Events

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

Joey Lakerdas-Gayle, University of Waterloo

Effective Algebra 1

We will begin learning about recursive groups following Chapter 8 of Yuri Manin's "A Course in Mathematical Logic for Mathematicians".

MC 5417

AJ Fong, University of Waterloo

The Markov equation and birational geometry

We will briefly talk about the basics on the Markov equation and its solutions, and producing Hirzebruch--Jung continued fractions from their weights. We will also describe some connections to certain degenerations of the complex projective plane. This talk is based on work of Urzúa and Zúñiga.

MC 5417