Events

Jack Jia, University of Waterloo

Categories of representations of groups are well-behaved: They are abelian (behave like module categories), symmetric monoidal (have tensor products), every object has a dual and is semi-simple, to name a few. A natural question to ask is whether every category that exhibits similar behaviour is a representation category. Deligne proved a remarkable theorem that shows every symmetric tensor category with some imposed growth condition is in fact a category of representations. Moreover, he constructed some symmetric tensor categories with faster-than-exponential growth-these are so-called Deligne categories, which can be interpreted as complex rank analogs of classical representation categories. In this talk, I will introduce the notion of symmetric tensor categories, state Deligne's Theorem, and construct some of the Deligne categories. 

MC 5403

Mainak Poddar, IISER Pune

Equivaraint principal bundles over toric DM stacks

We extend the classification of torus equivariant framed principal bundles over toric varieties using Tits buildings by Kaveh, Mannon, and Huang to such bundles over toric Deligne-Mumford stacks. A corollary is that any such bundle over a stacky projective line splits. This is a joint work with Chandranandan Gangopadhyay and Ramandeep Singh Arora.

MC 5479

Beining Mu, University of Waterloo

Degree of categoricity and treeable degrees

In this seminar, we will discuss treeable degrees and degree of categoricity. We will introduce past results on which degrees can or cannot be a degree of categoricity and when degrees of categoricity coincide with treeable degrees. We will also introduce a notion of \(\Pi^0_1\) singletons as an example of treeable degrees and their relation to degree of categoricity.

MC 5403

Julius Frizzell, University of Waterloo

Factors of Measure Preserving Systems

We will discuss the basic idea of homomorphisms of measure spaces and measure preserving systems and what it means for a measure preserving system to be a factor of another measure preserving system.

MC 5417

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