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?



Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...)
**(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](http://jlms.oxfordjournals.org/content/65/1/27.full.pdf+html) and [here](http://www.sciencedirect.com/science/article/pii/S0168007204001058) 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](http://math.berkeley.edu/%7Escanlon/papers/pd.pdf) 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](http://math.berkeley.edu/%7Escanlon/papers/bsl4ap00.pdf) 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](http://www.ams.org/journals/jams/1996-9-03/S0894-0347-96-00202-0/S0894-0347-96-00202-0.pdf). For more see [this book](http://link.springer.com/book/10.1007/978-3-540-68521-0).


**(5) Number theory:**


For example see the recent works of [Jonathan Pila](https://www.maths.ox.ac.uk/people/profiles/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](http://www.sciencedirect.com/science/article/pii/0003484376900061).


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](http://arxiv.org/pdf/1210.6485.pdf) and [here](http://arxiv.org/pdf/1309.0340v1.pdf)), **approximate groups**, etc. (for more see [here](http://arxiv.org/find/math/1/au:+Hrushovski_E/0/1/0/all/0/1?skip=0&query_id=a15dce4be5221189), [here](http://arxiv.org/find/math/1/au:+Scanlon_T/0/1/0/all/0/1), [here](http://arxiv.org/pdf/math/0510133.pdf) and [here](http://arxiv.org/find/math/1/au:+Moosa_R/0/1/0/all/0/1) )


**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!


