Events

Spiro Karigiannis, University of Waterloo

Schwarz Lemma for Smith maps

I will discuss a generalized Schwarz Lemma for Smith maps, proved recently by Broder-Iliashenko-Madnick, and explain how it generalizes the classical Schwarz Lemma from complex analysis.

MC 5403

Noé de Rancourt , Université de Lille

Big Ramsey degrees of metric structures

Distortion problems, from Banach space geometry, ask about the possibility of distorting the norm of a Banach space in a significant way on all of its subspaces. Big Ramsey degree problems, from combinatorics, are about proving weak analogues of the infinite Ramsey theorem in structures such as hypergraphs, partially ordered sets, etc. Those two topics, coming back to the seventies, have quite disjoint motivations but share a surprisingly similar flavour. In a ongoing work with Tristan Bice, Jan Hubička and Matěj Konečný, as a step towards the unification of those two topics, we developped an analogue of big Ramsey degrees adapted to the study of metric structures (metric spaces, Banach spaces...). Our theory allows us to associate to certain metric structures a sequence of compact metric spaces quantifying their default of Ramseyness. In this talk, I'll present our theory and its motivations and illustrate it on the examples of the Banach space $\ell_\infty$ and the Urysohn sphere. If time permits, links with topological dynamics will also be discussed.

QNC 1507 or Join on Zoom

Jon Cheah, University of Hong Kong

An advertisement of cluster algebras

Cluster algebras have had many surprising links with many areas of mathematics beyond their original purpose in studying total positivity. In this expository talk, we consider two discrete dynamical systems, namely Markov triples and Coxeter--Conway friezes. While the study of these examples predated that of cluster theory, we will see how the latter provides a conceptual explanation for the intergrality and positivity phenomena.

MC 5479

(snacks at 16:00)

Alex Pawelko, University of Waterloo

Riemannian Geometry of Knot Spaces

We will review the construction of knot spaces of manifolds, specifically over G2 and Spin(7) manifolds. We will then see an explicit construction of the Levi-Civita connection of the knot space, and see what this can tell us about the torsion of the induced special geometric structures on knot spaces of G2 and Spin(7) manifolds.

MC 5403

Joaco Prandi, University of Waterloo

When the weak separation condition implies the generalize finite type in R^d

Let S be an iterated function system with full support. Under some restrictions on the allowable rotations, we will show that S satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. To do this, we will extend the notion of net intervals from R to R^d. If time allows, we will also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support.

QNC 1507 or Join on Zoom

Sergey Grigorian, University of Texas Rio Grande Valley

Geometric structures determined by the 7-sphere

The 7-sphere is remarkable not only for its rich topological and algebraic properties but also for the special geometric structures it encodes. In this talk, we explore how the symmetries and stabilizer subgroups of Spin(7) acting on the 7-sphere, regarded as the set of unit octonions, give rise to G2​-structures on 7-manifolds, SU(3)-structures on 6-manifolds, and SU(2)-structures on 5-manifolds. We will trace how these structures arise naturally via the inclusions of Lie groups and are reflected in the geometry of sphere fibrations. This perspective highlights the role of the 7-sphere as a unifying object in special geometry in dimensions from 5 to 8.

MC 5417

Jacques Gideon van Wyk, McMaster University

A Lucky Game of Yahtzee Over Christmas

Yahtzee is a game where you win points by rolling five dice into particular patterns, sort of like hands in Poker. The roll which wins you the most points, and which is the hardest to get, is the Yahtzee, the game's namesake, where all five die land on the same digit.

Over the Christmas holidays, I played Yahtzee with some of my family, and we had a game where we rolled five Yahtzees among five teams in one game. In our experience, this felt like quite the lucky game, as many games of Yahtzee end with no one rolling any Yahtzees at all.

In this talk, we're going to get to the bottom of how lucky we actually were: I'll explain how Yahtzee is played, we'll discuss some strategies you can employ to increase your chances at winning, and, assuming an optimal strategy, we'll figure out how likely it is to get at least five Yahtzees in one game.

MC 5479

(Refreshments will start at 16:30)

Spiro Karigiannis, University of Waterloo

Organizational Meeting

MC 5403

Pavel Zatitskii, University of Cincinnati

Extremal problems and monotone rearrangement on averaging classes

We will discuss integral extremal problems on the so-called averaging classes of functions, meaning classes defined in terms of averages of their elements, such as BMO, VMO, and Muckenhoupt weights. A typical extremal problem we consider involves an integral inequality, such as the John--Nirenberg inequality for BMO. One common way to formulate such questions is using Bellman functions. It turns out that such Bellman functions are solutions to specific boundary value problems, formulated in terms of convex geometry. We will also discuss the monotone rearrangement operator acting on the averaging classes, which arises naturally in this context and is useful when solving extremal problems.

MC 5417

Michael Gregory, University of Waterloo

Computability Relative to Random Sets

This presentation explores the interaction between algorithmic randomness and Turing degrees. We focus on 1-random sets and how randomness interacts with computable reducibility. Several fundamental results are discussed that illuminate the placement of random sets within the Turing degrees and the constraints that randomness imposes on computable reductions. In particular, the Kucera-Gacs Theorem is presented, which establishes that every set is weak truth-table reducible to a 1-random set.

MC 5403