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I learn and produce mathematics. In that process, I had to read quite a number of research articles. Question : What are some efficient ways to keep a note of results when reading a research article in mathematics? I keep a note of definitions (in detail) and results (with out proofs) for...

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Let $\Omega=[0,1]^2$ be the unit square, $\Gamma_1=[0,1]\times\{0;1\}$ its horizontal boundary and $\Gamma_2= \{0;1\}\times[0,1]$ its vertical boundary. I would like to know the optimal Poincaré constants $C$, defined by $$\forall u\in W^{1,2}(\Omega)\quad \int_\Omega u^2\le C\int_\Omega |\nab... 3 hours later… 3:08 AM 3 I am following the book Introduction to Quadratic Forms over Fields by T. Y. Lam. In section VI.2, the author proves that, over an arbitrary local field F, there is a unique quaternion division algebra, namely D=\left(\frac{\pi,u}{F}\right) where \pi is a uniformizer and u is such that F... 5 hours later… 8:01 AM 1 It is well known, that if V is a C^\infty G-module, then the differentiable cohomology groups H^*_{d}(G, V) are isomorphic to H^*(\mathfrak{g}, K, V). I am interested if the result can be strengthened to continous cohomology of unitary modules in the following way: Let V be a strongl... 2 hours later… 10:02 AM 18 John Horton Conway is known for many achievements: Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger 15-theorem, the Free-Will theorem—the list goes on and on. But he was so prolific that I bet he estab... 4 hours later… 1:35 PM 4 I have some general questions about the deformations of Galois covers of curves. Suppose we are given a G-Galois cover k[[z]]/k[[x]], where k is algebraically closed of characteristic p>0. Bertin and Mézard show that the deformation functor \text{Def}_{G} is represented by a versal def... 3 hours later… 4:29 PM 6 Assume A is an infinite-dimensional C^*\!-algebra and \tau_w is the Banach weak topology. I want to know when A has the \star -property:$$x_n\to 0 \ (\text{by}\ \tau_w)\ \ \ \ \Longrightarrow \ \ \ \ yx_n\to 0\ (\text{by norm}, \forall y\in A) \ \ \ (\star )  My attempts: Pr...

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A certain question on graph theory (about the existance of graphs with a certain coloring inherited by perfect matchings) can be translated into the satisfiability problem of a certain set of equations (formulated by Michael Engelhardt). Let $X_i$ be indexed by an integer $1\leq i\leq n$, and ...