Obstructions to symplectically embedding compact manifolds of dimension 44 or higher

It is known in Li’s paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds (X2n,ω) of dimension at least 2n≥4, an immersed symplectic surface represents a 2-homology class as long as that homology class has positive symplectic area. When 2n≥6, this immersion may even be taken to be an embedding. My question regards finding symplectically embedded submanifolds … Read more

Amalagamation of a sequence of closed immersions of schemes

Let (Xn)n≥0 be a family of schemes. Let X0→X1→X2→… be a sequence of closed immersions (which therefore gives rise to an ind-scheme). Under which (necessarly and/or sufficient) conditions is there a scheme S and a family of closed immersions Xn→S such that Xn⟶Xn+1↘↙S commutes? This is not always satisfied (see the comments). Answer AttributionSource : … Read more

Whitney-like embedding theorem for posets?

The Whitney embedding theorem says that any finite-dimensional smooth manifold can be embedded into $\mathbb{R}^n$ for some $n$. Is anything like this true for posets? I’m looking for conditions on a poset $P$, under which there exists an embedding of $P$ into a cube $L^n$ for some linear order $L$. I’m interested in both the … Read more

Finite approximations to the Kuratowski/Fréchet embedding

Let $(X,d)$ be a compact doubling metric space with doubling constant $C>0$. Let $\{\mathbb{X}_n\}_{n=0}^{\infty}$ be a sequences of finite subsets of $X$ with $$ \left\{B\left(x_k,\frac1{n}\right)\right\}_{k=0}^{\mathbb{X}_n} \mbox{covers } X \mbox{ and } \#\mathbb{X}_n \mbox{is the $\frac1{k}$-covering number of $X$}. $$ Fix some $x^{\star}\in X$ and consider the associated sequences of $1$-Lipschitz maps $$ K_n:\,x\mapsto \left(d(x,x_n)-d(x_n,x^{\star})\right)_{x_n\in \mathbb{X}_n}. … Read more

How to show that random graphs cannot be embedded with short edges

For each (not necessarily planar) embedding of a graph in $\mathbb{R}^k$ one can calculate the ratio $$\gamma = \frac{\textsf{mean Euclidean length of edges}}{\textsf{mean Euclidean distance between nodes}} \times \textsf{diameter}$$ where the diameter is the maximal Euclidean distance between nodes. An embedding consists in giving each node of the graph an arbitrary position under the restriction … Read more

Generalizations of Abhyankar-Moh theorem (embeddings of the line in the plane)

Abhyankar-Moh theorem says that if L is a complex line in the complex affine plane C2, then every embedding of L into C2 extends to an automorphism of the plane. It seems that one can replace C by any algebraically closed field k of characteristic zero. Are there ‘similar’ results for fields other than k, … Read more

Generalization of Gagliardo-Nirenberg Inequality

The standard Gagliardo-Nirenberg Inequality is ‖u‖Lnn−1(Rn)≤Cn‖∇u‖L1(Rn), and constitutes a key step to proving Sobolev Injection Theorems. Of course we may assume that u is smooth and compactly supported, the important point is that the constant Cn is dimensional and independent of u. Question 1. Are there some improvements of (∗) where the rhs is replaced … Read more

Is a Morse function always the height function of some embedding? [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 7 years ago. Improve this question Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of … Read more

Equivariant isometric embedding of manifolds in a Hilbert space under a noncompact group action

Given a Riemannian manifold M and a group of isometries G of M, I am interested in when there exists a isometric embedding ι:M→H, where H is a Hilbert space and a representation ρ:G→B(H;H) such that for each g∈G and x∈M, ι(g(x))=ρ(g)ι(x). When M is a compact manifold and G is a compact Lie group … Read more

Theoretical justification of time-series forecasting using Takens’ embedding

This is a cross-posting where I couldn’t get an answer. In the meantime I have tried to improve the original logic: As in Takens original paper about his embedding theorem, consider a compact m-dimensional manifold M and a smooth map f:xt∈M→xt+1∈M that expresses the state-evolution in a discrete-time state-space system. Consider a second. continuous map … Read more