How strong is your math background? I have taken real analysis 1, 2, machine learnig, bayesian inference. I am taking measure theory next term. I am worried thst i'm lacking so much though. Differential equations, stochatic calculus, forecasting and the list goes on. I know it may depend on the area of quant finance, what are the must-know topics?
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Your background is looking good mathematically speaking. I would make sure to load up on more stats classes. Time series, theoretical stats would eb good. Differential equations is mostly useless. Stochastic calculus is useful for the sell side but not as much on the buyside i don't think. Maybe for specific hedge funds that deal with more exotic products or market makers.
Programming is important. Become an expert in Python (numpy, pandas, scipy, TensorFlow). Become at least very proficient in R. Being able to code in C++ or Java is also often useful if you're going to be working closer to the production side of things.
Anyway, yeah, your math background is fine. You aren't lacking. The important thing is that you are able to comfortably read and understand research papers. Take a look at some stats or machine learning papers that have come out from the main journals recently. Can you grasp them pretty quickly, understand the conclusions and think of some potential applications? If so, you're doing fine. Usually the applicable papers you would read don't really require real analysis or measure theory. Taking those classes just allows you to more quickly understand the math because you're used to thinking in that way. But being able to understand or do complicated proofs isn't really necessary
I think people are overdoing it. All this quant shows mathematical aptitude/ ability/ interest, but basic programming and some quant skills are all you actually need. If you are applying to a quant-heavy place, just pick a couple of topics and get good... and not real analysis, topology, number theory, etc... something more applied. I have seen people mention Dastidar's investments book on the forum; that has intermediate level material for both quant and discretionary folks. Starting with that is a decent choice.
Quantitative finance is not all encompassing. Exotic derivatives people are quants. There are commodities people who are quants. There are options pricing people who are quants. There are HFT people who are quants. There are long only investors who are quants. There are fixed income people who are quants.
Stock brokers and investment bankers might both use Excel but that doesn't mean their jobs have all that much in common. Similarly, while quants across different areas of finance may both use math, they use different types of math.
Stochastic Calculus is very important to quants on the sell side who are pricing options and other derivatives. If you want to deal with options and other complex financial products, stochastic calculus is important. This is not my area of expertise. I actually know very little stochastic calculus.
Real analysis really just provides the language and structure for thinking, reading and writing mathematically. Knowing real analysis gives you a strong toolbox for reading mathematically heavy text. It isn't particularly useful itself. Crazy advanced probability theory is not often used. More commonly, it is a combination of upper level undergrad/lower level graduate probability and statistics. However, what makes the 'real world' and graduate school much more difficult than undergrad math classes is that you have to understand and use a combination of concepts from the undergrad level. They aren't silo'd off in separate classes.
Differential equations are used in stochastic calculus. I don't know much stochastic calculus so I can't speak to it. But generally, differential equations are used to model more deterministic systems. Obviously, financial markets are not deterministic systems.
In the quantitative equity world where I live, it all comes down to finding and/or generating useful/predictive signals and then coming up with a methodology for using those signals to rank/select stocks to long or short. For the most part, methods in this space use upper level undergrad/lowever level graduate probability, stats, time series analysis and more recently machine learning techniques.