The Colloquium
The Society of Mathematics Colloquium is a venue for students studying (and interested in) mathematics to give talks on topics of their choice to a general math audience. We will have colloquia weekly on Mondays at 6:30. We ask that talks be in the range of 60 minutes. Speakers will be asked to provide notes for their talk typeset in LaTeX to be included in a proceedings-style publication at the end of the semester.
We ask that topics of talks be accessible to, say, a student who has taken the calculus sequence and seen some linear algebra. However, this is not a strict requirement and will consider colloquium talk proposals on a case-by-case basis. If you're interested in presenting, please contact Zach in our discord or email him at zzobair@bu.edu.
Colloquium Proceedings (2025)
Edited by Zach Zobair
The 2025 proceedings for the colloquium are complete! Thanks to Zach Zobair for putting them together, and all our speakers for compiling such high-quality notes.Upcoming Talks
Sheaf Cohomology
Skyler Marks
Sheaves are universally present in modern geometry, from the most abstract and esoteric arithmetic and algebraic geometry to the most concrete differential geometry. They allow us to capture, in a very flexible way, the notion of various types of "functions on a space". For example, we can capture the idea of "differentials" or the "tangent space" to a shape using sheaves. In this talk we will explain the cohomology theory related to sheaves, and give some examples of computations therewith. We'll also see some of the cohomology theories we've been introduced to in the past appear as sheaf cohomology!Past Talks (2025 - 2026 Academic Year)
Abelian Categories
Grant Talbert
The theory of abelian categories arose as an axiomatization of homological algebra. In this talk, we will define abelian categories, and then construct homology and derived functors in the more general, categorical setting they provide.Homology and Cohomology in Mathematical Physics
Zoe Siegelnickel
In this talk, we will take a quick tour of some of the places that cohomology shows up in physics, and discuss why it's a useful tool in physics, as well as in math.Simplicial Cohomology
Alice Marchant
In this talk we will be covering a geometric way to view homological algebra. In some way, we will be going over what motivated the subject in the first place. We will begin by seeing that the idea of a “hole” can be captured using an algebraic invariant coming from homological algebra and then doing some computations of this invariant for some spaces.Cohomology as Measuring Obstruction
Zach Zobair
This talk will be an overview of how we can see cohomology as "measuring obstruction", with an overview of some basic category theory and abstract algebra. Group theory and group cohomology will provide the setting for most of the examples.Homological Algebra: Preliminaries
Skyler Marks
”Homological Algebra” is a collection of tools which are used throughout most of math. When the composition of two linear maps of vector spaces (or, more generally, abelian groups or modules) is zero, the image of the first map lies within the kernel of the second. Homological algebra can be thought of as the study of collections of vector spaces chained together with such maps, called ”differentials”. In particular, given such maps, we may take (for each map) the quotient of the kernel of that map by the image of the previous; this gives the homology or cohomology vector space (or group, or module). Somehow, in a diverse range of cases, chaining together vector spaces with differentials gives a way to express complicated and useful information from a particular math problem; we can associate these chain complexes in various ways to geometric, topological, number theoretical, and algebraic objects (among others), and each express different pieces of information about those objects. Taking cohomology, then, stratifies the (often intractable) data in these chain complexes into usable, numerical data. Generally, the information captured by the chain complex is contained in the specific details of the map, while taking homology yields vector spaces who’s dimension often encodes useful information. This talk is designed to be an intuitive discussion of homological algebra using vector spaces, and introducing participants to some of the topics our colloquia will discuss.Past Talks (2024 - 2025 Academic Year)
Smoothness of the Area Feature Function
Hunter Golemon
Does the area of the level set of a real valued function of two real variables vary smoothly as the function is moved around? In order to understand this question, we must first understand why we would want to ask it. This question arose from studying zebrafish pattern formation, where they took level sets of functions to generate patterns. They made a gene parameter space to vary the “genes” of the fish, and then drew curves in this parameter space to separate regions containing spots from stripes. However, to draw these curves, we needed the area of a level set to be smooth in the function space. We will learn about the derivative as a linear map, the Implicit Function Theorem, and then prove the area of a pattern varies smoothly in the infinite dimensional vector space of functions.An Introduction to the Theory of p-Adic Numbers and Local Fields
Zach Zobair
First introduced by Kurt Hensel in 1897, for a fixed prime p, the p-adic numbers are a completion of the rational numbers, similar to the real numbers, but with respect to a different sense of "distance" between two rational numbers. They are an example of a nonarchimedean local field. In this talk we will begin by constructing the p-adic numbers and exploring some of their analytic, algebraic, and topological properties, as well as remarking on how these properties manifest for general local fields (An Introduction to Braid Groups
Olivia Hu
A braid in math is not too far from how it sounds — a collection of strands interlacing and intertwining as they travel through paths in space. These braids are interesting creatures in their own right. Some braids look complex, but are just one twist from being untied; other braids can look simple, but are in truth much more obstinate. However, the study takes an unexpected turn upon discovering braids, like numbers, can be multiplied and divided to create new braids: they form an abstract structure called a group. By studying this group, we will realize that in studying braids, we are actually also studying continuous functions on a disk.CW-Complexes and Related Topics
Alice Marchant
Imagine building a tower. You have to start with some sort of foundation and then build a layer up and then another layer on top of that and so on. A CW complex is a way that we can do this same building process but for (almost) any space you can think of by just adding n-disks to a previous part. We will explore in more detail how this construction works, some of the constraints on it, and why it is such a useful construction in algebraic topology.An Invitation to Algebraic Geometry
Skyler Marks
In high school algebra 1 and 2, we study the theory of single variable polynomial equations. In linear algebra, we study systems of linear equations, or polynomial equations with no exponents greater than 1. Algebraic Geometry combines these disciplines to study polynomial equations (in particular, their solutions) in many variables. This theory is useful as it is specific enough that we can compute with it, yet general enough that it applies to many problems we care about. The promised invitation will be extended by way of plane conics and cubics. After some definitions and preliminaries, we will review a family of classical results regarding plane conics (quadratic polynomials in two variables). We will begin by classifying conics into families who’s members are “alike”. We will then leverage this classification to study the intersections of two conics. Our conclusion to this first act will be a detailed discussion of the number of conics passing through n points in the plane, introducing Moduli spaces and projective space.Root Systems and Lie Groups
Zoe Siegelnickel
A root system is a collection of vectors definied over a Euclidian vector space with certain geometric properties. Some of the isometries generated by these roots form a group called a Weyl group. These groups are used extensively to study various symmetries. We will do a brief introduction to Euclidean space, isometries, and root systems before discussing Weyl groups and Weyl chambers and how they appear in Lie Theory.