Alex Pawelko

Mathematics student and former video game programmer interested in differential geometry (and lots of other things) living in Waterloo, Canada.

About Me

Hi, I'm Alex (he/him), and welcome to my page! Here you'll find my rather inactive blog and information about me and my work. At a high level, I'm a Pure Mathematics major and Computer Science minor at the University of Waterloo, and a former professional programmer (in video games). My main mathematical interests are in differential geometry and geometric analysis, which you can read more about below. When I'm not doing math, I'm probably playing TTRPGs and video games, mostly cRPGs and singleplayer shooters. Occasionally, I write about video games, with my focus typically on the art of narrative in games. I also speedrun games, particularly Command & Conquer: Renegade, where I am a former world record holder. I have also taken many volunteer roles advocating for the undergraduate students of the University of Waterloo, both inside and outside of student government. I'm particularly passionate about student disability rights, and ensuring that the University systematically takes student voices and stances seriously. If you are a Waterloo student and would like to talk (in particular, if you believe I can help you with something), please send me an email. My website URL comes from my old nickname Notoh (pronounced No-toe, with less emphasis on the toe). My last name is spelled either Pawełko or Pawelko, pronounced "pa-vewl-ko" or the anglicized "pa-well-ko" (I am fine with both). If you'd like to reach out to me, my email is firstname<dot>lastname<at>uwaterloo<dot>ca (with a normal l in the address).

Research

My main mathematical interests are in the mathematical subject of differential geometry. Below you can find brief explanations of my interests aimed at both non-mathematical audiences and mathematical audiences. If you are interested in any of this and would like to talk, please send me an email! For everyone: In one sentence, differential geometry is the use of (multivariable) calculus to study shapes. In order to use calculus effectively, the shapes under consideration need to be "smooth", without sharp edges, like the surface of a doughnut or a sphere. Differential geometry has many applications in physics (though I'm nowhere near a physicist), most notably in general relativity where it is used to describe the shape of spacetime. My particular interests lie in using differential geometry to study what are called "special geometric structures". A geometric structure is a way of measuring certain geometric information about a shape. For example, you might be able to measure distances and angles on a shape (this is a geometric structure called a metric), or you might be able to tell which way is "outward" on a shape (this is a geometric structure called an orientation). A given geometric structure may or may not exist on a given shape, for example, orientations don't exist on a shape like a Möbius strip, but it turns out that metrics exist on every shape. As the name suggests, special geometric structures are those that are "special" in some way, often by being rare or having particularly neat properties. For example, given a metric and the ability to measure lengths and angles, one can measure how a shape bends and curves in space, and in an amazing turn of events, there are some exceptional seven and eight-dimensional shapes that curve like no others. Most of my research is about understanding these special shapes and the geometric structures they carry, and in particular trying to find new examples of them and figure out exactly what special properties they possess. For mathematicians: I am interested in the study of special geometric structures within differential geometry and geometric analysis. I am fascinated with the entire field on the whole, though most of my work thus far has focused on manifolds with exceptional holonomy (G2 and Spin(7) manifolds). My current research is focused on trying to apply certain ideas from symplectic geometry (in particular, geometric quantization) to the setting of special holonomy, and lately has crossed paths with some very interesting gauge theory and geometric representation theory. You can also read my research statement for more details. Outside of geometry, I have also done some research in programming language theory in computer science, where my focus was applying mathematical ideas (measure-theoretic probability theory, categorical logic) and computer formalization to probabilistic programming languages. Below, you can find links to various talk and class notes of my mine (click the names as links), though they certainly contain numerous errors, idiosyncrasies, and missing citations. Any errors are, of course, my own. Please send me an email if you have any corrections or confusions.

Talks

Calibrated Geometry of a Strongly Nondegenerate Knot Space

Notes for a board talk given September 18th, 2025 to Waterloo's Differential Geometry Working Seminar, based on my current research, J.-L. Brylinski's book "Loop Spaces, Characteristic Classes and Geometric Quantization", and Lee-Leung's paper "Higher dimensional knot spaces for manifolds with vector cross products". Despite the title, there is almost no calibrated geometry in this talk.

Gerbes of Coassociative Submanifolds and the First Chern Class

Notes for a board talk given August 14th, 2025 to Waterloo's Differential Geometry Working Seminar, based on G. Oliveira's paper "Gerbes on G2-manifolds", N. Hitchin's notes "Lectures on Special Lagrangian Submanifolds", and a small amount of my own work.

How Special is Curvature?

Slides for an expository talk on holonomy given June 26th, 2025 to the Canadian Undergraduate Mathematics Conference 2025.

The Formal Kaehler Structure of the G2 Knot Space

Notes for a board talk given May 22nd, 2025 to Waterloo's Differential Geometry Working Seminar, based on J.-L. Brylinski's book "Loop Spaces, Characteristic Classes and Geometric Quantization" and a small amount of my own work.

Differential Geometers Suck at Naming Things

Slides for a humorous talk on differential geometry (and its notation) given March 7th, 2025 to Waterloo's Pure Math Club as part of their 24-hour Short Attention Span Math Seminar.

Prequantum Line Bundles and Geometric Quantization

Notes for a board talk given November 20th, 2024 to Waterloo's Differential Geometry Working Seminar, primarily based on the exposition in Casey Blacker's symplectic geometry notes.

2-Outta-3 Ain't Bad

Notes for an expository board talk on U(n) and the linear algebra of Kaehler geometry aimed at undergrads given November 7th, 2025 to Waterloo's Pure Math Club as part of their Short Attention Span Math Seminar.

Strongly Nondegenerate Forms and their Associated Structures on Higher Knot Spaces

Notes for a board talk given August 7th, 2024 to Waterloo's Differential Geometry Working Seminar, primarily based on Lee-Leung's paper "Higher dimensional knot spaces for manifolds with vector cross products".

The Cross Product in R^7 (and Other Fun Adventures with Normed Division Algebras)

Notes for an expository board talk on the quaternions and octonions given March 14th, 2024 to Waterloo's Pure Math Club as part of their Short Attention Span Math Seminar.

Notes

CHEM 356: Quantum Chemistry (co-written with Kathryn Froese)

Polished notes from a first course on quantum mechanics and chemical applications, aimed at chemistry students.

PMATH 965: Harmonic Maps

Nearly complete notes from a graduate course on harmonic maps in Riemannian geometry.

PMATH 453: Functional Analysis

Complete notes from a first course in functional analysis.

PMATH 450: Lebesgue Integration and Fourier Analysis

Complete notes from a course in Hilbert space theory and a small amount of measure theory (unfortunately, very little Lebesgue integration or Fourier analysis was covered).