# Welcome to Pure Mathematics

We are home to 30 faculty, four staff, approximately 60 graduate students, several research visitors, and numerous undergraduate students. We offer exciting and challenging programs leading to BMath, MMath and PhD degrees. We nurture a very active research environment and are intensely devoted to both ground-breaking research and excellent teaching.

## News

## Two Pure Math professors win Outstanding Performance Awards

The awards are given each year to faculty members across the University of Waterloo who demonstrate excellence in teaching and research.

## Pure Math PhD student wins Amit and Meena Chakma Award for Exceptional Teaching

The award ($1000), which is given to up to four recipients annually, recognizes excellence in teaching by students, including intellectual vigour, skill in communication and presentation of subject matter, and concern for the needs of students.

## Spring 2023 Graduands

Congratulations to Clement Wan, MMath and Eric Boulter, PhD, who convocated in Spring 2023. Best of luck in your future endeavours!

## Events

## Geometry and Topology Seminar

**Nikolay Bogachev, University of Toronto**

Commensurability classes and quasi-arithmeticity of hyperbolic reflection groups

In the first part of the talk I will give an intro to the theory of hyperbolic reflection groups initiated by Vinberg in 1967. Namely, we will discuss the old remarkable and fundamental theorems and open problems from that time. The second part will be devoted to recent results regarding commensurability classes of finite-covolume reflection groups in the hyperbolic space H^n. We will also discuss the notion of quasi-arithmeticity (introduced by Vinberg in 1967) of hyperbolic lattices, which has recently become a subject of active research. The talk is partially based on a joint paper with S. Douba and J. Raimbault.

MC 5417

## Pure Math Department Colloquium

**Matthew Harrison-Trainor, University of Illinois at Chicago**

Back-and-forth games to characterize countable structures

Given two countable structures A and B of the same type, such as graphs, linear orders, or groups, two players Spoiler and Copier can play a back-and-forth games as follows. Spoiler begins by playing a tuple from A, to which Copier responds by playing a tuple of the same size from B. Spoiler then plays a tuple from B (adding it to the tuple from B already played by Copier), and Copier responds by playing a tuple from B (adding it to the tuple already played by Spoiler). They continue in this way, alternating between the two structures. Copier loses if at any point the tuples from A and B look different, e.g., if A and B are linear orders then the two tuples must be ordered in the same way. If Copier can keep copying forever, they win. A and B are isomorphic if and only if Copier has a winning strategy for this game. Even if Copier does not have a winning strategy, they may be able to avoid losing for some (ordinal) amount of time. This gives a measure of similarity between A and B. A classical theorem of Scott says that for every structure A, there is an α such that if B is any countable structure, A is isomorphic to B if and only if Copier can avoid losing for α steps of the back-and-forth game, that is, when A is involved we only need to play the back-and-forth game for α many steps rather than the full infinite game. This gives a measure of complexity for A, called the Scott rank. I will introduce these ideas and talk about some recent results.

MC 5501

## Number Theory Seminar

**David McKinnon, University of Waterloo**

How crowded can rational solutions be?

Say you've got the equation x^2-2y^2=z^4-1. Lots of rational solutions there, like (1,1,0). How are those solutions distributed in 3-space? In particular, how close can they get to (1,1,0)? This abstract has the questions, but the talk has the answers. Well, some of 'em.

MC 5479