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Let 
$X$ denote the ‘conifold smoothing’, the symplectic Weinstein manifold which is the complement of a smooth conic in 
$T^*S^3$ or, equivalently, the plumbing of two copies of 
$T^*S^3$ along a Hopf link. Let 
$Y$ denote the ‘conifold resolution’, by which we mean the complement of a smooth divisor in 
$\mathcal {O}(-1) \oplus \mathcal {O}(-1) \to \mathbb {P}^1$. We prove that the compactly supported symplectic mapping class group of 
$X$ splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category 
$D(Y)$ (the three-dimensional ‘affine 
$A_1$-case’). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the ‘other side’ of the mirror.
