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We consider the planar polynomial vector field $$(*) \;\;\;\begin{cases} \dot x= P(x,y) \newline \dot y =Q(x,y)\end{cases}$$ We replace the real variables $x,y$ with complex variables $x:=x_{1}+ …

Assume that $J$ is an almost complex structure on torus $\mathbb{T}^2$. Let $X$ be a non vanishing vector field on the torus. Let $g$ be a Riemannian metric with corresponding $LC$ connection …

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$. Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if the …

What is a precise example of the following situation: A compact Riemanian manifold $M$ admits two vector field $X,Y$ such that the the operator $$Z\mapsto R(X,Y)Z$$ Would be an elliptic operator and …

Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2} …

Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$. We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows: $T(f)= …

Let $M$ be a Riemannian manifold with boundary $\partial M$. Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of the …

In your question you mentioned the word "Fredholm index". So I would like to say that in the circle case there are two different interpretations of Fredholm index of certain lin …

Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting dy …

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which …

In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in th …