistics and everything related to data science will always be useful though.
From my understanding, it depends a lot on the sub-field you're in.
Data science, statistics and machine learning follow a different paradigm than calculus. Whereas data driven approaches to problem solving are generally concerned with developing predictive models of the underlying system, differential equation based approaches describe a mechanistic model.
These 'top down' and 'bottom up' approaches both have their benefits and drawbacks. On the data science side, it is very difficult to get a low-level understanding of the system. An oversimplification would be:
"The cell is a black box, so let's figure out the output for a given input to get a better understanding of what's going on inside."
On the calculus side, although mechanistic descriptions are a mathematical formalism of what's going on inside the system (and thus imply a complete understanding of the system - in theory), there is still so much that is unknown about most biological systems you'd care to model. This kind of modelling (from the discussions I've had) will be important in the future, when more is known about these biological systems, but for now it is constrained to small sub-systems. A good analogy might be:
"It's hard to figure out how an engine works when we don't know what half of the parts do."
Of course, there are many projects that bridge the gap between the above two methodologies.
So, to answer your question, statistics, computer science, machine learning, linear algebra, calculus, graph theory, set theory, optimization and combinatorics are all useful.