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Looking for some advising help (my advisor was not helpful)! I'm a Junior Math major interested in Applied Math/Statistics. I'm currently taking Real Analysis I and have yet to take Abstract Algebra I. Several places "highly recommend" two semesters of real analysis and abstract algebra. However, I'm not really interested in these topics and it appears that very few people (i.e. none of my friends) take them at my school. I'm moreso interested in mathematical modeling/scientific computing/numerical analysis/statistics/data science.

My question is: would I be behind or be limiting myself for grad school (Applied Math/Statistics) by not taking second semester real analysis and/or abstract algebra?

Here are the course descriptions for the analysis courses at my school:

Real Analysis I - Properties of the real numbers, basic topology of metric spaces, infinite sequences and series, continuity.

Real Analysis II - Differentiation and integration in n-space, uniform convergence of functions, fundamental theorem of calculus, inverse and implicit function theorems.

Theory of Functions of Real Variables - The theory of Lebesgue integration, Lebesgue measure, sequences of functions, absolute continuity, properties of LP-spaces.

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    $\begingroup$ Statistics and stochastic are related to measure theory! $\endgroup$ – user302982 Nov 29 '16 at 16:29
  • $\begingroup$ Very hard to put it in terms of if you will be behind or not. I personally feel that real analysis 2 gave me much needed exposure to proofs far beyond what I was comfortable with. But it allow to see many assumptions I took for granted earlier on. You should take real analysis 2, as for abstract algebra take part 1 if you don't like it or feels its a waste stop their. To each ones own. Again not every class you take will match that of another persons. $\endgroup$ – OLE Nov 30 '16 at 1:12
  • $\begingroup$ For any non-linear equation solving in numerical analysis you need multidimensional calculus, and to understand the methods also the implicit function theorem. Perhaps the basics of that are also covered in the num ana course, but if not you need the contents of the Analysis II course or you would be behind from the start. $\endgroup$ – Lutz Lehmann Nov 30 '16 at 12:53
  • $\begingroup$ What do Abstract Algebra 1 and 2 contain? $\endgroup$ – Alex M. Dec 22 '16 at 17:46
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PhD programs in statistics and data science at major universities differ in their preferences. I would say that a solid background in calculus through multiple integration and infinite series is expected by all. Real analysis and measure theory are clearly the more important than abstract algebra. Linear algebra is directly applicable. A post-calculus course in statistics and probability will make the first year easier. Computation is of increasing importance in statistical inference, probability modeling, and data science, so it is a good idea to know the basics of one computer programming language.

You should start now to look at the web sites of various departments to which you might apply. Some of them have specific information on the undergraduate courses they prefer. Almost all PhD programs will start with a measure theoretic course in probability and statistics that involves at least modest computing. These courses are supposed to be accessible to well-prepared math majors with no previous statistics or computing. However, you will likely be more at ease in your first year courses and get more from them if you have at least a little prior background in measure theory, statistics, probability, and/or computing.

Another reason to take an undergraduate statistics course is so you will understand what you are getting into. Statistics is a 'mathematical science' in that it consistently uses mathematical methods. But mathematics is mainly deductive in nature. (Start with axioms and prove what you can.) Statistics and data science are fundamentally inductive. (Start with data from the real world and speculate in a structured way on what they say about reality.) Many statistical ideas and methods are driven by applications, sometimes specific applications. I know of no 'famous century-old unsolved problems' in statistics. But there is a constant barrage of challenging and untidy real-world problems to be solved--or at least better understood--right away.

You should plan to apply to at least half a dozen PhD programs, including a couple that may be a reach and a couple were admission seems likely. There is a lot of randomness in how many PhD students any one program can afford to admit in any one year. In a statistics or data science PhD program you should expect to get a commitment for full support, contingent on your steady progress towards the degree. 'Full support' means just enough money for simple food and lodging and for tuition. This will probably require some teaching or assisting with instruction or research.

Not everyone should go on to a PhD program after a BS (or MS) degree. You need to have the motivation and background to do well. You may be used to being at the top of your undergraduate class, but so will most of the other PhD students. Some of my smart and well-prepared students who have entered PhD programs have used words like 'brutal' and 'incredibly demanding' to describe the pace of PhD coursework. Without strong motivation and an adequate background you may not make it through the first year. You need to have clear professional goals in mind from the start to sustain you through a PhD program.

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Anyone interested in graduate study in statistics needs analysis courses to include Measure Theory and Lebesgue Integration. Electrical Engineers and Computer Graphics specialists would do well to have at least two courses in Abstract Algebra - coding theory (encryption,etc.) depends on the study of Finite Fields which is a topic in the second Abstract Algebra course (ring theory.) Navigation and computer graphics use Integration on Continuous Algebras (Lie Algebras), and in recent years have used Quaternions - a topic in Abstract Algebra 2. Essentially if you want to be admitted to Graduate School in a STEM field, the more advanced math the better

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If you do Applied Math/Statistics i would recommend that you do Real Analysis2,also knowledge in Linear Algebra is important,but Abstract Algebra not necessarily(i think)

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