I am a final-year MSci student in Mathematics at Imperial College London, working in algebraic geometry, especially deformation theory, Hilbert schemes, moduli problems, and computational methods.
We prove that the one-step loci with Hilbert functions (1,5,6) and (1,5,7) are smoothable in embedding dimension five. The proof uses the Erman-Velasco smoothable regularity-two family and a finite-field differential rank test, placing these exceptional Shafarevich gap cases on the smoothable side.
We study the nested Hilbert scheme of points on the affine plane via deformation theory, torus actions, and Young diagram combinatorics. We identify tangent spaces at nested pairs, describe torus-fixed points by partitions with a removable corner, derive the tangent-weight formula using a Young-diagram shortening rule, and verify the formula computationally in Macaulay2.
We describe the GameTheory package version 1.0 for computing equilibria in game theory available since version 1.25.05 of Macaulay2. We briefly explain the four equilibrium notions, Nash, correlated, dependency, and conditional independence, and demonstrate their implementation in the package with examples.
A small software companion for the thesis: removable corners, addable monomial generator directions, and the row-and-column shortening rule for tangent weights.
Open PlaygroundA tiny lab with Macaulay2 starter snippets for Groebner bases, monomial ideals, Young diagram boundaries, and Betti tables.
Open Lab