Join us for a two-part workshop on communicating mathematics. In the first part, learn how to present mathematical arguments effectively and how to wield Mathematical English fluently.  We will discuss

• how to write clearly and how to structure mathematical content for different audiences;

• notation, conventions, and patterns in mathematical statements;

• development of good writing habits;

• common errors and how to fix them.

Following this presentation, we will continue with an introduction to LaTeX, a free document typesetting language used widely throughout mathematics and natural sciences for various purposes, including

• problem sets and assignments;

• posters and presentations;

• resumes and CVs;

• research papers, theses, and books.

You can register for the event here.  

Poster 

Introduction to Mathematical Writing:  slides

Introduction to LaTeX guide: LaTeX guide 

LaTeX symbol cheat sheet: Cheat sheet 

Classical Algebraic Geometry 

Algebraic geometry has gained a reputation of being a highly abstract and technical subject with a fairly challenging entry point. But its foundational ideas are surprisingly accessible. In this talk, we will give a gentle introduction to the basic ideas and concepts in algebraic geometry, primarily focusing on examples and

computations. We will also gently hint towards high level ideas that extend to its modern formalization. Only basic knowledge of rings and polynomials are required. 

Generalized Rank in Generalized Linear Algebra  

In linear algebra, one often studies a single matrix at a time. However, systems of matrices naturally arise in mathematics, for instance, as quiver representations in the study of modules, or as persistence modules in the context of topological data analysis. One is therefore led to ask what the natural notion of rank is for such a system of matrices. We will answer this question and discuss a surprising method for computing it. Time permitting, we will see what the meaning of this rank is for someone working in topological data analysis. In the spirit of the Grad Seminar, only basic linear algebra will be assumed. 

A brief history of algebraic geometry  

Algebraic geometry, like most things, cannot be fully understood without some understanding of its history. Starting with the very first notions of a conic section all the way up to the modern categoric perspective pioneered by Grothendieck and Serre, we will attempt to discuss the major developments that influenced the field. Very little background in algebra or geometry is needed to follow this talk. 

Niven's Theorem: Polygons and Quadratic Rings 

During the school year, the University of Waterloo publishes a Problem of the Month (POTM). These problems are aimed at grade 11 and 12 students across Ontario to be tackled throughout the month, and allow them to experience post-secondary level mathematics in some instances.

In this talk, we will discuss one such POTM and its solution. We will then see how this solution leads us to ask about rational points on special curves. This leads to Niven’s Theorem, a complete classification of these rational points. I will prove Niven’s Theorem, and discuss two of its corollaries: irrationality of π and how the POTM becomes disproving a single case.

Time permitting, I will discuss Niven’s theorem and its connection to algebraic number theory via quadratic rings.

One Conjecture to rule them all

Understanding prime numbers (and their generalizations) has been a driving force behind the development of many ideas across mathematics. During the 18th and 19th centuries, many mathematicians, including Gauss, Dirichlet, Legendre, and Chebyshev, attempted to formulate precise estimates for the number of prime numbers less than a given bound. This work culminated in one of the great mathematical achievements of the 19th century, now known as the prime number theorem. 

Near the turn of the 20th century, mathematicians such as Landau, Hardy, Littlewood, and Schinzel began asking questions regarding primes represented by polynomials. While nearly all of these conjectures remain open today, they have since been superseded by the Bateman–Horn conjecture. In this talk, I will discuss the history and development of the Bateman-Horn conjecture and comment on some of the known results related to it.