# 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

## Pure Math Department Colloquium

**Brent Nelson, Michigan State University**

Uniqueness of almost periodic states on hyperfinite factors

Murray and von Neumann initiated the study of "rings of operators" in the 1930's. These rings, now known as von Neumann algebras, are unital *-algebras of operators acting on a Hilbert space that are closed under the topology of pointwise convergence. Elementary examples include square complex matrices and essentially bounded measurable functions, but the smallest honest examples come from infinite tensor products of matrix algebras. These latter examples are factors—they have trivial center—and are hyperfinite—they contain a dense union of finite dimensional subalgebras. Highly celebrated work of Alain Connes from 1976 and Uffe Haagerup from 1987 showed that these infinite tensor products are in fact the unique hyperfinite factors. Haagerup eventually provided several proofs of this uniqueness, and one from 1989 included as a corollary a uniqueness result for so-called periodic states. This result only holds for some infinite tensor products of matrix algebras and is known to fail for certain other examples, but in recent joint work with Mike Hartglass we show that it can be extended to the remaining examples when periodicity is generalized to almost periodicity. In this talk, I will discuss these results beginning with an introduction to von Neumann algebras that assumes no prior knowledge of the field.

MC 5501

## Number Theory Seminar

**John Yin, Ohio State University**

A Chebotarev Density Theorem over Local Fields

I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

MC 5479

## Algebraic Geometry Working Seminar

**Kaleb D Ruscitti, University of Waterloo**

ntroducing the Log Canonical Threshold of a Singularity

Given a variety X, an ideal sheaf a, and a point p in X, the log canonical threshold of a at p is a birational invariant which generalizes the order of a. It appears in asymptotic expansions of certain intergrals, in the minimal model program for log-canonical pairs, and in many other algebraic geometry contexts. In this seminar, I will give an introduction to this invariant, following the IMPANGA Lecture Notes on Log Canonical Thresholds by Mircea Mustață.

MC 5403