## Is every indecomposable homogeneous continuum unicoherent?

• Continuum = compact connected metrizable space

• Indecomposable = not the union of any two proper subcontinua.

• Homogeneous = for every two points $$x$$ and $$y$$ there is a homeomorphism of the space onto itself which maps $$x$$ to $$y$$.

• Unicoherent = the intersection of every two subcontinua is connected.

Examples of indecomposable homogeneous continua include solenoids, the pseudoarc, and solenoids of pseudoarcs (a solenoid with each point is blown up into a pseudoarc). I think it’s an open problem to determine whether this list is complete, but so far all of the examples are unicoherent. Is there a theorem stating that such continua must be unicoherent?

## Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $$C P^{n-1}$$ can be lifted to a homogeneous Poisson structure on $$C^n$$? By homogeneous I mean a quadratic Poisson structure of the form $$\{z_i,z_j\}=q_{ij}^{kl}z_kz_l$$ where coefficients $$q_{ij}^{kl}$$ are constants.

I suspect that this is correct but do not know any reference nor any idea of possible proof. Could you please help me with these?

## Coordinate ring of an equivariant embedding of a homogeneous projective variety

Lie algebra: Let $$G$$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $$B$$. Let $$P$$ be a parabolic subgroup containing $$B$$. Let $$\lambda$$ be a dominant integral weight. It follows from the Borel-Weil-Bott theorem that the homogeneous coordinate ring $$A_\lambda(G/P)$$ is a sum of highest weight representations. Namely, \begin{align*} A_\lambda(G/P) = \oplus_{n \in \mathbb{N}} V(n\lambda). \end{align*}

Lie superalgebra: Do we have an analogue result of this in the super case? More precisely, let $$G$$ be a simply connected analytic supergroup with $$\mathfrak{g} = Lie(G)$$ (a basic classical Lie superalgebra) and a Borel subsupergroup $$B$$. Let $$P$$ be a subsupergroup containing $$B$$. I know that there is an analogue of Borel-Weil-Bott theorem for supercase, but I wonder if there is a result for the coordinate ring as above. I guess we might need to assume $$\lambda$$ is a typical weight.

## Is $x f(x,y)+y g(x,y)$ homogeneous if f and g homogeneous?

If $$f_{1}(x,y)$$ and $$f_{2}(x,y)$$ are two homogeneous polynomials then prove that : $$\frac{f_{1}(x,y)+f_{2}(x,y)}{xf_{1}(x,y)+yf_{2}(x,y)}$$ is homogeneous.

## Connected and homogeneous $T_2$-space not homeomorphic to a subset of $\mathbb{R}^n$

What is an example of an connected and homogeneous $$T_2$$-space $$(X,\tau)$$ with $$|X|=2^{\aleph_0}$$ such that for no $$n\in\mathbb{R}$$ the space $$(X,\tau)$$ is homeomorphic to a subspace of $$\mathbb{R}^n$$?