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Let $ G$ be a commutative group scheme of finite type over an algebraically closed field $ k$ of characteristic zero, with an embedding $ k\subset\mathbf{C}$ . Is every torsion element of $ G(\mathbf{C})$ defined over $ k$ ? That is, do we have $ G(\mathbf{C})_{\rm tor} = G(k)_{\rm tor}$ ?Read more

Assume That $ A$ is a commutative complex Banach algebra. Let $ G$ be the connected component of invertible elements containing the identity. Is there an smooth embedded curve $ c:(-\epsilon, \epsilon) \to A$ with $ c(0)=0$ and a neighborhood $ W$ of $ 0$ and a continuous (or Frechet differentiable or holomorphic* ) mapRead more

In PostgreSQL, we can ensure the relations stored inside a table a commutative-ly (or bidirectional etc) unique. For instance, CREATE TABLE foo ( a int, b int, UNIQUE ( greatest(a,b), least(a,b) ) ); INSERT INTO foo VALUES (1,2); — works INSERT INTO foo VALUES (2,1); — explodes I know SQL Server can create computed columns,Read more

I want to try to understand non commutative geometry by reading Connes’s book ..and I am discovering it is a hard book to read 🙂 as I miss a lot of background specially in operator algebra and homology theory ( my field is nonlinear PDE so I know a bit of functional analysis already- atRead more

If $ F:V\to W$ is a smooth at $ a\in V$ function between finite-dimensional vector spaces over $ \mathbb{R}$ , then we have $ $ F(x) = \sum_{k=0}^N\frac{1}{k!}(D^kF)(a)[(x-a)^{\otimes k}]+\text{remainder}, $ $ where $ (D^kF)(a):\mathrm{Sym}^k(V)\to W$ are $ \mathbb{R}$ -linear maps. Algebraically, this is a statement about writing tuples of homogeneous polynomials of same degree asRead more

Let $ R$ be a cofibrant commutative $ S$ -algebra (in the sense of Elmendorf-Kriz-Mandell-May; they call them “$ q$ -cofibrant”) and $ A$ be a cofibrant commutative $ R$ -algebra. Does $ A\wedge_R-:RMod→RMod$ preserve all weak equivalences? This could be rephrased as “$ A$ is a flat $ R$ -module”. It is true thatRead more