Events

Larissa Kroell, University of Waterloo

Partial C*-dynamical systems and the ideal structure of partial reduced crossed products

We study C*-algebras stemming from partial C*-dynamical systems. We develop equivariant injective envelopes associated to such systems, which allow us to obtain canonical connections to enveloping actions as well as results on the ideal structure of partial crossed products.

We extend the theory of equivariant injective envelopes pioneered by Hamana in the 1980s to partial C*-dynamical systems. To do so, we introduce the category of generalized unital partial actions by allowing for *-automorphisms acting on families of special hereditary subalgebras. Utilizing properties of injective envelopes and the notion of an injective unitization of partial C*-dynamical systems, we argue that it suffices to consider unital objects in our category. This also allows us to connect our theory to Abadie's notion of enveloping actions leading to a canonical relationship of their G-injective envelopes.

Utilizing properties of injective envelopes, we introduce novel non-triviality conditions for partial *-automorphisms inspired by global C*-dynamics. We contrast this notion with existing conditions in the literature. Lastly, we study a non-commutative generalization of stabilizer subgroups for pseudo-Glimm ideals. In particular, we show that for Glimm ideals in the C*-injective envelope, these stabilizer subgroups are in fact amenable --- a result which is crucial for our main theorems regarding the ideal structure of partial reduced crossed products.

Finally, our main application of the theory of G-injective envelopes is a characterization of the ideal intersection property for partial C*-dynamical systems subject to a cohomological condition as a generalization of the result for global group actions. To state this generalization, we utilize the dynamical conditions introduced previously and generalize the notion of equivariant pseudo-expectations to partial C*-dynamical systems. We also give a sufficient intrinsic condition in terms non-commutative uniformly recurrent partial subsystems utilizing pseudo-Glimm ideals. As a consequence of our results, we obtain a full characterization of the ideal intersection property for partial actions on commutative C*-algebras in terms of freeness of the partial action on the spectrum of the G-injective envelope.

MC5403

Robert Harris, University of Waterloo

Exotic constructions on covers branched over hyperplane arrangements

As a consequence of embedded surfaces and codimension two submanifolds coinciding in dimension four, many of the tools that are used in higher dimensions fail or are underwhelming when applied to 4-manifolds. For this reason, the development and advancement of techniques that are applicable to 4-manifolds are of particular interest and importance to low dimensional topologists. The general techniques of interest are those that either construct a 4-manifold in a novel way or those that provide ample control over the geometric data of the resulting 4-manifold.

In this talk, I will discuss my thesis, in which we investigate ways to construct 4-manifolds with positive signature. We also describe a construction that can guarantee the existence of algebraically interesting embedded symplectic submanifolds.

Specifically, we discuss how the combinatorial data of line arrangements and the algebraic data of their complements in rational complex surfaces can be utilized to construct symplectic 4-manifolds with arbitrarily large signatures through the method of branched coverings. In general, we not only show that these line arrangements can be used to provide asymptotic bounds for the existence of symplectic 4-manifolds but we also show that for any line arrangement, there exists symplectic branched covers with sufficiently nice geometric and topological properties. Namely, we show they contain embedded symplectic Riemann surfaces which carry their fundamental group.

Online presentation: contact r26harri@uwaterloo.ca for details on how to attend

Samantha Nadia Pater, Cuiwen Zhu and Hanwu Zhou

The Hasse Principle for Diagonal Forms via the Circle Method

The Hasse principle predicts that a Diophantine equation should have a rational solution whenever it has solutions in reals and p-adics for all primes p. For diagonal forms, this principle can be analyzed via the Hardy–Littlewood circle method. In this talk, we examine how the major and minor arc contributions are handled to establish asymptotic formulas for the number of integral solutions. Moreover, we would present a sketch of Jorg Brudern and Trevor D. Wooley's proof of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables.

MC 5417

Amanda Petcu, University of Waterloo

Cohomogeneity one solitons of the hypersymplectic flow

Given a hypersymplectic manifold X^4, one can give a flow of hypersymplectic structures. In this talk, we let X^4 be R^4 with an SO(4) action, and the hypersymplectic triple depend on three different functions h_k that depend solely on the radial coordinate. We will examine how the triple evolves under the hypersymplectic flow and given this initial structure, determine all possible solitons of the flow.

MC 5403

Anton Iliashenko, Beijing Institute of Mathematical Sciences and Applications

Hyperbolicity and Schwarz Lemmas in Calibrated Geometry

We will define two new notions of hyperbolicity for a general Riemannian manifold equipped with a calibration, which generalize the notions of Kobayashi and Brody hyperbolicity from complex geometry. For this we introduce a decreasing Finsler pseudo-metric that specializes to the Kobayashi-Royden pseudo-metric in the Kahler case; and derive the generalization of the classical Schwarz Lemma but for Smith immersions. We will talk about how these notions of hyperbolicity relate to one another and will see some examples. This is joint work with Kyle Broder and Jesse Madnick.

MC 5417

Alex Pawelko, University of Waterloo

Gerbes of Coassociative Submanifolds and the first Chern class

We will define (bundle) gerbes, a generalization of principal S^1-bundles, and define connections on gerbes, whose corresponding forms over a trivialization are 2-forms and whose curvatures are 3-forms. Then, we will build a gerbe from coassociative submanifolds of a G_2 manifold, and study its analogue of the first Chern class, an integral cohomology class in degree 3.

MC 5403

Xuemiao Chen, University of Waterloo

Space of lines-II

I will continue to talk about some related constructions on the space of oriented lines in the three dimensional Euclidean space.

MC 5403

Noah Slavitch, University of Waterloo

Generic Absoluteness in Set Theory

We give an overview of the study of generic absoluteness for V in set theory, including a discussion of projective absoluteness and Sealing.

MC 5403

Beatrice-Helen Vritsiou, University of Alberta

On the Hadwiger-Boltyanski illumination conjecture for convex bodies with many symmetries

Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever we want, to illuminate its entire surface. What is the minimum number of light sources that we need? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases.

The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). In this talk I will briefly discuss some of its history, and then focus on recent progress towards verifying the conjecture for all 1-symmetric convex bodies and certain cases of 1-unconditional bodies.

MC 5501

Spiro Karigiannis, University of Waterloo

Organizational Meeting

The Differential Geometry Working Seminar is an opportunity for all participants (students, postdocs, and faculty) to learn new things, to teach each other new things, and to get more practice in giving talks. It is very informal and confusion is common/encouraged. That's how we learn.

As usual, we'll start the term by attempting to decide on as much of the schedule as we can for the coming term. We'll have one speaker per week (most weeks) on Thursdays at 2:30pm.

MC 5403