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We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs, and apply this invariant to various problems in algebraic and symplectic geometry. On the algebro-geometric side, we introduce a new method based on symplectic flexibility and mirror symmetry to bound the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most $3$ . As an application, we resolve a well-known conjecture of Orlov for new classes of examples (e.g. toric $3$ -folds, certain log Calabi–Yau surfaces). We also discuss applications to non-commutative motives on partially wrapped Fukaya categories. On the symplectic side, we study various quantitative questions including the following. (1) Given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly supported Hamiltonian diffeomorphism? (2) What is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers? We give lower bounds for these quantities which are to the best of the authors’ knowledge the first to go beyond the basic flexible/rigid dichotomy.

We study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds and pairs. We describe upper and lower bounds for this invariant in terms of various quantities of interest in symplectic topology. By combining these bounds, we get information about concrete geometric questions such as: (1) given a Weinstein manifold, what is the minimal number of intersection points between the skeleton and its image under a generic compactly-supported Hamiltonian diffeomorphism? ( 2) what is the minimal number of critical points of a Lefschetz fibration on a Liouville manifold with Weinstein fibers?Using homological mirror symmetry, we also obtain improved upper bounds for the Rouquier dimension of derived categories of coherent sheaves on certain complex algebraic varieties and stacks. These bounds are sharp in dimension at most 3. As a result, we resolve a well-known conjecture of Orlov for a large class of new examples, including all toric 3-folds and certain log Calabi-Yau surfaces. The proof of these upper bounds crucially relies on the arborealization theorem of Álvarez-Gavela-Eliashberg-Nadler and is therefore an application of symplectic flexibility to a problem in algebraic geometry.

For any closed symplectic manifold, we show that the number of 1-periodic orbits of a nondegenerate Hamiltonian thereon is bounded from below by a version of total Betti number over Z of the ambient space taking account of the total Betti number over Q and torsions of all characteristic. The proof is based on constructing a Hamiltonian Floer theory over the Novikov ring with integer coefficients, which generalizes our earlier work for constructing integervalued Gromov-Witten type invariants. In the course of the construction, we build a Hamiltonian Floer flow category with compatible smooth global Kuranishi charts. This generalizes a recent work of Abouzaid-McLean-Smith, which might be of independent interest. Contents 1. Introduction 1 2. Recap of the FOP natural transformation and multiplicativity 11 3. Abstract constructions of chain complexes and maps over the integers 22 4. Proof of the integral Arnold conjecture 52 5. Global Kuranishi charts on Floer moduli spaces 69 6. Smoothing 118 7. Constructions for PSS, SSP, and the homotopy 146 Appendix A. Product of canonical Whitney stratifications 155 Appendix B. Relative stable equivariant smoothing 160 Appendix C. A proof sketch of Proposition 5.64 163 References 166

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