The Colloquium
The BU 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 bi-weekly on Tuesdays at 6:30. There will be a 15-minute tea (and snacks) before the talks begin. We ask that talks will be in the range of 30-60 minutes, but this is by no means strict (of course don't go too far over/under). 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.
Upcoming Talks
Recent Talks
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.
A follow up talk may consist of an attempt to generalize this theory to the case of plane cubics; polynomials of degree three in two variables. Through this endeavor we will introduce results from classical algebraic geometry; these may (if time permits) include Bézout’s theorem, intersection theory, and blowups; of course, for these developments, we must introduce projective geometry and some more advanced language. Finally, our third act will discuss in vague terms the position of stacks and moduli spaces in the picture, together with the categorical and algebraic foundation they necessitate.
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.