every simple $A$-module naturally has a simple $A/J(A)$-module structure and every simple $A/J(A)$-module naturally has a simple $A$-module structure, where natural means whatever you think it means
as far as what probability? Too early to call. I just enjoy reading it so far. But I'll get back to you on this over winter break
yes, like harmonic analysis is just purely quantitative. It lacks analogies, and riddled with "tricks". I can't say that it's my cup of tea. Although I'm not bad in it.
I meant that there are many different areas, like statistical learning theory, random geometry and random graphs, markov processes, SDE, point processes
percolation
@JoeShmo oh, but probability is not necessarily better
It's a whole bunch of tricks and inequalities as well
Why can't I just start with $N$ as a simple $T_n(k)$-module, and let $k$ act on $N$ via constant diagonal matrices. Won't this turn $N$ into a $k$-vector space, and $\{aI \cdot n \mid a \in k, n \in N \}$ is a nonzero submodule, so by simplicity $N = \{aI \cdot n \mid a \in k, n \in N\}$, so $N$ is $1$-dimensional...Is there something wrong with this?
@user193319 if we have a simple module $M$ over a product ring $A\times B$, then one can show that either $AM = 0$ or $BM=0$ and in the first case $M$ is a simple $B$-module and in the second case a simple $A$-module
so simple modules over $A\times B$ are the union of simple modules over $A$ with simple modules over $B$
you can apply that inductively to $k^n$
the only simple module over $k$ is one-dimensional
But I learned enough to satisfy my philosophical interests, and would have liked to apply this to more current problems arising from biological organisms or linguistics, but that approach wasn't very fruitful in the past and it seems the logic methods are currently more adapted to computer science questions
@user193319 actually any module whatsover over a product ring $A \times B$ is of the form $M \oplus N$ where $M$ is an $A$-module and $N$ is a $B$-module and $A$ acts just on $M$ and $B$ just on $N$
now it's easy to see that $M\oplus N$ is simple iff $M=0$ and $N$ is simple or the other way around
so I'm now doing a lot of stochastic analysis and probability, which I am gradually getting better at. Among that stuff, a course on stochastic processes in neuroscience. My current plan is to maybe go into stochastic dynamical systems related to neuroscience. I'm still fascinated by turing's philosophical analysis that led to his definition of the turing machine, and by the fact that all realistic computability concepts led to the same class of computable functions.
