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80

It's sort of like the inverse function theorem, and that is why it is so strong. If you have $n$ functions vanishing at the origin of $k^n$ and want to know if they give a local coordinate system, you ask if their differentials are independent at the origin. Or equivalently if their differentials generate the cotangent space at the origin. So in a [not ...


57

This is indeed not a typical math overflow question, but never mind that. Of course you can learn mathematics at the age of 30 after having stopped studying it at the age of 18! Examples are abundant -- in almost every math department I've ever been in, there are at least one or two older graduate students that took some years off (after high school, after ...


52

The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S_r|$ of the $r$-neighbourhoods $S_r$ of $S$ at $r=0$: $$ |\partial S| = \frac{d}{dr} |S_r| |_{r=0}.$$ Thus, for instance, the boundary $\partial D_r$ of the disk $D_r$ of radius $r$ has circumference $\frac{d}{dr} (\pi r^2) = 2\pi r$. More generally, one ...


50

The most immediately obvious relation to category theory is that we have a category consisting of types as objects and functions as arrows. We have identity functions and can compose functions with the usual axioms holding (with various caveats). That's just the starting point. One place where it starts getting deeper is when you consider polymorphic ...


41

Mnemonic: $\quad M=IM \Rightarrow m=im$ The version of Nakayama described: If $I$ is an arbitrary ideal of an arbitrary commutative ring $A$ and if a finitely generated module $M$ satisfies $M=IM$, then there exists $i\in I$ such that for all $m\in M$ we have $m=im$. Please notice: no noetherian nor local assumption on $A$, no assumption at all on $I$.


39

There must be many ways to think of this. Here's one: The symmetric group is involved with homogeneous polynomials of degree $n$ because they correspond to symmetric multilinear functions of $n$ variables, and division by $n$ factorial appears when recovering the former from the latter. For example, a homogeneous polynomial map $f:V\to W$ of degree $3$ ...


38

If you want to teach something intriguing, you should do something that introduces a new geometric idea while also involving algebra in an essential way. I recommend that you give an introduction to the projective plane, showing the other students that it is a natural extension of ordinary space which makes some geometric properties more uniform (such as ...


36

It's easiest to understand for local rings, so let $R$ be one with residue field $k$. Nakayama's lemma just says that a finitely generated $R$-module is zero if and only if the induced $k$-vector space is. Through the magic of abelian categories, this implies that a map of $R$-modules is surjective if and only if the induced $k$-linear map of $k$-vector ...


34

A little bit more information about your background and situation would be helpful. Are you: a graduate student, a post-doc, a tenure-track professor? Are you teaching at a university? Are you teaching undergraduate or graduate courses? (I will assume that by "analysis" you mean something which is at the advanced undergraduate level, at least.) I ...


34

I feel as though this question may have come up before. Anyhow, the $\ell_4$ norm, and more generally the $\ell_{2k}$ norm for any positive integer $k$, come up naturally in Fourier analysis, since the $\ell_{2k}$ norm of the Fourier transform of $f$ equals the sum of $f(x_1)...f(x_k)\overline{f(y_1)...f(y_k)}$ over all $x_1+...+x_k=y_1+...+y_k$. That sort ...


34

The original Liouville's number is probably the easiest, but most of the proofs tend to invoke calculus (because why not?), so let me try to show it in a more 7th-grade friendly way. I'll call this the swaths-of-zero approach. So we know that Liouville's number $L$ looks like this: .1100010000000000000000010... with a 1 in the $n!$ places. When we square ...


33

Dear bitrex: your enthusiasm is heart-warming! I have had students much older than you and they have always been a joy to teach: their maturity more than compensated for their potential knowledge-gaps and they fared very well on their exams. The nicest success story is a professional cellist who didn't even have the "baccalauréat", a French diploma for the ...


32

I learned about the "secretary problem" when I was about 10 years old from one of Martin Gardner's books. Though I thought is was cool and amazing, I don't think it gave me much insight into $e$. Here's a way to introduce $e$ with only addition and multiplication, in the form of a game. Tell him he's got a "budget" of say, 100 to work with, and his goal ...


29

To me, the explanation for the appearance of div, grad and curl in physical equations is in their invariance properties. Physics undergrads are taught (aren't they?) Galileo's principle that physical laws should be invariant under inertial coordinate changes. So take a first-order differential operator $D$, mapping 3-vector fields to 3-vector fields. If ...


29

Huge chunks of the theory of nonlinear PDEs rely critically on analysis in $L^p$-spaces. Let's take the 3D Navier-Stokes equations for example. Leray proved in 1933 existence of a weak solution to the corresponding Cauchy problem with initial data from the space $L^2(\mathbb R^3)$. Unfortunately, it is still a major open problem whether the Leray weak ...


28

The Graded Nakayama's Lemma My intuition for Nakayama's lemma is rooted in the graded version. (Graded Nakayama's Lemma) Let $R$ be a $\mathbb{N}$-graded algebra, and let $R_+$ be the 'irrelevant' ideal of positive degree elements. Let $M$ be a finitely-generated $\mathbb{Z}$-graded $R$-module. If $I\subseteq R_+$, and $IM=M$ then $M=0$. I find this ...


28

An animation of your first example, $y=x(x−1)(x+1)$. The animation is flawed in that it jumps jarringly when the tick labels reach the boundary. I removed the ticks and labels, which gains smoothness at the expense of magnitude feedback. (The previous version is here.) And I limited the number of frames so that the file would not be too huge (it is ~1MB ...


25

Tim, I've got two words for you: interpolation theorems (e.g., Riesz-Thorin and Marcinkiewicz interpolation theorems). Such theorems let you pass from information about some operators on $L^1$ and $L^\infty$ to some operators on $L^2$ using all the intermediate exponents $p$. The point here is not that one actually cares about $L^{37.24}$ for its own ...


25

A partial answer to Q1 -- apologies if this is obvious, but I don't see it written here yet, and this is the thing that made me sit up and take notice of the fact that there's some sort of connection between the boundary operator $\partial$ and differentiation. If $X$ and $Y$ are two topological spaces and $A \subset X$, $B\subset Y$ are closed, then they ...


24

Quick answer: No, you are not too old. Yes, such people do exist. It sounds like you're off to a good start. Don't let your age worry you. I dropped out of college when I was 21 to work as a software engineer. Admittedly my work was technical, but my background in abstract mathematics was basically nonexistent. Two+ years ago, when I was 34, I decided ...


24

Sure, why not? But let me suggest a better plan. What you need is a faculty mentor. If you have a favored topic that you are working on, then first simply go and speak about it with a knowledgable professor about it in person. It will be extremely helpful to you to have a faculty mentor in your mathematical work. You can explain your idea and result and ...


24

In PDEs, various values of p arise as degrees of regularity. The Sobolev embedding theorems let you "trade in" generalized derivatives for classical derivatives. You might need the exponent p to be above a certain threshold to get a desired regularity result. Still, I agree with your observation that much of the time the values of p that matter are 1, 2, ...


23

I'm not sure how good an idea this would be. I happen to be in a position where I read many applications of students wishing to do a PhD in my group. Some applicants have a very definite idea of what it is that they want to do, but this is usually for lack of exposure to other topics. It is not unheard of that they end up doing their PhD is a completely ...


23

For me the Nakayama lemma (even though maybe not in its strongest form) simply says that: If $\mathcal{F}$ is a coherent sheaf over the (locally noetherian) scheme $X$, then the dimension of the fiber of $\mathcal{F}$ at a closed point $x\in X$ is equal to the rank of the stalk, and a basis of the fiber lifts to a system of generators of the stalk.


23

Don't you mean to say that $f^{(n)}(a)$ is $n!$ times the lead coefficient? For instance, take $f=x^n$. The functions $f_k(n)=\frac{n!}{(n-k)!k!}$ satisfy a discrete version of the derivative. $f_k(n)-f_k(n-1)=f_{k-1}(n)$. If we view Taylor's theorem as being about approximating with a very fine version of these functions, we just have to see where and why ...


22

Finding an anti-derivative of $x\tan x$ amounts to finding an anti-derivative of $f=\frac{x}{e^x+1}$. Consider the field $K=\mathbb C(x,e^x)$. Note that $K$ is closed under taking derivatives. If $f$ is elementary integrable, then Liouville's Theorem gives elements $u_i\in K$, $\gamma_i\in\mathbb C$, $v\in K$ with \begin{equation} ...


21

If you can explain $\pi$, then you can also explain radians, and this leads to a geometric interpretation of $e$ that is not in the wikipedia article Martin Brandenburg referred to. Get a big sheet of paper and make a polar coordinate grid, and ask your ten-year-old to put a pencil down somewhere away from the origin, and try to draw a continuous curve ...


21

I keep meaning to write this up nicely, but one can prove the transcendentality of Liouville's number in a very, very elementary way. Write $L$ for the Liouville number. Suppose $p(L)=0$ for some polynomial with integer coefficients. Then $p_+(L)=p_-(L)$ where $p=p_+-p_-$ and polynomials $p_+$ and $p_-$ have only positive coefficients and no terms of the ...


21

In algebraic geometry, this construction is known as the tangent cone to the graph. More generally, suppose we have the zero set of any polynomial $f(x,y) = 0$, and assume $f(0,0)=0$. Then we can write $f(x,y) = a_m (x,y) + a_{m+1}(x,y) +a_{m+2}(x,y) +\cdots$ where $a_i(x,y)$ is a homogeneous polynomial of degree $i$ and $a_m$ is nonzero. The zero set ...


21

One way is to use a combinatorial definition of the derivative. Let $A(z) = \sum a_n z^n$ be a power series. In combinatorics, where $A$ is likely to be an ordinary generating function, $a_n$ is likely to count the number of possible combinatorial structures of some kind on an ordered set with $n$ elements. For example when $a_n = 1$ the structure is "being ...



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