Events

Soham Chakraborty, École Normale Supérieure

Measured groupoids and the Choquet-Deny property

A countable discrete group is called Choquet-Deny if for every non-degenerate probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Recently a complete characterization of Choquet-Deny groups was obtained by Frisch, Hartman, Tamuz and Ferdowsi. In this talk, we will look at the extension of the Choquet-Deny property to the framework of discrete measured groupoids. Our main result gives a complete characterisation of this property in terms of the associated measured equivalence relation and the isotropy groups of the groupoid. This talk is based on a joint work with Tey Berendschot, Milan Donvil, Mario Klisse and Se-Jin Kim.

MC 5417 or Join on Zoom

Xuemiao Chen, University of Waterloo

The Sphere

I will make a long story regarding the two dimensional sphere.

MC 5403

Joaco Prandi, University of Waterloo

Bounding the Local Dimension of the Convolution of Measures

Let mu be a finite measure on a metric space X. Then the local dimension of the measure mu at the point x in the support of mu is given by

dim_{loc}mu(x)=lim_r ln(B(x,r))}\ln(r)

In a sense, dim_{loc}mu(x) represents how much mass there is around the point x. The bigger the local dimension, the less mass there is. In this talk, we will explore how the local dimension of the convolution of two measures mu and nu can be bounded by the local dimension of one of the measures. This is based on joint work with Kevin Hare.

MC5417

Laura DeMarco, Harvard University

The (algebraic) geometry of the Mandelbrot set

One of the most famous -- and still not fully understood -- objects in mathematics is the Mandelbrot set. By definition, it is the set of complex numbers c for which the recursive sequence defined by x_1 = c and x_{n+1} = (x_n)^2+c is bounded. This set turns out to be rich and complicated and related to many different areas of mathematics. I will present an overview of what's known and what's not known about the Mandelbrot set, and I'll describe recent work that (perhaps surprisingly) employs tools from number theory and algebraic geometry.

MC 5501

Kaleb Domenico Ruscitti, University of Waterloo

Organization Meeting

We will be scheduling talks for the term, please join us!

MC 5403

Enric Solé-Farré, University College London

The Hitchin and Einstein indices of cohomogeneity one nearly Kahler manifolds

Nearly Kähler manifolds are Riemannian 6-manifolds admitting real Killing spinors. They are the cross-sections of Riemannian cones with holonomy G2. Like the Einstein equation, the nearly Kähler condition has a variational interpretation in terms of volume functionals, first introduced by Hitchin in 2001.

The existence problem for nearly Kähler manifolds is poorly understood, and the only currently known inhomogeneous examples were found in 2017 by Foscolo and Haskins using cohomogeneity one methods. For one of their examples, we establish non-trivial bounds on the coindex of the Hitchin-type and Einstein functionals. We do this by analysing the eigenvalue problem for the Laplacian on coclosed primitive (1,1)-forms under a cohomogeneity-one symmetry assumption.

MC 5417

Zhenchao Ge, University of Waterloo

An additive property for product sets in finite fields.

Lagrange's Four Square Theorem states that every natural number can be written as a sum of four squares, i.e. squares form an additive basis of order 4. Cauchy observed that in a finite field F with q elements, squares form an additive basis of order 2. Bourgain further generalized the problem and proved that for any subset A in F, writing AA={aa': a,a'∈ A}, we have 3AA=F whenever |A|>q^{3/4}. 

In general, for subsets A,B in F with |A||B|>q, one might ask that how many copies of AB are enough to cover the entire space? The current record of this problem is due to Glibichuk and Rudnev. Using basic Fourier analysis tools, they achieved 10AB=F unconditionally and 8AB=F assuming symmetry (or anti-symmetry).

In this talk, we will (hopefully) go through the paper of Glibichuk and Rudnev.

MC 5417

Justin Fus, University of Waterloo

The KKS Form and Symplectic Geometry of Coadjoint Orbits

A compact Lie group acts on its Lie algebra dual via the coadjoint representation. In this talk, we will explore how the coadjoint orbits of this representation carry a natural symplectic structure called the Kirillov-Kostant-Souriau (KKS) form. The KKS form is preserved by the action. If time permits, we will show that there is a moment map for the action that coincides with the inclusion map of the orbit. A worked example for SU(2) will be performed.

MC 5403

Facundo Camano, University of Waterloo

Convergence Results for Taub-NUT and Eguchi-Hanson spaces

We define multi-Taub-NUT and multi-Eguchi-Hanson spaces and look at Gromov-Hausdorff convergences involving these spaces.

MC 5403

Jiahui Huang, University of Waterloo

Motivic integration for schemes, DM stacks, and Artin stacks.

We give an overview of motivic integration and its generalization to stacks. Early motivations for motivic integration involve singularity theory and the monodromy conjecture. We will explain how the change of variable formula works, and how it generalizes to the stack case. Motivic integration for stacks will use twisted or warped arcs, and we shall give a summary of the construction of the twisted arc space for DM stacks.

MC 5403