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