Arnold conjecture

The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.

Strong Arnold conjecture
Let $$(M, \omega)$$ be a closed (compact without boundary) symplectic manifold. For any smooth function $$H: M \to {\mathbb R}$$, the symplectic form $$\omega$$ induces a Hamiltonian vector field $$X_H$$ on $$M$$ defined by the formula

$$\omega( X_H, \cdot) = dH.$$

The function $$H$$ is called a Hamiltonian function.

Suppose there is a smooth 1-parameter family of Hamiltonian functions $$H_t \in C^\infty(M)$$, $$t \in [0,1]$$. This family induces a 1-parameter family of Hamiltonian vector fields $$X_{H_t}$$ on $$M$$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms $$\varphi_t: M \to M$$. Each individual $$\varphi_t$$ is a called a Hamiltonian diffeomorphism of $$M$$.

The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of $$M$$ is greater than or equal to the number of critical points of a smooth function on $$M$$.

Weak Arnold conjecture
Let $$(M, \omega)$$ be a closed symplectic manifold. A Hamiltonian diffeomorphism $$\varphi:M \to M$$ is called nondegenerate if its graph intersects the diagonal of $$M\times M$$ transversely. For nondegenerate Hamiltonian diffeomorphisms, one variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a Morse function on $$M$$, called the Morse number of $$M$$.

In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field $${\mathbb F}$$, namely $\sum_{i=0}^{2n} \dim H_i (M; {\mathbb F})$. The weak Arnold conjecture says that

$$\# \{ \text{fixed points of } \varphi \} \geq \sum_{i=0}^{2n} \dim H_i (M; {\mathbb F})$$

for $$\varphi : M \to M$$ a nondegenerate Hamiltonian diffeomorphism.

Arnold–Givental conjecture
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, gives a lower bound on the number of intersection points of two Lagrangian submanifolds $L$ and $$L'$$ in terms of the Betti numbers of $$L$$, given that $$L'$$ intersects $L$ transversally and $$L'$$ is Hamiltonian isotopic to $L$.

Let $$(M, \omega)$$ be a compact $$2n$$-dimensional symplectic manifold, let $$L \subset M$$ be a compact Lagrangian submanifold of $$M$$, and let $$\tau : M \to M$$ be an anti-symplectic involution, that is, a diffeomorphism $$\tau : M \to M$$ such that $$\tau^* \omega = -\omega$$ and $$\tau^2 = \text{id}_M$$, whose fixed point set is $$L$$.

Let $$H_t\in C^\infty(M)$$, $$t \in [0,1]$$ be a smooth family of Hamiltonian functions on $$M$$. This family generates a 1-parameter family of diffeomorphisms $$\varphi_t: M \to M$$ by flowing along the Hamiltonian vector field associated to $$H_t$$. The Arnold–Givental conjecture states that if $$\varphi_1(L)$$ intersects transversely with $$L$$, then

$$\# (\varphi_1(L) \cap L) \geq \sum_{i=0}^n \dim H_i(L; \mathbb Z / 2 \mathbb Z)$$.

Status
The Arnold–Givental conjecture has been proved for several special cases.


 * Givental proved it for $$(M, L) = (\mathbb{CP}^n, \mathbb{RP}^n)$$.
 * Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.
 * Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number.
 * Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono proved it for $$(M, \omega)$$ semi-positive.
 * Urs Frauenfelder proved it in the case when $$(M, \omega)$$ is a certain symplectic reduction, using gauged Floer theory.