Linked Questions

66
votes
30answers
12k views

The concept of Duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come. ...
36
votes
15answers
7k views

Examples of “unsuccessful” theories with afterlives

I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and ...
26
votes
7answers
2k views

Why are we interested in permutahedra, associahedra, cyclohedra, …?

The following families of polytopes have received a lot of attention: permutahedra, associahedra, cyclohedra, ... My question is simple: Why? As I understand, at least the latter two were ...
32
votes
1answer
3k views

Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...
12
votes
2answers
1k views

Combinatorics of the Stasheff polytopes

First a little background for those unaware. The Stasheff polytopes (or associahedra) are certain convex polytopes that arise in the theory of $A_\infty$-algebras. There is one polytope for each $n\...
7
votes
2answers
698 views

“MultiCatalan numbers”

Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient $$ \binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...} $$ is ...
4
votes
3answers
2k views

An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

What are the roles that the classic number arrays-- Eulerian, Narayana--play in the application of totally non-negative Grassmannians, or amplituhedrons, to string / twistor scattering theory? (This ...
23
votes
0answers
2k views

Why do polytopes pop up in Lagrange inversion?

I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
9
votes
0answers
425 views

Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series; 1) multiplicative inversion for quadratic algebras, 2) compositional inversion for quadratic operads, 3) ...
6
votes
0answers
306 views

Guises of the Noncrossing Partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: "Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples ...