Intersection of representation theory with other subjects I have yet to take a class in representation theory (will be doing so during the spring semester), but none the less I read a bit of rep theory here and there out of curiosity. Almost at the beginning of the wikipedia page there is the following quote:
"Second, there are diverse approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology."
And was wondering wether somebody had sources/reccomendatopn to read about the interesections/applications of representation theory with the above subjects. There are some subjects where I have a vague idea how they relate to representation theory, but others... (analytic number theory or algebraic combinatorics)
In addition if anybody knows of other fun intersections/application of rep theory to other parts of math
 A: 
One intuitive reason there's representation theory in many subjects is that: group actions (on topological spaces, vector spaces, or other groups) show up in lots of situations, and so do vector spaces; representation theory is a marriage of the two: it studies group actions on vector spaces, say given $\rho: G\to \text{GL}(V)$ for $G$ a group and $V$ a vector space, $g\in G$ acts on $V$ by $g\cdot v:=\rho(g)(v)$. As for some concrete subjects...
Something about relation to algebraic combinatorics: Young tablaeu (wikipedia) are combinatorial objects recording irreducible representations of the symmetric groups $S_n$. Some textbook references are Fulton's "Young Tableaux" (Part II) and "Representation Theory" by Fulton-Harris (Chapter IV).

If you've seen Riemann surfaces and fundamental group, monodromy representation of the fundamental group (via symmetric group) relates several different subjects. The starting point is to take $$f:X\to Y$$ a degree $d$ map of compact, smooth Riemann surfaces.
Away from a finite set $B$ of points of $Y$ ("the branch points"), for all $y\in Y\setminus B$, $f^{-1}(y)$ should be $d$ distinct points on $X$. Take such a $y$, a loop $$\ell: [0,1]\to Y$$ in $Y$ based at $y$ (starts and ends at $y$). If we pick a fixed $y'\in f^{-1}(y)$, in general we can find a well-defined path $$\tilde{\ell}:[0,1]\to X$$ such that $$\tilde{\ell}(0)=y',$$ and $$f(\tilde{\ell}(t))=\ell (t)$$ (so $\tilde{\ell}$ is a lift of $\ell$ up in $X$, along the map $f:X\to Y$). But we only know that $\tilde{\ell}(1)\in f^{-1}(y)$, it doesn't have to be $y'$; in other words, we can think of the lift as "mapping" $y'\mapsto \tilde{\ell}(1)$. Thinking of all the $d$ points $y'\in f^{-1}(y)$ gives us a permutation of the $d$ points, ie., an element in $S_d$.
Now we can do this to all (homotopy classes of) loops based at $y$, and get a group homomorphism $\pi_1(Y, y)\to S_d$. This is what's called the monodromy representation. It is not a representation by vector spaces ("linear representation") but is nevertheless useful: for instance, it sheds light on "counting" Riemann surfaces and gives a recipe of reconstructing Riemann surfaces. A reference is the book "Riemann Surfaces and Algebraic Curves" by Cavalieri and Miles (Chap 5-8).
