We construct an exotic monotone Lagrangian torus in CP 2 using techniques motivated by mirror symmetry. We show that it bounds 10 families of Maslov index 2 holomorphic discs, and it follows that this exotic torus is not Hamiltonian isotopic to the known Clifford and Chekanov tori. 53D12; 53D37, 53D40

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP 2 #CP 2 , CP 2 #2CP 2 , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in CP 2 #kCP 2 for k = 0, 3, 4, 5, 6, 7, 8. We name these tori. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that CP 2 #CP 2 also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP 2 #2CP 2 . Finally, the Lagrangian tori n 1 ,n 2 ,n 3 p,q,r

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Related to each degeneration from CP 2 to CP(a 2 , b 2 , c 2 ), for (a, b, c) a Markov triple (see (1.1)) there is a monotone Lagrangian torus, which we call T (a 2 , b 2 , c 2 ). We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other. Contents

Symplectic flux measures the areas of cylinders swept in the process of a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangian submanifold that we define using its Fukaya algebra. The main geometric feature of the invariant is its concavity over isotopies with linear flux.We derive constraints on flux, Weinstein neighbourhood embeddings and holomorphic disk potentials for Gelfand-Cetlin fibres of Fano varieties in terms of their polytopes. We show that Calabi-Yau SYZ fibres have unobstructed Floer theory under a general assumption. We also describe the space of fibres of almost toric fibrations on the complex projective plane up to Hamiltonian isotopy, and provide other applications.