We survey a selection of Yasha Eliashberg's contributions to the philosophy of the h-principle, with a focus on the simplification of singularities and its applications. Contents 1. Introduction 1.1. Flexible mathematics 2. Removal of singularities 2.1. Immersion theory 2.2. Main inductive step 2.3. Conclusion of the argument 3. Holonomic approximation 3.1. The holonomic approximation lemma 3.2. Wiggling into the codimension 4. Surgery of singularities 4.1. Maps between manifolds of the same dimension 4.2. Direct and inverse surgeries 4.3. Inverse surgeries via direct surgeries 4.4. Proof of the existence h-principle for S-immersions 5. Wrinkling 5.1. Wrinkling of mappings 5.2. Soft and taut S-immersions 5.3. Wrinkling of functions 5.4. Wrinkling of embeddings 5.5. Further applications 5.6. Universal holes 6. The arborealization program 6.1. Singularities of wavefronts 6.2. The arborealization program 6.3. Relation between arborealization and flexibility of caustics References

Given a symplectic surface (Σ, ω) of genus g ≥ 4, we show that the free group with two generators embeds into every asymptotic cone of (Ham(Σ, ω), dH), where dH is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds. * This paper was the outcome of the authors' work in the computational symplectic topology graduate student team-based research program held Hofer's geometryLet (M, ω) be a symplectic manifold. Given a smooth function H :where H t (p) := H(t, p). Let ϕ t H : M → M be the flow of the ODEẋ(t) = X t (x(t)), making sufficient assumptions to ensure that the flow is globally defined on the time interval [0, 1] (for example, we could take M to be compact). Inside the group Symp(M, ω) = {φ ∈ Diff(M ) : φ * ω = ω} of symplectomorphisms we have the subgroup of Hamiltonian diffeomorphisms Ham(M, ω), which consists of the time-one maps ϕ 1 H : M → M of flows as above. The group Ham(M, ω) is equipped with a geometrically meaningful bi-invariant metric introduced by Hofer. The resulting metric group is an important object of study in symplectic geometry. For φ ∈ Ham(M, ω), we define the Hofer norm φ H = inf H 1 0

We prove an "h-principle without pre-conditions" for the elimination of tangencies of a Lagrangian submanifold with respect to a Lagrangian distribution. The main result says that the tangencies can always be completely removed at the cost of allowing the Lagrangian to develop certain non-smooth points, called Lagrangian ridges, modeled on the corner {p = |q|} ⊂ R 2 together with its products and stabilizations. This result will play an essential role in our forthcoming paper on the arborealization program [AGEN].

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Let X be a Weinstein manifold. We show that the existence of a global field of Lagrangian planes in T X is equivalent to the existence of a positive arboreal skeleton for the Weinstein homotopy class of X. Contents 1. Introduction 1 2. Wc-manifolds and cotangent buildings 15 3. Positivity of Lagrangian planes 44 4. Arboreal models 50 5. Arboreal Lagrangians and their stability 64 6. Symplectic neighborhoods of arboreal Lagrangians 83 7. Positivity of cotangent buildings 98 8. Ridgification of Lagrangians 104 9. Arborealization of skeleta 110 References 124