We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main classes: one denoted by $\A_{n,t}$ called of type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the geometric model of cluster categories of type $A, $ or type $D$, we give a geometric description of (co)torsion pairs in $\A_{n,t}$ or $D_{n,t}$ respectively, via defining the periodic Ptolemy diagrams. This allows to count the number of (co)torsion pairs in these categories. Finally, we determine the hearts of (co)torsion pairs in all finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting via quivers and relations.

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the$C$^{*}-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces$A$^{2}(Ω) for a wide class of plane domains Ω⊂$C$, and in Fock spaces$A$^{2}($C$^{N}),$N$≧1.

We give a classification of birational transformations on smooth projective surfaces which have a Zariski-dense set of noncritical periodic points. In particular, we show that if the first dynamical degree is greater than one, the union of all noncritical periodic orbits is Zariski-dense.

We are interested in the 3-Calabi-Yau categories D arising from quivers with potential associated to a triangulated marked surface S (without punctures). We prove that the spherical twist group ST of D is isomorphic to a subgroup (generated by braid twists) of the mapping class group of the decorated marked surface S_Delta. Here S_Delta is the surface obtained from S by decorating with a set of points, where the number of points equals the number of triangles in any triangulations of S. For instance, when S is an annulus, the result implies that the corresponding spaces of stability conditions on D are contractible.

No abstract uploaded!