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
Cy Maor, Hebrew University of Jerusalem
Stability of isometric immersions and applications
An isometric immersion f:M→N between two Riemannian manifolds of the same dimension is very rigid—the values of f(p) and Df(p) at one point p∈M completely determine f. But what can be said about maps that are "almost" isometries (in a precise sense)—must they be close to true isometries? In this talk, I will survey this question from its origins in the 1960s to recent developments, and discuss its applications to non-Euclidean elasticity, where one seeks the “most isometric” immersion even when exact isometric immersions do not exist. Based on joint works with Raz Kupferman.
MC 5417