Nils’s Notes (on math)

These are assorted essays on math, all hopefully of wide interest (I haven’t gotten to the specialized topics yet).

If you’re a novice to math, here’s how to learn math.


Vector Bundle notes for WOMP 2007 (tex); properly at WOMP 2007

legible : general

cool observations

Kodaira dimension and curvature (4 pages)
Kodaira dimension is an invariant in algebraic geometry that corresponds roughly to curvature.
The lower floors of the Postnikov tower, with torus bundles (4 pages)
The universal cover is a point-wise realization of the 1st stage of the Postnikov tower. You can also realize the 2nd stage (partially) point-wise, via a torus bundle that kills the free part of π2.
Index p normal subgroups are a projective space (2 pages)
Self-explanatory: a cool geometric structure in finite group theory.
Boy’s surface and sphere eversion (1 page)
(A form of) sphere eversion follows immediately from the existence of Boy’s surface. This was overlooked for over 50 years.


This is both practical philosophy of math and applications of math to philosophy (including to philosophy of math); it is not ontology or foundations of mathematics.

Eschatology (the study of ends)
The structure and eventual end of mathematics (8 pages)
I see math as a digraph of connected subfields, with a highly interconnected core. I expect it to someday end, mostly by petering out (as is “ω: with a whimper”, below), but the core to have a more satisfying closure (as in “Ω: with a bang”, below). This is an overview; more interesting are ω and Ω.
ω: with a whimper: messy fields (7 pages)
A common end to fields of science is to degenerate into details: there is nothing more to understand. I elaborate and illustrate, with examples including chess and antenna design.
Ω: with a bang: meta-mathematical closure (4 pages)
I imagine that core mathematics may have a particularly satisfying end, via the development (and solution) of a meta-mathematically closed theory.
Meta-meta and Monads (3 pages)
Frequently there is no field of meta-meta-something, because it is already contained within meta-something. This is a strong kind of closure, related to the categorical notion of monad.
Constraints yield structure (2 pages)
Conversely to the elegance and power of meta-closed fields, often fields are easier to analyze if they are less powerful, more constrained; notably, not self-referencing. I illustrate with constrained programming languages, including Backus’s function-level programming.
Explicit and implicit (5 pages)
A great organizing principle in math is to think of descriptions of objects as either explicit or implicit presentations.


These are largely concrete (and a bit tedious) essays on how to be a math student and how to be a math teacher (or at least, what I found worked for me).

How to learn math, and what that means (6 pages)
Learning math is not hard, but requires time, and hence commitment: it’s on the same order as learning to play the piano. If you’re interested (it is interesting!), call your local university and say “I want to learn Galois theory”, and take it from there.
How to write math (6 pages)
How to write an answer to a math problem: it’s not just “showing steps”, but rather, “write a coherent argument”.
How to format homework (6 pages)
Pedantic details, but worth following.
Misc admin (7 pages)

practical / career

Career advice for mathematicians
I am not a professional mathematician, but here are some pros to take advice from.

interesting problems

not written up – sketchy notes in src directory

I demonstrate not how to solve problems, but how to analyze and understand a solution deeply.
These are elementary problems that illustrate principles (material or problem-solving) that I find interesting. I give detailed solution, and discuss the solutions, showing what’s going on.

heat flow on cube
discrete Laplacian, cubical harmonics Special meaning for me Requires linear algebra.
random walk on cube
Surprisingly nice answer; recurrance, exponential decay.

legible : specific fields

linear algebra

Working on spectral theorems.

not written up – sketchy notes in src directory

Trace (5 pages)
Trace: easy to define, subtle to interpret.


None yet. Working on sups and infs via lattices (connection to lcm and gcd).

not written up – sketchy notes in src directory

68: Algorithmic Information Theory

Why I credit Chaitin
I frequently praise Gregory Chaitin’s work in Algorithmic information theory; here I elaborate.

Copyright: public domain

All my writings here (within this directory and its subdirectories) I release into the public domain, and do not claim any rights over. You can do whatever you want with them (at least as far as I understand US law). I’ve included the source, so you can use portions and make changes easily.

less legible

Here are the raw directories, including source and PDF output; these also include all notes that I haven’t worked through.


70 pages and counting!