I’m interested in theories involved or would be involved in current or near-future biophysics/mathematical biology, so I’m wondering that
1.what modern mathematics/theoretical physics one can use in mathematical biology/biophysics?
From my research online, it seems to be statistical physics, random process, dynamical systems/ergodic theory, fluid dynamics, statistics, probability and measure theory, fractal geometry, and possibly algorithmatic methods (I'm not familiar with these fields, so there may be overlaps.)
For example, some of the mathematics are related to noises and signals and brownian motions and so they are useful for biophysics.
Is there any topic of theoretical physics or mathematics that needs to be added to this list?
2.A closely related question is: is the above modern mathematics indeed necessary, or at least important, for making discoveries in biophysics?
I guess some biophysicists perhaps mainly proceed with physical intuitions and some with complicated mathematics, and some with both. But what’s the case for real biophysics nowadays?
Edits:
I intended to ask what type of theoretical learning prepared a student best for biophysics research, and I guess it’s a question about scientific education, which is perhaps better to be asked elsewhere. Anyway I guess the answer is not much different from what prepares a student best for physics research.
I’m so far interested in biophysics in genetic expression and microscopy. So if anyone is doing research in these fields and uses any mathematical and physical theory I would like to know. Instead of asking what prepares a student for the general field, I think it would be useful to know some specific cases.
Edits:
The following is my opinions.
These partly answer my question: https://physics.stackexchange.com/a/19223/273056 https://physics.stackexchange.com/a/19241/273056 https://physics.stackexchange.com/a/19311/273056
There's a chance that not having some maths could limit discoveries (and you possibly don't know exactly which piece of maths), ways of thinking, and fields in physics that one can learn. I'm not sure to what extent this apply to biophysics, but if it does apply, probably one can't tell which piece of maths is very useful in the long term, especially considering that there are some theoretical advancement in biophysics now and that the field could be very theoretical in the future (in the short term some fields of maths is obviously useful.) And the maths mentioned above seems already good place to start?
Since the future is difficult to predict, and new fields emerge any time, my question is particularly about maths useful nowadays in genetic expression and super resolution microscopy.
The following seems to answer my question: How much pure math should a physics/microelectronics person know and How should a physics student study mathematics?
I would say different people have very different views on this and I would not try to find a universal answer.
Overall I have the impression that surprisingly most experimentalists go on without learning much mathematics (possibly they learn some equations and even some methods, but not further details), and that theorists may for a while need learning mathematics much more than physics. Physics is essentially a mixture of two styles (while chemistry and biology are mostly experimental).
As a note for myself, though physics does have lots of theories, and equations, for non-theoretical physicists, most of the theories are (or learned as) non-mathematical theories, or sort of explanations of things. For non-theoretical physicists, it might have many more equations than chemistry and biology, (and there are lots of mathematics behind them), but its theories (at least as it is learnt) are not fundamentally different from theories in chemistry, biology, economics, and computer science, etc. and are very different from the kind of theories mathematics care about.
For example, genetics actually involve some complicated statistics, namely the invention of variance and covariance from the scratch (it’s much more complicated than it appears to be), but the way it’s taught now makes people feel that it’s pretty simple maths only.