Events

Zhihao Zhang, University of Waterloo

Translation-Invariant Function Algebras of Compact Groups

Let G be a compact group and let Trig(G) denote the algebra of trigonometric polynomials of G. For a translation-invariant subalgebra A of Trig(G), one can consider the completions of A under the uniform norm and the Fourier norm. We show in Chapter 2 using techniques developed by Gichev that both completions have the same Gelfand spectrum, answering a question posed in a paper of Spronk and Stokke. In the same paper, a theorem describing of the Gelfand spectrum of the Fourier completion of finitely-generated such algebras A was given. In Chapter 3, we extend this theorem to the case of countably-generated, translation-invariant subalgebras, A. In Chapter 4, we give a brief overview of the Beurling--Fourier algebra, a weighted variant of the classical Fourier algebra studied by Ludwig, Spronk and Turowska. The addition of a weight for these particular algebras invites new spectral data in contrast to its classical counterpart. In Chapter 5, we show for Beurling--Fourier algebras of compact abelian groups, G, that its weight can be used to construct a seminorm on a real vector space generated by the dual of G that remembers the spectral data of the algebra.

MC 2009

Joaquin G. Prandi, University of Waterloo

Iterated Function Systems and the Local Dimension of Measures

Given an iterated function system S in R^d, with full support and such that the rotation in it fixed the hypercube [-1/2,1/2]^d , then S satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. With this in mind, we extend the notion of net intervals from R to R^d. We also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support.

We study the local dimension of the convolution of two measures. We give conditions for bounding the local dimension of the convolution on the basis of the local dimension of one of them. Moreover, we give a formula for the local dimension of some special points in the support of the convolution.

We study the local dimension of the addition of two measures. We give an exact formula for the lower local dimension of the addition based on the local dimension of the two added measures. We give an upper bound to the upper local dimension of the addition of two measures. We explore the special case where the two measures satisfy the convex additive finite-type condition that we introduce.

We introduce the notion of graph iterated function system. We show that we can always associate a self similar to the graph iterated function system.

MC 5417

Spencer Kelly, University of Waterloo

Proper Group Actions and the Slice Theorem in Finite Dimensions

In this talk we will begin by reviewing important properties of group actions on manifolds, and characteristics of proper actions. We then define isotropy and orbit types, discuss the slice theorem (on finite dimensional manifolds), and go over non-trivial examples of slice bundles. This will set us up to conclude with the principal orbit theorem and the stratification of the orbit space.

MC 5403

Siyuan Lu, McMaster University

Interior C^2 estimate for Hessian quotient equation

In this talk, we will first review the history of interior C^2 estimates for fully nonlinear equations. As a matter of fact, very few equations admit this property, not even the Monge-Ampère equation in dimension three or above. We will then present our recent work on interior C^2 estimate for Hessian quotient equation. We will discuss the main idea behind the proof. If time permits, we will also discuss the Pogorelov-type interior C^2 estimate for Hessian quotient equation and its applications.

MC 5417

Jashan Bal University of Waterloo,

Strong convergence of random permutations

We will start proving that i.i.d random permutations strongly converge to Haar unitaries.

MC 5479

Viktor Majewski, University of Waterloo

Dirac Operators on Orbifold Resolutions

In this talk we discuss Dirac operators along degenerating families of Riemannian manifolds that converge, in the Gromov-Hausdorff sense, to a Riemannian orbifold. Such degenerations arise naturally when analysing the boundary of Teichmüller spaces of special Riemannian metrics as well as moduli spaces appearing in gauge theory and calibrated geometry. Here sequences of smooth geometric structures on Riemannian manifolds may converge to an orbifold limit. To understand and control these degenerations, we introduce smooth Gromov-Hausdorff resolutions of orbifolds, that are, smooth families (X_t,g_t), which collapse to the orbifold (X_0,g_0) as t goes to 0.

The central analytic problem addressed in this paper is to understand the behaviour of Dirac operators along such resolutions, in particular in collapsing regimes where classical elliptic estimates fail. We develop a uniform Fredholm theory for the family of Dirac operators on the Gromov-Hausdorff resolution. Using weighted function spaces, adiabatic analysis, and a decomposition of X_t into asymptotically conical fibred (ACF), conically fibred (CF) and conically fibred singular (CFS), we obtain uniform realisations of the model operators and prove a linear gluing exact sequence relating global and local (co)kernels. As a consequence, we construct uniformly bounded right inverses for D_t, and derive an index additivity formula.

The theory developed here provides the analytic foundation for nonlinear gluing problems in gauge theory and special holonomy geometry, including torsion-free G-structures, instantons, and calibrated submanifolds of Riemannian manifolds close to an orbifold limit.

MC 5403

Zhihao Zhang, University of Waterloo

Spectra of Beurling Algebras of Discrete Abelian Groups

We will discuss a variant of the group algebra, called the Beurling algebra. These algebras differ from their classical counterpart through the addition of a weight function modifying the norm. The Gelfand spectrum of the group algebra of absolutely integrable functions on an abelian group, G, is well known to be the Pontryagin dual of G. In the case of a Beurling algebra, the Gelfand spectrum can be much larger for suitable weights. We will focus on the Beurling algebra of a discrete abelian group, G, and give a description of its Gelfand spectrum in terms of a seminorm constructed from a symmetric weight.

MC 5417 or Join on Zoom

Viktor Majewski, University of Waterloo

Filling Holes in the Spin(7)-Teichmüller Space and String Cohomology

In this talk, I apply the analytic results from the first talk to study the boundary of the Spin(7) Teichmüller space. Using compactness results for Ricci-flat metrics together with known examples of Spin(7) manifolds, it is known that Spin(7) orbifolds with SU(N) isotropy arise as boundary points of the moduli space.

Building on the resolution scheme for Spin(7) orbifolds that I discussed in 2024, and which I will briefly review, we show how this boundary can be removed by requiring Spin(7) orbifolds to encode information about their resolutions. In this way, the Teichmüller space is enlarged to include orbifold limits together with their compatible resolutions, thereby filling in the boundary.

Finally, we explain how this perspective is related to a Spin(7) analogue of the crepant resolution conjecture from string cohomology, providing a geometric interpretation of the obstruction complex discussed in the linear gluing analysis in the first talk.

MC 5403