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Hey Theo --- I don't think it is reasonable to expect Lagrangian fibrations to be cotangent bundles globally. Easy example: take a 2d torus, give it a symplectic form (equivalently a volume form in t …

Configuration space is, by definition, the position space of your particles. Phase space, on the other hand, is the space of pairs (position, momentum). The latter has a symplectic structure; the form …

Here is an attempt. Let $X$ be a submanifold of $Y$, and let $i : X \to Y$ be the inclusion. In general, we have the exact sequence of vector bundles $$ 0 \to N^\ast X \to i^\ast T^\ast Y \to T^\ast …

For which manifolds or varieties is quantum cohomology known to converge? Are there any manifolds for which quantum cohomology is known to not converge? I seem to have the impression that quantum coho …

See Dmitri's comment.

There are a few comments about this in the notes from Kontsevich's talk (Internet Archive) at the Arbeitstagung (Internet Archive). I also attended the Seidel talk that people are referring to. I shou …

Following, e.g., Wikipedia's definitions, the (small) quantum cohomology ring of $X$ is defined over a "Novikov ring" consisting of formal power series of the form $$ \sum_{\beta \in H_2(X;\mathbb{Z}) …

Here is the simplest example that I can think of... The ordinary cohomology ring of $\mathbb{CP}^n$ is given by $\mathbb{C}[a]/(a^{n+1})$. The structure of this ring can be thought of as describing t …

What are some examples of compact Kaehler manifolds (or smooth complex projective varieties) that are not isomorphic as complex manifolds (or as varieties), but are isomorphic as symplectic manifolds …

In (for example) Semisimple Frobenius structures at higher genus (section 1.2) and Gromov-Witten invariants and quantization of quadratic Hamiltonians (section 6.8), Givental gives a conjectural formu …

Let X be a compact symplectic manifold. There is an idea, I think probably originally due to Kontsevich, that we should be able to get Gromov-Witten invariants of X out of the Fukaya category of X. On …

I just skimmed a bit of this fresh-off-the-press paper on homological mirror symmetry for general type varieties. One thing that intrigued me was statement (ii) of Conjecture 3.3. It suggests that, j …

I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to …

I hope I'm not getting any details wrong, but I think the Narasimhan-Seshadri theorem asserts that the moduli space of $G=U(n)$ representations of $\pi_1$ is the same as the moduli space $M(n)$ of sem …

I think the basic example is when you have a symplectic manifold $M$ with a Hamiltonian $H : M \to \mathbb{R}$. Then take a regular value $a$ of $H$, and look at the hypersurface $N := H^{-1}(a)$, whi …