What math classes are relevant for machine learning? I'm a math & cs student interested in machine learning, especially the areas of (deep) reinforcement learning and NLP.
According to the current state of research, which advanced math subjects (apart from the basics like calculus, linear algebra and probability) seem like they are going to be the most important for future theoretical work on ML? (convex optimization? game theory? information theory?)
 A: A great professional of machine learning should know a mix of the ingredients that make up the Computer Scientist and Statistician. Furthermore, It should also know some math to be able to realy learn these subjects.
Math
1) Calculus in $\mathbb{R}^n$ and Optimization (diferential calculus, interior optimization, properties of gradients, constraint optimization, integrals (to know how to calculate expected values)). 
Calculus
2) Math analysis
Mathematical analysis
3) Linear Algebra (The more you know, the better. Solutions of linear systems, inverses and pseudo inverses, decompositions, vector spaces, linear transformations) 
Linear algebra
Linear algebra and optimization
4) Numerical Calculus and analysis (solution of linear and non-linear systems, numerical solutions of eigenvectors/eigenvectors, several gradient methods)
Scientific computing
Numerical analysis
5) Notions of Applied Functional Analysis (for example, Banach Fixed Point Theorem, Hilbert Spaces Projection Theorem)
Functional analysis
6) Convex optimization and optimization (theory and numerical analysis)
Convex optimization
Optimization
7) Measure theory (to understand the details of probability theory)
Concise and great measure theory
Related knowledge:
1) Probability Theory and Statistics
Probability theory
Introduction to probability
Statistical inference
Statistical learning
Monte Carlo methods
Bayesian theory
Bayesian analysis
2) Multivariate statistics (for example, principal component analysis)
Multivariate statistics
3) Regression models (linear regression, binary response)
Introductory econometrics
Intermediate econometrics: lots of interesting models
Generalized linear models
4) Time series
A very good introduction
Hamilton: The bible
With deep math
5) Neural network models (classic problems and also those that include Deep Learning)
Classical neural networks (preparation for deep learning)
Deep learning
6) Lots of computer programming (structured, functional and object oriented programming, pattern design)
Think python
Functional programming
OOP and Design pattern
A great book of design pattern
7) Algorithms (algorithms and algorithm complexity).
Levitin
Cormen
Kleinberg
8) Reinforcement learning (Dynamic Programming and Monte Carlo methods)
Putterman: Math
Suton and Barto: Computation
9) Languages: Python and R
Think python
Advanced R
10) Database: MySQL, PostgreSQL and Hadoop
Data base systems
Beautiful database in python
More database
Finally, you need to have practical experience. Nothing better than connecting with a community that has the same interests as you. You can also face competitions.
Most of this answer comes from a previous answers 1 and 2 that I gave to this (Brazilian) site.
I must have forgotten lots of great references... Sorry about that. 
