Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...) Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many applications in algebraic geometry and Diophantine geometry.
(A) I wonder to know if there are any nontrivial applications of set theory in branches like algebraic geometry, Diophantine geometry, K-theory or number theory (algebraic or analytic)? In particular:
1) are there statements in these fields which are independent from $ZFC$? 
2) Are there $ZFC$ provable statements  in these fields whose proofs are known just using set theoretic methods?
(B) On the other hand are there any results in set theory whose proofs are based on some techniques from the above quoted fields?
Giving references is appreciated.
(C) Are there any connections between model theory and algebraic or analytic number theory?
 A: I would like to point out a work that may be an answer to your question (B).
Misha Gavrilovich constructs a certain model structure on a category of sets (rather sets of sets), and argues that the covering number (of Shelah's PCF theory) can be obtained as a value of certain derived functor (in the sense of Quillen) with respect to this model structure.
A: I suppose this counts as algebraic geometry, so it would be an example of (A) 1). 
Let $R$ be a ring and $D(R)$ its unbounded derived category. Let $D^c(R)$ be the full subcategory of compact objects (in the explicit example below it is spanned by bounded complexes of f.g. projective modules). We say that $D(R)$ satisfies Adams representability if any cohomological functor $D^c(R)^{op}\rightarrow Ab$, i.e. additive and taking exact triangles to exact sequences, is isomorphic to the restriction of a representable functor in $D(R)$ (in particular it extends to the whole $D(R)$), and any natural transformation between restrictions of representable functors $D^c(R)^{op}\rightarrow Ab$ is induced by a morphism in $D(R)$ between the representatives.
Let $\mathbb C\langle x,y\rangle$ be the ring of noncommutative polinomials on two variables. The statement '$D(\mathbb C\langle x,y\rangle)$ satisfies Adams representability' is equivalent to the continuum hypothesis. 
You can make similar statements with commutative $R$, they are related to $|\mathbb C|=\aleph_n$ for $n>1$ (still independent of ZFC), this is why I prefered the previous explicit example.
All this follows from :
Failure of Brown representability in derived categories
J. Daniel Christensen, Bernhard Keller, Amnon Neeman
Topology 40 (2001) 1339}1361
A: (C)
Recently applied model theorists have touched many areas of algebra, algebraic geometry, number theory and even analysis structures.
(1)  Exponential fields:
Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s:
Given any $n$ complex numbers $z_1,\dots,z_n$ which are linearly independent over the rational numbers $\mathbb{Q}$, the extension field $\mathbb{Q}(z_1,\dots,z_n, \exp(z_1),\dots,\exp(z_n))$ has transcendence degree of at least $n$ over $\mathbb{Q}$.
In 2004, Boris Zilber systematically constructs exponential fields $K_{\exp}$ that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinal. Zilber axiomatises these fields and by using the Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proves that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. See here and here for more.
(2) Polynomial dynamics:
The connection between algebraic dynamics and the model theory of difference fields was first noticed by Chatzidakis and Hrushovski. A series of three papers entitled "Difference fields and descent in algebraic dynamics".
It seems that the first-order theories of algebraically closed difference fields where the automorphism is "generic" are quite nice. See here for more result by Scanlon and Alice Medvedev.
(3) Diophantine geometry:
Hrushovski, Scanlon and their students have worked on model theory and its application in Diophantine geometry.
See here for information about applications of model theory in Diophantine geometry.
(4) Algebraic geometry:
The Mordell-Lang conjecture for function fields: Let $k_0\subset K$ be two distinct algebraically closed fields. Let $A$ be an abelian variety defined over $K$, let $X$ be an infinite  subvariety of $A$ defined over $K$ and let $\Gamma$ be a subgroup of "finite rank" of $A(K)$. Suppose that $X\cap \Gamma$ is Zariski dense in $X$ and that the stabilizer of $X$ in $A$ is finite. Then there is a subabelian variety $B$ of $A$ and there are $S$, an abelian variety defined over $k_0$, $X_0$ a subvariety of $S$ defined over $k_0$, and a bijective morphism $h$ from $B$ onto $S$, such that $X=a_0 + h^{-1}(X_0)$ for some $a_0$ in $A$.
This theorem is proved by Hrushovski in 1996, see here. For more see this book.
(5) Number theory:
For example see the recent works of Jonathan Pila.
(6) Analysis:
Traditionally model theory is consistent with algebra. But recently, model theorists have been interested in continuous structures that appears in analysis, for example Banach spaces. For more see here.
Model theory has many other application in other fields of mathematics, such as geometric group theory, differential algebra, Berkovich spaces (see recent works of Hrushovski, Loeser, Poonen here and here), approximate groups, etc. (for more see here, here, here and here )
Note: Model theorists have many important and interesting problems in their fields and I believe that the goal of model theory is not necessary to solve the problems of the other fields!
