I am an undergraduate student in Mathematics and I would like to continue my postgraduate studies in the harder, more mathematical aspects of Linguistics. What exactly would that include is unknown even to me, but possible areas of interest would include the mathematical aspects of syntax and semantics, as well as computational linguistics, natural language processing, artificial intelligence or machine learning.
As of now, the courses that I have taken have been purely mathematical: real, complex and functional analysis, measure theory, point-set topology, differential equations, group and ring theory, linear algebra. I understand that the analytic courses are not very pertitent in linguistics where data is rarely "continuous" or as nicely behaving as the functions studied in analysis. I would like to hear the community's suggestions about courses that should constitute good preparation for postgraduate studies in the aforementioned areas. The courses need not be mathematical. They can be linguistics, statistics, computer science courses or courses from any other discipline that would best prepare me for the task.
I kindly request that you do not close, delete, flag or anything of the sort, this question. It is quite important for me to read opinions from people who know better than me. If you could somehow promote the question so that I can potentially receive more opinions, this will be greatly appreciated!
UPDATE: If there are other people interested in this question, some more answers have been given here. Moreover, I found this blog, which is extremely rich regarding topics of mathematical and computational linguistics as well as natural language processing. This post as well as this should be of much interest to those in the mathematical side of linguistics.
There are a few things you would like to try.
Working in linguistics you will surely work with computers (to test your hypotheses, to gather and manipulate data, to process result statistically).
That begin said, it is not necessary, but it would be very helpful for you to learn Python and familiarize with Natural Language Toolkit. Good place to start is the NLTK book, full text is available freely online, and is a great source of information (many working examples) on computational linguistics. Even if you won't work as a computational linguist, this area has a great influence nowadays and basic understanding about what and how things are done there is a must.
Of course, probability theory. There are many results that linguists use from there, e.g. Bayesian rules or hidden Markov models.
Naturally, statistics, I suspect this does not need any comments.
Formal languages, including Chomsky hierarchy, formal grammars, automata, also a bit of abstract algebra (e.g. see syntactic monoid). This courses usually contain some information on computability theory (e.g. this) and theory of information (e.g. Kolmogorov complexity), it useful to have some understanding of basic concepts and results in both.
Logic, including predicate calculus, lambda calculus, inference systems. Be aware, that there are (at least) two terms "logic" available: in philosophy and in mathematics. Note, also that mathematical logic is very broad, and, for example, you don't need much model theory (which could a whole domain on its own).
Semantics, including the formal way (e.g. denotational semantics, etc.), but also ontologies and Co.
I know, this is a lot (and still incomplete!), but I just wanted to sketch the area. Most of it you will pick along the way (you don't need everything from the very beginning). Please note, that this is more from computational linguist side, and there is a whole range of themes outside of it which probably will be useful to you. However, it would be best to ask some specialist, why don't you try http://linguistics.stackexchange.com?
Good luck!