Homological algebra of singularities : from mirror symmetry to deformations of fat points

This thesis investigates the derived endomorphism algebras of singular geometric objects at extremal points of their moduli spaces. Motivated by homological mirror symmetry and the local deformation theory of the Hilbert scheme of points, we compute the homological structures governing these spaces, including their Ext algebras, associative Yoneda products, and differential graded Lie algebra (DGLA) obstructions. We first analyze the B-model mirror to the pair of pants, the 1-dimensional singularity X = Z(xy) ⊂ ℂ². Using periodic free resolutions for the structure sheaves of its irreducible components, we compute their Ext groups and Yoneda pairings. The resulting algebra matches the wrapped Floer cohomology and the binary m₂ composition of the mirror A-model, verifying the algebraic structures of the partially wrapped Fukaya category. The thesis primarily focuses on the deformation theory of the 0-dimensional fat point in ℂ³, corresponding to the quotient module M = R/mᴷ over R = ℂ[x,y,z]. While vertex formalisms compute the dimensions of the virtual tangent and obstruction spaces, determining the full multiplicative Ext algebra is limited by the impractical computation of large syzygy matrices. To resolve this, we develop a coordinate-free approach using sl₃(ℂ)-equivariant representation theory. We construct the k-linear minimal free resolution of M via finite-dimensional isotypic decompositions, reformulating the syzygies using the exterior algebra of polynomial spaces. This equivariant framework allows us to lift Ext classes to explicit chain maps and compute their associative products through invariant tensor calculus. We decompose Ext*R(M,M) to classify first-order deformations geometrically into unobstructed spatial translations and non-commutative internal shape deformations. Using the perfect trace pairing of graded local duality, we derive coordinate formulas for the binary and triple Yoneda products. Finally, we extract the totally symmetrized trace component to derive the primary Maurer-Cartan equation, expressing abstract DGLA obstructions as explicit polynomial constraints governing the critical locus of the associated local Calabi-Yau superpotential.