Semester 2 2025
Seminar meeting: Wednesday 2:30-4:00pm in HN 2.48 (MSI board room)
The "What is...?" seminar is for honours+ students to explain a concept or a tool that they might take for granted in their area of study in pure mathematics that others may not be well acquainted with.
In the same spirit as the Informal Friday Seminar at USyd, the rules for the seminar are the following:
Any question/additional explanation from the audience is allowed.
Ego out the window.
Examples, examples, examples.
As such, talks will likely be disconnected and dependent on the interests of the speaker. Each talk should be around 30-45 minutes, with remaining time for exercises and discussion.
This seminar will be for STUDENTS ONLY, targeted towards honours+, but undergraduates are welcome to join and encouraged to give a talk about something they find interesting! Unless otherwise stated, prerequisites will assume a strong second-year background.
Light refreshments will be provided! (maybe...)
NOTE: Due to a large proportion of the audience being in Honours, talks may continue in the second half of semester, depending on interest
Please email any of the organisers at firstname.lastname@anu.edu.au if you would like to join the mailing list or give a talk.
Organisers: Yangda Bei, Tobin Carlton Van Buizen, Brice Gardner
Abstract: Why did we study group theory in Algebra 1? It turns out that many objects admit a group action, and studying these actions can give us more information about the objects. We will explore G-sets, G-spaces and other G-“stuff” through some definitions and lots of examples. Strong understanding of Algebra 1 and Analysis 1 content is required.
Abstract: Products, preimages, pushforwards, pullbacks... in all its obtuse terminology, it's sometimes hard to tell if a category theorist is doing mathematics or the hokey pokey. But this is not for nothing: the goal of category theory is to reason in the most abstract way, the upshot being the ability to combine and recognise different constructs as instances of the same concept.
In this introductory talk, we will begin by defining the category basics to the beginner, before formulating numerous common constructions in mathematics (e.g. cartesian products, disjoint unions) as limits and colimits of diagrams. We will rely heavily on examples and exercises, so please stick around afterwards to work through and verify a range of things you know to be true, but probably never knew why! (I'm looking at you, direct sum vs. infinite product)
Prerequisites: We will assume Algebra 1 and Analysis 1 (i.e. topological spaces), but no more. Students who have taken higher courses such as Differential Geometry, Algebraic Topology, Algebraic Geometry or Foundations of Mathematics will have access to/appreciate more of the exercises.
(Lecture notes): Please email me if you notice any errors or typos in the notes!
(Exercises): Solutions may be published in future upon request.
Abstract: Reflections aren’t just for mirrors - they’re the building blocks of Coxeter systems. In this lightning introduction, we’ll explore the world of Coxeter theory: What is a braid group? What is the Hecke algebra? And why does cos(π/mst) keep showing up? If time allows, we’ll touch on the Kazhdan-Lusztig conjectures. A strong understanding of Algebra 1 is required, and ideally some familiarity with representation theory.
Abstract:
If you take algebraic topology, you will run into lens spaces as an example of spaces with easy to compute homology groups. But in the world of 3-manifolds, they're far more than just pleasant exercises: lens spaces are the first real family of 3-manifolds beyond S^3 and S^1 x S^2, and are a perfect excuse to tour some big ideas in low-dimensional topology.
We'll meet lens spaces from several angles, starting with their descriptions as certain quotients of the 3-ball (familiar to those who have taken algebraic topology). From there, we'll use them to introduce two big ideas in 3-manifold theory: Heegaard splittings and surgery on knots. This will be a chance to see topology from a perspective rarely taught at ANU: the geometric side, where spaces are built, cut apart, and stitched back together, reminding us that topology is more than just homology and homotopy groups!
Prerequisites: Comfort with manifolds and standard topology notions (homeomorphisms, quotient spaces).
Abstract:
In the contemporary academic landscape, mathematicians are endlessly plagued by questions about the applications of their work. Representation theory has the misfortune of facing the same questions from disciples of even its sibling disciplines in algebra. So what is the point of representation theory?
It turns out that there isn't a single application of rep theory. If anything, it's a philosophy that mathematicians use to better understand different objects through corresponding objects from linear algebra (representations, if you will). In the Algebra 1 ASE, this approach is employed with finite groups, but the original objects need not be algebraic in nature.
A quiver is a directed graph, and it's representation theory is well-studied, delving into landscapes as diverse and intriguing as homological algebra, the representation theory of Artin algebras, and even triangulated categories. We will explore the very beginnings of this story, with a view towards these strong, and sometimes unexpected, connections.
Prerequisites: Confidence in the material of Algebra 1 and 1115/1116 linear algebra is required. Experience in category theory and homological algebra is a plus.