# All Questions

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### A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
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### Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...
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### What is the topology of teichmuller space of a given hyperhahler manifold

From verbitsky's paper, I read that the teichmuller space of a given hyperkahler manifold is not hausdorff but it's smooth. I am a bit confused about how to put a natural topology on the teichmuller ...
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### Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...
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### Schur multiplier of $Sp(2g, \mathbb{Z}_2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}_2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}_2$. With respect to ...
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### Continuation of an almost periodic sequence on the Bohr group

A sequence $x={(x_k)}_{k \in \mathbb{Z}}$ of real numbers is a uniformly continuous function on the discrete topological group $\mathbb{Z}$. Therefore it admits a unique continuous extension $\bar x$ ...
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### Duality argument to get $L^\infty-L^2$ inequality

In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that $$\lVert e^{-Ht}f \rVert_2 \leq c_1t^{-\mu/ 4}\lVert f \rVert_1$$ where the norms are $L^p$ norms. He states ...
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### General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics. I have been interested in some technical properties of the ...
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### $K$ theory and singular cohomology

For cell complexes${}^1$ $X$ we have an isomorphism $$K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}),$$ which is induced by the Chern character. What is the analogous statement for $KO(X)$? ...
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### Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, ...
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### Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

Wikipedia claims (permanent link) without reference: Testing whether one planar graph is dual to another is NP-complete. Another claim with reference: For any plane graph G, the medial graph ...
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### Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x$ is a complex ...
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### Holomorphic vector fields on compact complex manifolds with trivial canonical bundle

Let $M$ be a compact complex manifold whose canonical bundle $K_M$ is holomorphically trivial. Is it possible for $M$ to admit a non-zero holomorphic vector field with zeroes? Equivalently, using a ...
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### Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$ $$\pi^1(H)\to\pi^1(X)$$ is an isomorphism, ...
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### to find topological properties under a metric on a set [on hold]

we define a metric d on a set of composition operators on L2. I would like to find connected component and path connected component and other topological properties by d . Is there any book or paper ...
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### Exercise 5, chapter 4 in Do Carmo's “Riemannian Geometry” [on hold]

This is probably a silly question and maybe in the end the answer is trivial but I can't see it. The problem is the following. Let $M$ be a Riemannian manifold, $\gamma \colon[0,l]\to M$ be a ...
### The homology of $\varinjlim SO(p,q)$
Is there a way to explicitly compute the homology of the space $$\varinjlim_{(p,q)} SO(p,q)^+,$$ where each $SO(p,q)$ is the indefinite special orthogonal group, and $SO(p,q)^+$ its identity ...