What are examples of good toy models in mathematics?

Question

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A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to understand it better, but I don't know of too many examples since I don't see too many people talk about them. What examples are common in your field, or what examples do you personally think are very revealing? Here's what I've got so far, starting with Terence Tao's example. Feel free to modify any of these examples if I'm not stating them correctly and to elaborate on them in answers if you want.

• $F_p[t]$ is a toy model for $\mathbb Z$.
• $F_p[[t]]$ is a toy model for $\mathbb Z_p$.
• Simplicial complexes are a toy model for topological spaces.
• $\mathbb Z/n\mathbb Z$ is a toy model for $\mathbb Z$ (for the purposes of additive number theory).
• The DFT is a toy model for the Fourier transform on the circle.

Which properties of the original objects carry over to your toy model, and which don't? As usual, stick to one example per post.

Answer 1

2x2 matrices are a toy model for general square matrices.

"If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." -- Olga Tausky-Todd

Answer 2

One can think of toric varieties as "toy examples" of algebraic varieties. A lot can be said about them via combinatorial data, but they definitely are special (they are always rational varieties, for example).

Answer 3

Topological quantum field theories are toy models for real quantum field theories.

This is a funny example though, as TQFTs later became "real" again, when it was discovered that they describe the physics of the Fractional Quantum Hall Effect. Toy models sometimes pay off big!

Answer 4

Shift spaces are examples of topological dynamical systems, but they also serve as a toy model for topological dynamical systems.

Answer 5

The best examples I've come up with come from rational homotopy theory--commutative differential graded Q-algebras as a toy model for spaces and chain complexes of Q-vector spaces as a toy model for spectra--though really this is an instance of "toy examples" because we can build actual spaces/spectra corresponding to these algebraic data. It feels a bit like a toy model, though, I guess because those spaces aren't very geometric.

Answer 6

One can think of Grothendieck ring of varieties as a toy model for motives.

Answer 7

Your first example of F_p[t] being a toy model for Z is a facet of the well known (to number theorists) analogy between Q and F_p(t). This analogy extends fabulously to a more general connection between the arithmetic of number fields, i.e. finite extensions of Q, and of function fields, i.e. finite extensions of F_p(t).

Answer 8

Relations are a toy model for linear maps. In fact, they can be thought of as matrices over the boolean semi-ring.

Answer 9

Perhaps this doesn't count as a toy model, rather a toy example. A nice basic example for GIT is n points in CP¹ under the action of SL(2,C). A lot of the usual elements of the theory look nice in this picture. For example, the Hilbert Mumford criterion shows that a collection of n points is semi-stable iff all points have multiplicity less or equal to n/2 whilst a collection of n points is stable iff all points have multiplicity strictly less than n/2.

It's also a nice example of the equivalence between symplectic reduction and the GIT quotient. If you fix a Fubini-Study metric on CP¹ and look at the action of the corresponding SU(2) you can ask for a moment map. Thinking of CP¹ as a coadjoint orbit in su(2)^*, the moment map takes n points to their centre of mass. Now the equivalence of symplectic and GIT quotients says that, provided we don't have two points each of multiplicity n/2, you can move n points in CP¹ by an element of SL(2,C) so that their centre of mass is zero if and only if all multiplicities are strictly less than n/2. (The case when two points each have multiplicity n/2 is special because of the additional C^* stabiliser.) In one direction this is obvious, but in the other I think this is a neat non-trivial statement (at least when n is large).

Answer 10

Turing machines are in some sense toy models for real computers (which have parallel processors, networks, file systems, etc).

Answer 11

2-bridge knots (rational knots) are a toy model for knots. If you think something should be right, first test it on 2-bridge knots. This happens so much in knot theory that it's not even funny.

Answer 12

The Noncommutative torus serves as a toy model for noncommutative differentiable manifolds.

Answer 13

Derived categories of finite dimensional hereditary algebras for other derived categories. (Though under good circumstances other derived categories can turn out to be equivalent to derived categories of f.d. hereditary algebras.)

Answer 14

In Differential Geometry/Physics, Homogeneous spaces can be considered as nice examples of manifolds, where one can test conjectures.

Answer 15

This is not so much a new example of toy model but rather a complement to the excellent post by Tao on dyadic models. Like most good ideas, it has been rediscovered many times, in many different areas, and given different names. In quantum field theory and in particular in the framework of Wilson's renormalization group, many models have a Euclidean version (on $\mathbb{R}^d$) and a hierarchical version (for instance on $\mathbb{Q}_p^d$). The so-called hierarchical model in physics is another instance of what Tao calls "dyadic models". Wilson himself referred to it as "the approximate RG recursion" and it played a key role in his path to discovery of his RG theory (see quote by Wilson on page 8 of this paper). A good reference on the hierarchical model in physics is this review by Meurice. In probability theory, the dyadic model is the branching random walk or Brownian motion/Mandelbrot cascades used as a toy model for the Gaussian free field, see the references in this MO post.

Answer 16

Monounary algebras (that means, a set equipped with a single unary operation) are toy models for universal algebra.

Answer 17

Countable models of $ZFC$ (see, for example, Joel David Hamkins' blogpost "Upward closure in the toy multiverse of all countable models of set theory" on www.jdh.hamkins.org/upward-closure-in-the-toy-multiverse-of-all-countable-models-of-set-theory and his research on the set-theoretic multiverse).

Answer 18

The Theory of Metric Spaces is a toy model for the General Topology.

Answer 19

There is a powerful analogy, developed over the last ~25 years, between graphs and algebraic curves (1-dimensional algebraic varieties, over $\mathbb{C}$ say, i.e., Riemann surfaces). The first paper to propose this analogy may be "The lattice of integral flows and the lattice of integral cuts on a finite graph" by R. Bacher, P. de la Harpe, and T. Nagnibeda, 1997. A big milestone was the Riemann-Roch theorem for graphs of M. Baker and S. Norine. This analogy lets us use graphs as a toy model for curves. Again, the idea is that we have something "combinatorial" in place of something more purely algebraic/algebro-geometric (so this answer is similar to my other one about Stanley-Reisner rings, and the one about toric varieties). Note that sitting between graphs and curves, and in some sense interpolating between them, are the metric graphs.

Answer 20

Quantum graphs have become popular in the 1990s as toy models for quantum chaos, and they have been quite successful ever since, as the BGS-conjecture has been settled in the graph context.

Answer 21

Trees (resp. labelled trees) are a toy example for Young diagrams (resp. Young tableaux). Feel free to elaborate (I don't have time to).

Answer 22

Galois representations as toy models for varieties (more generally, motives). Conjecturally this isn't even a toy model, but for the moment it's a lot easier to work on the Galois side.

Answer 23

Semigroups of shifts on some function space are toy models for one-parameter $C_0$-semigroups.

On the one hand they are easy: Whereas in the general theory one often thinks of an operator $A$ as given ("the generator") and wants to study the unknown $C_0$-semigroup $(\mathrm{e}^{tA})_{t\geq 0}$, in the case of shift semigroups both objects are given by explicit formulas (the generator being the "first derivative"). On the other hand semigroups of shifts are interesting enough: By varying the underlying function space one can already produce many (counter-)examples which highlight crucial parts of the general theory.

Answer 24

Boolean algebras are toy models for distributive lattices, which in turn are toy models for lattices in general (and partially ordered sets).

Answer 25

The example that comes to mind is the "Toric Code" topological quantum field theory. It is in a strong sense the simplest non-trivial TQFT. The toric code is a testing ground both mathematically for people trying to understand quantum topology, and also physically for people who try to make these things in the lab.

In a Simon's Foundation interview on the topic, Kitaev (the inventor of the toric code) said this:

"Throughout my career I have been successful inventing toy models, some simple models that capture important features of a more complex problem" - Alexei Kitaev

Answer 26

Stanley-Reisner rings have a combinatorial structure which makes them amenable to computation and experimentation. Because of their simple structure, they can serve as toy models of general quotients of polynomial rings (or even more generally finitely generated modules over commutative rings). This is sort of similar to the way that toric varieties can be toy models of general algebraic varieties, as was mentioned in another answer to this question. Hailong Dao has a nice talk explaining this perspective: https://www.msri.org/seminars/25063.

Answer 27

Often homogeneous trees (connected acyclic graphs such that each vertex has the same valence) are considered a good toy model for real hyperbolic space.

Consider for example the behaviour of the heat semigroup https://www.ams.org/journals/tran/2000-352-09/S0002-9947-00-02460-0/S0002-9947-00-02460-0.pdf in comparison to the hyperbolic space.

Furthermore the Dirichlet space on the Poincarè disc has a lot of similarities with the discrete Dirichlet space on a tree. See for example http://www.dm.unibo.it/~arcozzi/rafrot_nyjm.pdf

Answer 28

Sequences of real numbers, and their sums and differences, as a toy model for real-valued functions, integrals and derivatives.

Answer 29

In the field of KPZ universality, the particular model of TASEP has led to a number of formulas and in particular the KPZ-fixed point formula that served as guide for finding analogous formulas for other models in the KPZ universality class.

It originates from the approach of using automata to model turbulence, interface models and biological models.

Answer 30

finite sets of points and the line segments connecting pairs of them are toy model for graphs and very popular for developing and testing heuristics for combinatorial optimization, most prominently the TSP.