Representations of (boxes) bocses and coherent sheaves

In my talk I present the result of my PhD Thesis and discuss
the classification problem of simple coherent sheaves on certain
degenerations of elliptic curves.
Indecomposable vector bundles on smooth elliptic
curves were classified in 1957 by Atiyah.
In works of Burban, Drozd and Greuel it was shown that the categories
of vector bundles and  coherent sheaves on cycles of projective
lines are tame.

It turns out,  that all other degenerations of elliptic curves
are vector-bundle-wild.  Nevertheless, we discover that the category of coherent sheaves of any reduced
plane cubic curve, including cuspidal and tacnode cubic curves  and  three concurrent
lines, is brick-tame. The main technical tool of our approach is
the representation theory of bocses. In my talk I am going to illustrate
its computational potential for investigating the tame
behavior in wild categories.

In particular,  this technique  allows to prove
that a simple vector bundle on a reduced cubic curve
is determined by
its rank, multidegree and determinant, generalizing Atiyah's
classification. Our approach
leads to an interesting class of bocses,
which can be wild but are brick-tame.