Abstract These are lecture notes for a mini-course given in Banff in June 2024. They discuss the problem of bounding the number of $$\delta $$ δ -incidences $$\mathcal {I}_{\delta }(P,\mathcal {L}) := \{(p,\ell ) \in P \times \mathcal {L} : p \in [\ell ]_{\delta }\}$$ I δ ( P , L ) : = { ( p , ℓ ) ∈ P × L : p ∈ [ ℓ ] δ } under various hypotheses on $$P \subset \mathbb {R}^{2}$$ P ⊂ R 2 and $$\mathcal {L} \subset \mathcal {A}(2,1)$$ L ⊂ A ( 2 , 1 ) . The main focus will be on hypotheses relevant for the Furstenberg set problem.