Events

Facundo Camano, University of Waterloo

Classifying ALF Gravitational Instantons of Cyclic Type

I will go over the paper ”Rigidity for Multi-Taub-NUT Metrics” by Vincent Minerbe.

MC 5417

Benoit Charbonneau, University of Waterloo

The Dirac operator on homogeneous manifolds

MC 5417

Joey Lakerdas-Gayle, University of Waterloo

Dominant functions and high sets

We prove Martin's theorem that \(A'\geq_T\emptyset''\) if and only if $A$ computes a function that bounds every total computable function almost everywhere. Then we will begin discussing the infinite injury priority tree construction of Sacks' theorem that there exists a high incomplete c.e. set.

MC 5403

Julius Frizzell, University of Waterloo

Examples of Factors and Conditional expectation

We will discuss examples of factors of measure preserving systems arising from products and skew-products. We will also briefly define regular and separable extensions. We will then introduce the concept of Conditional expectation and prove some of its basic properties.

MC 5417

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

Spiro Karigiannis (University of Waterloo)

Decomposition of Riemann curvature in 4 dimensions (and beyond?)

This is a continuation of my earlier talk from May. We showed that the Riemann curvature tensor of a Riemannian metric decomposes into three orthogonal components: the scalar curvature, the traceless Ricci curvature tensor, and the Weyl curvature tensor. We will see that in dimension 4 we can say more. The Weyl curvature decomposes into two pieces, and we will explicitly determine the decomposition the curvature operator (as a self-adjoint operator on 2-forms). As an application, we will discuss the Singer-Thorpe Theorem characterizing 4-dimensional Einstein metrics in terms of this decomposition. If time permits, we will briefly discuss a generalization of these ideas to G2-geometry in seven dimensions.

MC 5417

Alex Pawelko (University of Waterloo)