I think that Joseph gave a very comprehensive answer to that question. Undeniably, at the top of that list are the common packages you would encounter as a developer in the field.
However, what none of these comments really touches on is the length of time it takes to become familiar with each of these topics. They are actually quite deep and bad answers that results in 100% classification accuracy are common due to over-fitting.
If you just want an entry-level position I don’t see the need to know anything past Naive Bayes, LDA, and SVM which you can get from most common numpy + scipy packages. Linear algebra will help, but if you understand principal component analysis and singular value decomposition then these things above are 80% of the theory you will need in practice.
I would study harder on learning how to get good features more than anything else to be honest. Classifiers are a well-understood mathematical science at this point. Knowing how to get good features and build good kernels is where the biggest bang for your employment buck might be in more advanced roles.
Since you are probably going to want to implement these you should know your basic programming languages that most people who recruit use, like those mentioned above.
I seriously doubt knowing any fancy mathematics will matter very much. Knowing about reproducing kernel hilbert spaces is interesting, but often in most startups will just want you to keep implementing until it works right on the deployment end. Doing this means you just have to get something to work with the data you have been given. Perhaps this is slightly un-intellectual, but it is often how early-stage ventures work.
If you want to be a data-science person and treated as a scientist, or as anything more than a highly paid technician, and work at the management level you should consider formal education in machine learning and computer science. A bright 15 year old can get a job at a startup if he can do the work, but I wouldn’t ask him to lead a team unless they had some other extraordinary capabilities that are actually quite rare.
