Which areas of mathematics can be formally studied in a cognitive science major? Or: which areas of mathematics can support the study of brain. Some areas that seem relevant would be: mathematical logic, graph theory, linear algebra. Is any element of topology relevant? Please direct to links/sources if possible. Is network science being used in these fields? I am asking this question, as I want to study further in mathematics, and would love it if I could do so while pursuing cognitive science.
Here's my list of math subjects that support the study of brain (from a computational neuroscientist's perspective):
- Linear algebra
- to understand high dimensions, to compute things quickly, foundation for other math
- basics for everything continuous valued
- to analyze any data, you need stats!
- basis for modeling, regression, clustering, classification, and all
- Differential equations (basis for dynamical system)
- Dynamical system
- intuition for neural dynamics (deterministic approximation)
- modeling single neuron, synapse, small network
- Statistical physics
- modeling large scale noisy neural dynamics
- Information theory
- quantify how much "information" is coded in neural signal
- Numerical computation
- your data and algorithm needs to be implemented in a computer
- Convex optimization
- your statistics/model requires optimization
- Probability theory (basis for stochastic process and statistics)
- Stochastic process
- model of neural signals, decision process (diffusion), basis for advanced statistics
- point process theory is useful for dealing with neural spike trains
- Time series analysis (your data is a time series!)
- Signal detection theory
- psychophysics is often designed to be a detection task
Brain is quite noisy, you need tools to deal with noise. More applied math than pure math is needed.
I have only seen topology being used a handful of times, and they were not very useful nor impressive. I love set theory and mathematical logic, but sadly never used it nor seen it being used.
In addition, real/complex/functional analyses are also useful, just in general.
I will deviate from the other answers and give more pessimistic response based on my experience as a mathematician and theoretical computer scientist that spends some of his time in a psychology department.
- In cognitive science, neuroscience, and psychology (like in most sciences) you will never do mathematics in the definition, lemma, theorem, proof sense.
- Instead, you will rely on using (or modifying) mathematical tools others built, and doing calculations with them. (for a list of tools, see Memming's answer and more detail here on dynamic systems)
- Most of your colleagues will not know what you are doing (since the usual extent of training is linear regression),
- and even the ones that are trained in mathematical tools will usually use them in a rather black-box way without making deep connections to other parts of mathematics.
If you want to do mathematics and prove theorems then your are better off in one of the following fields allied to cognitive science:
- Mathematical linguistics -- although most people here still focus on tool use, there are some that prove things using ideas from category theory, mathematical logic, and a bit of topology (through Curry-Howard-Lambek crrespondence and Stone duality).
- Economics -- most of the proofs here seem to be performances, but sometimes deep results are proven, usually related to game theory and decision theory. A solid grasp of analysis will be useful and topology will be relevant since most results follow from fixed-point theorems.
- Computational learning theory -- this subfield of theoretical computer science often can use very deep mathematics (usually from graph theory, combinatorics, optimization, discrete Fourier analysis, probability, and information theory) in its proofs, and has a history of interaction with cognitive science which has unfortunately waned in recent years.
I would say that the maths that are most useful in cognitive science are the ones that have to do with decision theory. So I would include linear algebra (with its matrixes, and "transition" or changes of state analysis), as well as probability and statistics, with their "expected values" and resulting decision trees. Computational and information analysis, with their "data structures" might also be helpful. The last thing I would include is optimization.