webnotice

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

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

Mark Poor, Cornell University

Some results about the pseudoarc and its homeomorphism group

It is known that the so called pseudoarc can be represented as a quotient of a zero-dimensional compact "prespace" under an appropriate equivalence relation (which is an inverse limit of linear graphs), and the automorphisms of this prespace densely embeds into the homeomorphism group of the pseudoarc. Although this embedding is only continuous, not a homeomorphic embedding, we can actually characterize the topology inherited from the homeomorphism group intrinsically, only in terms of the prespace. Using this characterization we show that not all homeomorphisms are conjugate to an automorphism, and we give a second proof to Kwiatkowska's conjecture, namely that there exists a homeomorphism with a dense conjugacy class.

This is joint work with S. Solecki.

MC 5479

Amanda Wilkens, Carnegie Mellon University

Poisson–Voronoi tessellations and fixed price in higher rank

We briefly define and motivate the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discuss the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group. We give some intuition for the proof. No prior knowledge on fixed price or higher rank will be assumed.

MC 5417

Kaleb Domenico Ruscitti, University of Waterloo

Organization Meeting

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

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

Xuemiao Chen, University of Waterloo

The Sphere

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

MC 5403

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

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

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