Welcome to Pure Mathematics

Viktor Majewski (University of Waterloo)

Degenerations of exotic Calabi-Yau metrics through Atiyahs flop

The small resolution of the three-dimensional ordinary double point carries the classical Candelas–de la Ossafamily of asymptotically conical Calabi–Yau metrics. In this talk, I will describe the construction of a new one-parameter family of complete Calabi–Yau metrics on the same complex manifold. The construction begins withan explicit family of U(2)-invariant Kähler metrics adapted to the geometry of the exceptional curve and theasymptotic conical end. After establishing uniform Sobolev, curvature, and volume-growth estimates, one solvesthe noncompact complex Monge–Ampère equation to obtain Ricci-flat metrics in the corresponding Kählerclasses. A central issue is to understand the behaviour of this family as the area of the exceptional curve tends tozero. I will explain how a combination of weighted pluripotential estimates, Chern–Lu inequalities, symmetry,and analysis on the conifold yields uniform control away from the exceptional set and produces a singularCalabi–Yau metric on the conifold. The resulting degeneration has a different local geometric profile from theclassical Candelas–de la Ossa degeneration, providing an exotic family of Calabi–Yau metrics on the resolvedconifold

MC 5417

Faisal Romshoo (University of Waterloo)

Torsion-free hypersymplectic structures

We know that a torsion-free \(\mathrm{G}_2-\)structure determines a Riemannian metric with holotomy contained in \(\mathrm{G}_2\). However, a torsion-free hypersymplectic structure in dimension \($4$\) is not necessarily a hyperkähler structure. We will look at a counterexample and see under what conditions does a torsion-free hypersymplectic structure determine a hyperkähler structure.

MC 5417

Joey Lakerdas-Gayle, University of Waterloo

High incomplete c.e. set

We use an infinite injury priority tree construction to prove Sacks' theorem that there exists a high incomplete c.e. set.

MC 5403