TITLE: What is the relationship between t-structure and Torsion pair?
QUESTION [13 upvotes]: I am away from Torsion theory in abelian category for some while. So it might be a stupid question. 
The definition of a torsion pair in the category of modules is as follows:
Definition:
A pair $(\mathcal T,\mathcal F)$ of full subcategories of $A-\mathrm{mod}$ is called a torsion pair if following conditions hold:

$\mathrm{Hom}_{A}(M,N)=0$ for all $M \in \mathcal T,
    N \in \mathcal F$.
$\mathrm{Hom}_{A}(M,-)|_{\mathcal F}=0 \Rightarrow M \in \mathcal T$.
$\mathrm{Hom}_{A}(-,N)|_{\mathcal T}=0 \Rightarrow N \in \mathcal F$.

Condition 2) and 3) means that the pair $(\mathcal T,\mathcal F)$ is maximal for $\mathrm{Hom}_{A}(M,N)=0$.
This definition is from the book elements of representation theory of associative algebras
I found this definition is similar to the definition of t-structures in derived category. I just quote the definition from Dimca, Sheaves in topology as follows:
A t-structure on a triangulated category $\mathcal D$ consists in two strictly full subcategories: $\mathcal D^{\leq 0}, \mathcal D^{\geq 0}$ such that the following conditions hold:

$\mathrm{Hom}(X,Y)=0$ if $X \in \mathcal D^{\leq 0}, Y \in \mathcal D^{\geq 1}$.
$\mathcal D^{\leq 0} \subseteq \mathcal D^{\leq 1}, \mathcal D^{\geq 1} \subseteq \mathcal D^{\geq 0}$.
For any $X \in \mathcal D$, there is a distinguished triangle
$$A\rightarrow X\rightarrow B\rightarrow A[+1], \qquad A \in \mathcal D^{\leq 0}, B \in \mathcal D^{\geq 1}.$$

Although the axiom 3) for t-structures looks different to the axioms of torsion pairs. However, there is a proposition of torsion pairs establishing the similar formula:
Let a pair $(\mathcal T,\mathcal F)$ be a torsion pair in $A-\mathrm{mod}$, and let $M$ be an $A$-module. Then there exists a short exact sequence 
$$0 \rightarrow tM \rightarrow M \rightarrow M/tM \rightarrow 0, \qquad tM \in \mathcal T, M/tM \in \mathcal F,$$
where $t$ is the idempotent radical (it behaves like radical of module).
My questions

Is there any relationship between these two constructions?
Is there a definition of torsion theory in triangulated categories? If there exists, does it coincide with t-structures in triangulated categories?
t-structures played important roles in reconstruction schemes (or go back to abelian category) from derived category. So, is torsion theory in abelian category playing similar roles? (I suspected very much, so it might be stupid.)

Thank you in advance!

REPLY [5 votes]: As I understand it, all torsion classes correspond to t-structures in the way described by Greg, but there is almost always more t-structures than torsion classes (even taking into account the shifts).
I think t-structures are closer to some kind of filtration on the abelian category. 
There's a paper on stability conditions (Gorodentsev, A. L.; Kuleshov, S. A.; Rudakov, A. N. $t$-stabilities and $t$-structures on triangulated categories. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117--150; translation in Izv. Math. 68 (2004), no. 4, 749--781) that talks about this a bit.