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We study derived categories of Gorenstein varieties 
$X$ and 
$X^+$ connected by a flop. We assume that the flopping contractions 
$f\colon X\to Y$, 
$f^+ \colon X^+ \to Y$ have fibers of dimension bounded by one and 
$Y$ has canonical hypersurface singularities of multiplicity two. We consider the fiber product 
$W=X\times _YX^+$ with projections 
$p\colon W\to X$, 
$p^+\colon W\to X^+$ and prove that the flop functors 
$F = Rp^+_*Lp^* \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$, 
$F^+= Rp_*L{p^+}^* \colon {\mathcal {D}}^b(X^+) \to {\mathcal {D}}^b(X)$ are equivalences, inverse to those constructed by Van den Bergh. The composite 
$F^+ \circ F \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X)$ is a non-trivial auto-equivalence. When variety 
$Y$ is affine, we present 
$F^+ \circ F$ as the spherical cotwist of a spherical couple 
$(\Psi ^*,\Psi )$ which involves a spherical functor 
$\Psi$ constructed by deriving the inclusion of the null category 
$\mathscr {A}_f$ of sheaves 
${\mathcal {F}} \in \mathop {{\rm Coh}}\nolimits (X)$ with 
$Rf_*({\mathcal {F}} )=0$ into 
$\mathop {{\rm Coh}}\nolimits (X)$. We construct a spherical pair (
${\mathcal {D}}^b(X)$, 
${\mathcal {D}}^b(X^+)$) in the quotient 
${\mathcal {D}}^b(W) /{\mathcal {K}}^b$, where 
${\mathcal {K}}^b$ is the common kernel of the derived push-forwards for the projections to 
$X$ and 
$X^+$, thus implementing in geometric terms a schober for the flop. A technical innovation of the paper is the 
$L^1f^*f_*$ vanishing for Van den Bergh's projective generator. We construct a projective generator in the null category and prove that its endomorphism algebra is the contraction algebra.
The pentagram map takes a planar polygon 
$P$ to a polygon 
$P'$ whose vertices are the intersection points of consecutive shortest diagonals of 
$P$. This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if 
$P$ is a Poncelet polygon, then the image of 
$P$ under the pentagram map is projectively equivalent to 
$P$. In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.
In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical 
$p$-Kazhdan–Lusztig polynomials.