16
votes
0answers
429 views

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

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it. Recently I encountered in a class the fact that there is a generating function of ...
6
votes
1answer
196 views

Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
13
votes
1answer
688 views

Is there a Seiberg-Witten version of Donaldson-Thomas theory?

Donaldson invariants are a count of instantons (the solutions to a particular elliptic PDE) on 4-manifolds. One thing which makes the theory difficult is a lack of compactness for the moduli spaces of ...
0
votes
0answers
263 views

Pullback of harmonic forms.

If $f \colon X \to Y$ is a holomorphic map between Kaehler manifolds, then the pullback of a harmonic form on $Y$ is not necessarily harmonic on $X$, even if $f$ is an immersion. This came up during a ...
6
votes
5answers
860 views

partial differential equation for ruled surfaces

We say that a surface $f(x,y,z)=0$ is ruled if for each point $p$ in the surface there is a line that passes through $p$ and is contained in the surface. See http://en.wikipedia.org/wiki/Ruled_surface ...
16
votes
4answers
2k views

Classification of PDE

Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for ...
29
votes
5answers
2k views

Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...
5
votes
2answers
1k views

Characteristic surface for systems of PDE

Despite the title, this is probably actually a question in linear algebra or algebraic geometry. Let me write the question(s) first, before I explain the background. Problems Let $h^{\mu\nu}_{ij}$ ...
59
votes
8answers
7k views

What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem. The ...
19
votes
4answers
844 views

What is the relationship between various things called holonomic?

The following things are all called holonomic or holonomy: A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the ...
8
votes
3answers
541 views

Equations for Integrable Systems

So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further ...