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Get me excited to learn Algebra and Analysis this summer!

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A big goal of abstract algebra is to get Galois theory, which will allows you to prove a number of famous impossibility theorem: impossibility of 2 out of 3 classical straightedge-and-compass construction problem, and impossibility of solving quintic equation using radicals. From abstract algebra you can go and study number theory, especially algebraic number theory, and algebraic geometry.

Real analysis will get you the tools to set up for many intuitive facts you learned from calculus. Why does a curve that go on both sides on a line must intersect that line? What does "real number" really mean? If you're interested in how we can logically define everything and work from there (the same way Euclid made a system of axioms) you would be interested in real analysis. Real analysis serves as important stepping stone for functional analysis, harmonic analysis (which go into PDE) and topology.

Linear algebra is not part of studying abstract algebra. It's considered a separate topic. But there are certainly overlap, linear algebra overlap significantly with both algebra and analysis, simply because it's so useful it get applied everywhere. But depends on what exactly do you want to use it for, the focus of study in linear algebra can change significantly. Any given linear algebra class will often try to get a balance of both sides, but there will be preferences that change the syllabus somewhat. An analysis-focused linear algebra class will work over real and complex number, have application to differential equations, like to use Jordan normal form, like to use either finite-dimensional space or inner product space, and focus on spectral theorems and related theorems. An algebra-focused linear algebra class will work over arbitrary fields with possible focus on finite fields for applications, prefer algebraic quantities like trace and discriminant, have general theorems that is even applicable to infinite dimensional space, and like rational canonical form.

Complex analysis is used to study complex geometry, which is something that get abstracted out into algebraic geometry. However, algebraic geometry are often taught without complex analysis (because they are all abstracted out), so it only serves as motivations. However, a big connection is in the GAGA theorem and GAGA-style results.

C^* -algebra and Banach algebra is pretty much on the analysis side. They're not part of abstract algebra.
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I think algebra and analysis are the core of mathematics and the place where most people are introduced to real mathematical thinking. I don't think you need to have any further goals in mind, considering that learning these two subjects alone will change you from someone who doesn't know mathematics into someone who does.

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