User spencer - MathOverflow

Answer by Spencer for Applications of Rademacher's Theorem

2013-03-21T17:47:59Z

The application I am most familiar with is that it is used in the proof of the following result:

Suppose $f : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz. For any $\epsilon > 0$, there exists a $C^1$ function $g$ such that the Lebesgue measure of the set { $f\neq g$ } $\cup$ { $D f \neq D g $ } is at most $\epsilon$.

One reason I know for this result being useful is that it gives the `approximate' tangent bundle structure on a countably rectifiable set:

A countably $n$-rectifiable subset of Euclidean space is usually defined as a set (almost all of) which is contained in a countable union of Lipschitz images of $\mathbb{R}^n$. The preceeding proposition is used to show that one can replace " Lipschitz images of $\mathbb{R}^n$ " with "embedded $n$-dimensional $C^1$ submanifolds" in this definition.

Since $C^1$ submanifolds have tangent spaces, one is now only a couple of checks away from the fact that one can define the intrinsic derivative of a locally Lipschitz function at almost every point of a countably rectifiable set.

Having this kind of differentiable structure for objects and functions so weak is essential for studying various GMT-esque regularity problems e.g. understanding the singularities of stationary varifolds.

What is the "real osculating space" of a (minimal) immersion?

2012-11-28T11:40:15Z

In a differential geometry paper from 1979 I have come across some terminology which I have not found explained anywhere else.

We have an immersion $x : S^2 \to S^n$. In the paper, it is a minimal immersion but I'm not sure it matters. It goes on to say

"Let $T_k(x)$ denote the real osculating space of order $k$ of $x$".

A) What is the precise definition of the real osculating space of an immersion in moden differential geometric language?

B) What does it mean intuitively?

[The paper is "An Extrinsic Rigidity Theorem for Minimal Immersions of S^2 into S^n" by J.L.M. Barbosa]

(I asked this on Stack Exchange originally)

Divergence form Elliptic PDE Removable Singularity/Regularity Question

2012-11-26T14:47:55Z

Idea

Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) continuous. If you have better regularity away from one isolated point, say you are $C^1$ on the puncutered ball, can the solution still fail to be differentiable at that isolated point?

The situations I am most familiar with tend to be ones in which one can go from boundedness and continuity to smoothness via Schauder theory and bootstrapping. Here, we already have continuity: But can differentiability fail at a single point? TIt seems out of reach of all the theorems I've seen, which makes me suspect it is false, but I cannot be sure without a counterexample. Does anyone have any ideas?

Details

In the specific situation I am interested in, I know a bit more. I am considering the following:

$u \in W^{1,\infty}(B_1(0)) \cap C^{1,1}_{loc}(B_1(0)\setminus ${0}$)$ satisfies weakly the equation

$D_i(A_{ij}(x)D_ju) = D_ig^i$

in $B_1(0)$, where $A_{ij},g^i \in L^{\infty}(B_1(0))\cap W^{1,2}_{loc}(B_1(0)\setminus ${0}$)$.

Questions

1) Must $u$ in fact be a $C^1$ solution on $B_1(0)$?

2) What about just being differentiable at 0?

3) How about even just $u \in W^{2,p}(B_1(0))$ for some $p > 1$?

Computer Science for Mathematicians

2011-01-05T16:49:19Z

This is a big-list community question, so I'm sorry in advance if it is deemed too soft but I haven't seen anything similar yet.

I've seen computer scienctists post questions looking to learn things from pure maths. This is basically the other way around... My ignorance may prevent me from being as specific as I think I would like to be and so I have separated my main question into two.

What good books - readable introductions - are there for mathematicians to learn about computer science?

By this I really mean the science of how computers work. There are perhaps some books out there which are written in a style which mathematicians can relate to - e.g. not practicality-focussed, starting from the more abstract fundamentals and building up (I may be wrong but I'm under the impression that a lot of books in other disciplines shy away from presenting things this way around, whereas mathematicians (for better or worse) are accustomed to it). Partly to illustrate what the first question is not asking, the second question is

What good books - readable introductions - are there for mathematicians looking to learn about theoretical computer science, as it is as a subfield of maths?

Here is where my ignorance prevents me from explaining the question any more because I can only assume these two aren't the same thing...

It seems quite frustrating that I have made it to grad school and know very little about computers and theoretical CS.

Standard "one recommendation per post" is probably appropriate, + a few sentences about what the books did for you. Also, maybe I should say that I'm not looking to ditch my current interests and become a computer scientist, so things being readable is a fairly strong condition. I'm not looking to become an expert, just to deal with my own ignorance. Thanks in advance.

Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P?

2011-06-27T17:33:09Z

I am an analyst struggling through some geometry used in physics.

Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection 1-form on $P$ (a "principal connection"). This is a Lie algebra-valued 1-form.

As for the curvature two-form, either you see definitions with no explanation at all, e.g. "The curvature is given by $\Omega = d\omega + \frac{1}{2}[\omega,\omega]$". This is obviously less than ideal for improving one's intuitive appreciation. Or one defines something called the exterior covariant derivative $D$ (see wiki) and then the curvature is simply the exterior covariant derivative of the connection one-form.

The issue: I can't get round the following observation though: From the point of view of the manifold $P$, $\omega$ is just a one-form with values in some vector space which happens to be $\mathfrak{g}$. Usually when you need to covariantly differentiate such an object, you would need a connection in a bundle $E \to P$ with fibre $\mathfrak{g}$, no? Then $\omega$ would be an $E$-valued one-form on $P$, i.e. in $\Gamma(E) \otimes\Omega^1(P)$, and you can differentiate covariantly in the normal way using the connection. Why is this scenario different?

The exterior covariant derivative $D$ satisfies $D^2\phi = \Omega\wedge\phi$... So you have some sort of covariant differentiation $D$ which differentiates forms $\eta$ taking values in $\mathfrak{g}$ and for which $D^2$ is some sort of curvature... but of $P \to M$ and not of the bundle in which $\eta$ is taking values. Isn't this strange? Or is this indeed just how things are? This prompts my more precise question:

Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of some vector bundle over P with fibre $\mathfrak{g}$?

Why is the harmonic oscillator so important? (pure viewpoint sought). How to motivate its role in Getzler's work on Atiyah-Singer?

2010-03-04T22:05:00Z

I'm in the process of understanding the heat equation proof of the Atiyah-Singer Index Theorem for Dirac Operators on a spin manifold using Getzler scaling. I'm attending a masters-level course on it and using Berline, Getzler Vergne.

While I think I could bash my way through the details of the scaling trick known as `Getzler scaling', I have little to no intuition for it.

As I understand it, one is computing the trace of the heat kernel of the ("generalized") Laplacian associated to a Dirac operator. The scaling trick reduces the problem to one about the ("supersymmetric" or "generalized") harmonic oscillator, whose heat kernel is given by Mehler's formula. I am repeatedly assured that the harmonic oscillator is a very natural and fundamental object in physics, but, being a `pure' analyst, I still can't sleep at night.

What reasons are there for describing the harmonic oscillator as being so important in physics?

Why/how might Getzler have thought of his trick? (Perhaps the answer to this lies in the older proofs?)

Is there a good way I could motivate an attempt to reduce to the harmonic oscillator from a pure perspective?

(i.e. "It's a common method from physics" is no good). I'm looking for: "Oh it's simplest operator one could hope to reduce down to such that crucial property X still holds since Y,Z"...or..."It's just like the method of continuity in PDE but a bit different because..."

Thanks.

Answer by Spencer for Taking "Zooming in on a point of a graph" seriously.

2011-10-04T22:27:06Z

As Rbega says in the comments, if you are really keen to see this rescaling idea put to use in a more rigorous or advanced way, then you can look at some Geometric Measure Theory. While it will look very technical compared to this (because it is designed for potentially badly-behaved or very weakly-defined geometric objects), this sort of homothetic blowing up is standard for defining tangent objects to things. You get a weak kind of convergence of the rescalings of your original object to the tangent object, which, depending on the circumstances, may well (or perhaps will hopefully) display some sort of rigidity, e.g. it may be have to be a cone. It is rigorous and yes you can indeed end up with things like the union of two lines as your tangent object.

In the special case of the graph of a differentiable function, the tangent object at a point will indeed be the graph of the affine function associated with the derivative at the point.

I don't know of any books which take this approach pedagogically, in the development of calculus though.

Answer by Spencer for Mathematical habits of thought and action which would be of use to non-mathematicians

2011-09-07T15:04:39Z

This is really just an extension of James D. Taylor's comment on the question, but recognizing the value of definitions, and the inherent ambiguity without them, is ridiculously helpful.

Define your terms!

I recently saw a talk on how to teach students to write mathematics well. The advice "think like a lawyer" was given, which I totally agree with when writing mathematics. Anyone here who has read a quality legal document will know the similarities which this analogy is getting at. Both mathematicians and lawyers define their terms clearly at the beginning (of a debate/proof/court case) in order to eliminate as much ambiguity as possible from the words being used.

This is a great skill to have in order to cut through the BS in lots of other situations. E.g. Time and time again you see opinion pieces in which writer has no real point, but just trivially exploits the lack of a definition for a certain word. This really annoys me. Or you can be much better at seeing when an argument is a genuine difference of opinion or just a confusion arising from two people having different definitions.

If you take this too far, you end up as a bit of nihilist in that respect though: There's nothing to argue about because either people have different opinions (and there will always be people with different opinions) or people have different definitions... and arguments end trivially and inconclusively (I know I've done this many times to end boring arguments:) "If we accept your definition of [e.g.] feminism, then you're right and if we accept mine, then I'm right". (I suppose seeing to the core of an argument like this is similar to David White's first point).

On the other hand, you can debate quite freely things you have no clue about, just by deciding on a few axioms/vaguely reasonable assumptions and working from the definitions! (this is sort of the skill of debating competitions).

Answer by Spencer for Learning roadmap for harmonic analysis

2011-06-03T09:44:34Z

To my mind, the classical subject is quite different from the modern, evolved form of the subject

I started on the classical side with Yitzhak Katznelson's An Introduction to Harmonic Analysis: This is in the classical camp: Lots on Fourier Series. Very clear; very nice proofs. You will learn lots of gems about trigonometric series. In this classical camp, Zygmund's treatise Trigonometric Series (two volumes) deserves a mention. This is also a very beautiful book.

For 'harmonic analysis' as a modern field, you ought to get your hands on a copy of Stein's books as in Peter's answer. The late Tom Wolff has a very useful set of notes in this regard, available (I think, still) from Izabella Laba's homepage.

I also second the recommendation to look at Tao's old dvi/pdf notes on his website and later on on his blog. For example, I remember finding his post on interpolating $L^p$ spaces very nice.

Answer by Spencer for Which book would you like to see "texified"?

2011-05-14T01:09:06Z

Leon Simon - Lectures on Geometric Measure Theory

Answer by Spencer for Dimension Leaps

2011-04-18T20:50:09Z

An $n=7$ cut-off appears in the theory of minimal surfaces.

Bernstein's problem on golabl minimal surfaces: A global solution to the minimal surface equation on $\mathbb{R}^n$ is necessarily an affine function for $n \leq 7$, but there are counterexamples in all greater dimensions.

Also, an $n$ dimensional minimal surface in $\mathbb{R}^{n+1}$ is regular outside a singular set whose dimension is at most $n - 7$. The Simons cone, given by the set of points $x \in \mathbb{R}^8$ such that

$x_1^2 + x_2^2 + x_3^2 + x_4^2 = x_5^2 + x_6^2 + x_7^2 + x_8^2$,

is minimal and therefore shows that this is optimal - because it has an isolated singularity at the origin.

So, curiously, the set of points $x \in \mathbb{R}^6$ such that $x_1^2 + x_2^2 + x_3^2 = x_4^2 + x_5^2 + x_6^2$, just isn't minimal.

A set for which it is hard to determine whether or not it is countable.

2011-03-21T23:23:54Z

I got thinking recently, while trying to come up with a problem, that I did not know of any sets which were reasonable to define but for which it was very difficult to determine whether or not they were countable or uncountable.

When one first learns these concepts, it can be difficult, but with some experience, a mathematician can look at most sets which he or she meets in day-to-day and say almost immediately 'countable' or 'uncountable'.

What examples of sets are there for which determining whether or not they are countable is a difficult problem?

I won't define 'difficult' too rigorously but ideally I'm looking for something which any grad student can think about but which most would still be thinking about after 10 minutes.

Answer by Spencer for What should be taught in a 1st course on smooth manifolds?

2011-03-20T23:46:01Z

This is in agreement with Igor's comment on Anton's answer, but became too long.

I'd say whatever approach you ultimately take, for a first-year grad course it surely has to be done 'properly', i.e. starting from intrinsic definition of a smooth manifold and using the 'modern' language and general definitions of tensor bundles, connections etc.

Absolutely crucially (and here's what inspired this comment), the course simply has to teach people that there is more to manifolds than 2D surfaces because that's why the theory is quite so useful and so prominent in modern mathematics. The whole point is surely the sheer diversity of objects amenable to geometric thought (whatever that means). The job of the teacher would then be to maintain the intuition of "surfaces in R^3" while using general definitions. I believe this can be done. If it cannot, then what on Earth are we all doing?

By the look of the books mentioned in the question, it certainly looks like a course on what I would call "Differential Topology". Sure, there is nothing wrong with a good course on Differential Topology! However, it doesn't seem to me to be synonymous with "A First Course on Smooth Manifolds". My go to book for the latter is John Lee's Introduction to Smooth Manifolds.

Answer by Spencer for Extremal curves with a "should pass through" constraint

2011-03-19T15:20:56Z

Partial Answer/Too long for comment. If you are just working in the plane, then intuitively you know already that a length-minimizer exists (so long as you allow self-intersections). Such a solution will be a geodesic. This means you know (ahead of designing your functional) that a solution won't have any curvature - so one ought to focus on the length minimzation.

From here, it seems to me (I have not done any detailed calculation) that some sort of reasonable convex penalization on your curves for not going through a given point $q$ will result in the right critical points. For example, if you work in a class of piecewise $C^1$ curves $\gamma :(0,1) \to \mathbb{R}^2$, then consider

$F(\gamma) = \text{length}(\gamma) + \inf_{x \in (0,1)}|p-q|^2$.

You can then calculate the Euler-Lagrange equation for this functional. You do need to justify differentiating through the infimum, though.

The more interesting question is that of visiting multiple points in some order. I have not thought about this.

Answer by Spencer for Is the derivative of a Lipschitz function better than L^\infty

2011-03-04T23:19:09Z

Lipschitz functions are exactly $W^{1,\infty}$ (See 'Sobolev space' on wikipedia - under other examples and perhaps the bit about absolute continuity on lines). This means the short answer to your question is no.

Answer by Spencer for Proving theorems by using functions with fixed points.

2011-01-23T13:33:49Z

I am familiar with a good example from the theory of 2nd order elliptic PDE. Technicalities omitted...

A special case of the Leray-Schauder Theorem says the following:

Let $T$ be a compact mapping of a Banach space X into itself and suppose there exists a constant $M$ such that $\|x\| \leq M$ for all $x$ in the set {$ x \in X : Tx = \sigma x\ \text{for some}\ \sigma \in [0,1]$}. Then $T$ has a fixed point.

One proves this by applying a sort of infinite-dimensional Brouwer's fixed-point theorem. The clever bit comes next:

Say you want to solve the Dirichlet problem for the 2nd order quasiliniear elliptic PDE

$Qu = a^{ij}(x,u,Du)D_{ij}u + b(x,u,Du) = 0$,

and you know from more basic (Schauder) theory how to solve linear problems.Then, I define an operator $T$ by sending $v \in C^{1,\beta}(\overline{\Omega})$ to the unique solution $u$ of the linear problem

$Qu = a^{ij}(x,v,Dv)D_{ij}u + b(x,v,Dv) = 0$.

(I won't bother writing in the boundary conditions). Then a fixed point of this map is exactly a solution of the quasilinear problem! The Leray-Schauder theorem thus advocates the apriori bound philosophy: To prove the existence of a solution, you can assume it exists and then just bound it in the relevant Banach space. The task is getting the bound $\|u\|_{C^{1,\beta}(\overline{\Omega})} < M$ for solutions of

$Qu = a^{ij}(x,u,Du)D_{ij}u + \sigma b(x,u,Du) = 0$

This is in Gilbarg and Trudinger.

Answer by Spencer for Why should one still teach Riemann integration?

2011-01-22T14:57:52Z

Although not a direct answer to the question, this may be relevant to the discussion:

I learnt basic measure theory and the theory of Lebesgue integration in a course called "Probability and Measure" at Cambridge. As the title suggests, the course uses probability as the natural motivation for measure theory, which I think is a pretty good idea. As I now understand is quite different to how things are in the US, I've never taken a 'graduate real analysis'-type class. To comment on the discussion following fedja's answer, the book for the course is really the first half of David Williams' Probability with Martingales and is a good English-Language intro to measure theory and isn't too long. Unfortunately, it's called Probability with Martingales so all this isn't terribly obvious, though I've needed no other basic measure theory text and am not a probabilist myself.

The pedagogical aim would be to convince students that measure theory is a good setting for a rigorous theory of probability, which, before that stage, will only have been taught naively. There are plenty of explainable things that could help here: Why can't all sets be measurable? Why do we need countable additivity? What's the deal with "almost surely"?

The need for a rigorous theory of expectation, conditional expectation and various notions of convergence of random variables and so forth leads to the Lebesgue integral relatively naturally, if you ask me. Integrals of simple functions are basically just expectations of Bernoulli random variables... If monotonicity of measures is understood, then the monotone ,and hence dominated, convergence theorems (some of Lebesgue's main bonuses over Riemann) start seeming more natural.

Answer by Spencer for Ingenuity in mathematics

2010-12-15T18:32:16Z

Mathematicians are used to seeing these sorts of problems and are generally more intrigued by their solutions, but I came across this recently:

Colour every point in the plane either black or white.

Fix any positive distance $d$.

Can I find two identically coloured points which are a distance $d$ apart?

Answer: Of course. Look at the vertices of an equilateral triangle of side length $d$.

When is the group of homeomorphisms of a compact space locally compact?

2010-12-01T14:50:38Z

When is the group of homeomorphisms of a compact space locally compact?

I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate topology e.g. 'weak' or compact-open) is a locally compact space.

What extra conditions might we be able to put on $X$ to ensure that it is so?... What if $X$ is, say, a metric space and we ask when the isometry group is locally compact?

Answer by Spencer for Is it possible to partition $\mathbb R^3$ into unit circles?

2010-11-27T14:56:50Z

Péter's proof is very clever and, while there is no real need to resurrect this thread, the following is quite straightforward in case one is not inclined to hunt for it in the literature on this subject:

Observe that you can cover a two-punctured sphere with circles. Now consider a family of circles lying in the $xy$ plane, radii 1, centred at the points $(4k+1,0,0)$ for $k \in \mathbb{Z}$. Each sphere about the origin intersects this family in exactly two places.

Answer by Spencer for Classification of surfaces composed of circles

2010-11-27T09:38:20Z

I recently spent an unhealthy amount of time thinking about whether or not you can cover R3 with circles. Turns out you can. For your own search it might be worth looking at Szulkin, Andrzej, 'R 3 is the Union of Disjoint Circles', Amer. Math. Monthly 90 (1983) and then move on to more recent papers which cite it, to get some idea of where people went next.

I don't know how well known this is, but for those who are dying to know: Observe that you can cover a two-punctured sphere (Falls under 2. In Joseph's list). Now consider a family of circles lying in the xy plane, radii 1, centred at the points (4k+1,0,0). Each sphere about the origin intersects this family in exactly two places.

Inclusions of $C^{k,\alpha}$ spaces

2010-10-23T14:30:42Z

When is $C^{k,\alpha}(\bar{\Omega})$ a subset of $C^{k',\alpha'}(\bar{\Omega})$?

Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \alpha < k' + \alpha'$". They then give one example of a cusped domain for which $C^{1}(\bar{\Omega})$ is not a subset of $C^{\alpha}(\bar{\Omega})$, but don't elaborate further. Have the inclusions been characterized precisely in the form of a necessary and sufficient condition?

Brownian motion, martingales, Markov Chains - Rosetta Stone

2010-09-29T17:30:12Z

What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related?

I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a confusing blur, but I'm starting with a grounding in the basic theory of markov chains, martingales and Brownian motion. While I've done a fair amount of analysis, I have almost no experience in these other matters and while understanding the definitions on their own isn't too difficult, the big picture is a long way away.

I would like to gather together results and heuristics, each of which links together two or more of Brownian motion, martingales and Markov chains in some way. Answers which relate probability to real or complex analysis would also be welcome, such as "Result X about martingales is much like the basic fact Y about sequences".

The thread may go on to contain a Big List in which each answer is the posters' favourite as yet unspecified result of the form "This expression related to a markov chain is always a martingale because blah. It represents the intuitive idea that blah".

Because I know little, I can't gauge the worthiness of this question very well so apologies in advance if it is deemed untenable by the MO police.

Answer by Spencer for Freshman's definition of sin(x) ?

2010-09-27T17:50:28Z

When I was first taught analysis, I do remember things being in a slightly strange order. What annoyed me most was how we were expected to do an exercise involving say, the sine function before we had rigorously defined it. I remember thinking that we were being very careful and slow about all our elementary definitions of reals and sequences and series and yet jumping ahead and being asked whether or not $\sin\tfrac{1}{n}$ converged, but I turned out OK in the end. The point obviously was that everybody knew what sine was and what it did already and it would be silly to try to hide that fact and thinking about $\sin\tfrac{1}{n}$ would test our knowledge in a perfectly reasonable way.

When things were arrived at rigorously, the layers were added on to to our definition. e.g. Once power series are done 'properly', you can then say "here's a definition of sine". Once you point out that the theory is the same for complex numbers you could say "here's another" (the im part or exp as you say). Then once you have differentiation you can write down the ODE and say here's another.

Looking back now though, I'd say don't be too hard on yourself as a teacher: Perhaps your students needn't have all the answers at every step of the way. It's OK if you "cheat" in a couple of places, surely? They might come away once or twice feeling like you haven't told them everything but you just can't hope to. Hopefully they invest enough effort along the way that at the end they have all the different facts/definitions to reconcile.

Answer by Spencer for Moments of a distribution - any use for partial or higher moments?

2010-09-20T18:43:24Z

The typical example I remember being given at school is that of kurtosis, which uses fourth moment and is easy to understand.

Wiki says: "Higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations."

Answer by Spencer for Expressing any f(x,y) using only addition and unary functions?

2010-09-01T12:27:36Z

This is "due in successively more exact forms to Kolmogorov, Arnol'd and a succession of mathematicians ending with Kahane", to quote T.W. Korner on the subject.

I am informed that the proof I met is prepared using:

J.-P. Kahane Sur le treizieme probleme de Hilbert, le theoreme de superposition de Kolmogorov et les sommes algebriques d'arcs croissants in the conference proceedings Harmonic analysis, Iraklion 1978 Springer 1980

G. G. Lorenz, Approximation of functions Chelsea Publishing Co. 1986 (First Ed. 1966)

A. G. Vituskin On the representation of functions by superpositions and related topics in L'Enseignement Mathématique, 1977, Vol 23, pages 255-320

[This is all from these skeleton notes (no proofs) here (Links to pdf; See Chapter 1 and Chapter 11 for references)]

Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.

2010-07-17T15:07:44Z

What is the point of $\pi$-systems and $\mathcal{D}$ / Dynkin / $\lambda$-systems?

I am an analyst in the process of consolidating my measure theory knowledge before moving on to harder/newer things, having been first introduced to measure theory in a course with a probability as opposed to an analysis viewpoint. So far, everything that I've needed from elementary measure theory for analysis can be done (and is done in all of my analysis textbooks) without mention of the $\pi$-systems and $\mathcal{D}$-systems which were used in my first course. Do these set systems belong strictly to probability and not analysis? Heuristically, are they useful or important in any way? Why?

The characteristic (indicator) function of a set is not in the Sobolev space H1

2010-05-18T16:34:39Z

Is it true that the characteristic (indicator) function of a subset of Euclidean space with finite positive measure is never in the Sobolev space $H^1 = W^{1,2}$ And if so, what is the best/easiest/most elementary way to see this?

Context:

I have this on good authority (it is stated in a decent textbook). However, I have had no joy in showing this to be the case myself. Due to its placement in aforementioned textbook as the first exercise at the end of a chapter about $H^1$, it feels like it oughtn't be difficult to show, but a group of my friends and I had no luck. It is bugging me now.

[Though I am a student taking a course based on the textbook, this is not `homework'; I will not be graded on it in any way and I have attempted the problem myself.]

Answer by Spencer for What is a reasonable finitary analogue of the statement that harmonic functions are smooth?

2010-07-28T11:25:09Z

There are a few tidbits in this blog post of Tao's from a few months back. He talks about Lipschitz harmonic functions on groups (it's in the context of Gromov's theorem about finitely generated groups of polynomial growth). There is mention of 'maximum principles' and elliptic regularity but having not been through the details myself I can't say that I am confident with the analogy.

Answer by Spencer for What subfields of mathematics better lend themselves to visualization?

2010-07-06T10:17:18Z

To start with something of an anti-answer, when I took courses in algebra (not linear algebra - I mean groups, rings, modules etc.) or representation theory as an undergraduate, I found it practically impossible to get anywhere by trying to visualize what was going on. I suppose, concurrent with other answers, the important thing is that I understood the material some other way.

By contrast, and as you say, some parts of combinatorics are certainly very visual. And (obviously?) topology (not so much first-course point-set stuff, but the real stuff) and differential (at least) geometry can both be very visual subjects. It can be a lot of fun trying to find ways to use geometric inituition to attack something that is ostensibly out of reach visually (e.g. in 4 dimensions or something not embedded in $R^3$ etc.)

At the moment I'm interested in geometric analysis, where I have come across some of the most pleasingly visual things yet.

Comment by Spencer

2013-03-30T20:51:55Z

Probably my favourite answer on MO.

Comment by Spencer

2013-03-11T23:12:23Z

I'm not sure I quite appreciate exactly what is being asked. You lay out two cases and then say that what is essentially just a third case is an example of the other two not working. You say that the Sobolev embedding theorem "fails" or "goes wrong" when $k=n/p$, but one might say that it is simply neither of the two cases you lay out at the start. Nothing "fails", it just happens to be its own special case.

Comment by Spencer

2013-01-25T10:38:00Z

Sorry - so do you know how the osculating space is then defined using this polynomial map?

Comment by Spencer

2012-05-29T09:59:05Z

Aaah! Please don't spread the myth that things like top scores in Putman, Olympiad etc. are some sort of necessity for research at a high level! The infamous `many a website' is a notoriously unreliable source.

Comment by Spencer

2012-05-13T23:34:32Z

To clarify, the rest of Tilli's paper - i.e. the non-so-new stuff - looks, at a glance, to be the same as the old argument that's in Han and Lin. So it seems a perfectly good source.

Comment by Spencer

2012-05-13T23:31:45Z

The new point about this paper Mircea seems to be that they avoid the iteration altogether, by basically `differentiating' the quantity which is usually treated discretely and iterated. A proof along the lines which Connor describes in his initial answer, i.e. for pure divergence form equations, using iteration but not John-Nirenberg existed right `at the start' as it were - it is due essentially to De Giorgi himself. You can find it the book of Han and Lin, Elliptic PDE. The John-Nirenberg Lemma was used by Moser to prove his Harnack inequality, which itself was not crucial for regularity.

Comment by Spencer

2012-05-01T13:52:14Z

Did you find what you wanted?

Comment by Spencer

2011-10-25T19:40:57Z

It is true that a distribution T whose distributional laplacian is zero, $\Delta T = 0$, is actually $T = T_f$ for some smooth harmonic function $f$. What is S, sorry?

Comment by Spencer

2011-10-14T10:29:59Z

-1 While in some sense I would actually like to see a really good answer to this, I feel I must discourage this question on the grounds that (from FAQ): "MathOverflow is not the appropriate place to ask somebody to write an expository article for you" and that this is a little bit along the lines of (from how to ask): "what's the deal with algebraic geometry?". So to continue quoting from these advice pages: "The great answer you're hoping for doesn't exist because there isn't a precise question". But you could have a look at arxiv.org/abs/math/0304396

Comment by Spencer

2011-09-07T14:33:04Z

+1. Yes!; well-articulated. I do this all the time when talking with my girlfriend and she nearly always argues that the extremity of my example makes it invalid and irrelevant to the discussion, whereas I see the extremity as potentially highlighting the salient points of the general case.

Comment by Spencer

2011-08-03T19:25:19Z

You seem to be asking for suggestions without much motivation. The feeling of MO is that questions should have concrete answers unless specifically asked as big list communit wiki questions. FAQ reading is recommended, and then perhaps think of a way to ask a single question with a question mark at the end. You will already have made progress if you can do this.

Comment by Spencer

2011-07-25T12:05:54Z

Cauchy in which norm?

Comment by Spencer

2011-06-28T21:31:33Z

@Deane If you're referring to the bit I think you're referring to then I am either still very confused or have failed to get my point across again: I have a connection on $P$ = principal connection, in the sense of smooth $G$- covariant choice of horizontal subspaces. This is equivalent to a one-form on $P$ with values in the lie algebra and some other properties. Then I discuss the possibility of another, different, connection in a vectore bundle $E \to P$. I was not supposed to suggest $\omega$ actually is said connection in $E$.

Comment by Spencer

2011-06-27T18:57:01Z

@Spiro, Thanks for your comment. I think my trouble comes from the fact that from the point of view of the manifold $P$, $\omega$ is just a certain one-form with values in some vector space which happens to be $\mathfrak{g}$. I see how this object is a connection in the bundle $P \to M$ but what does this have to do with differrentiating $\mathfrak{g}$-valued forms on $P$? [@Jose Corrected, thanks.]

Comment by Spencer

2011-06-12T23:52:59Z

Yes this seems to be precisely why finding such a measure will fail. If one wishes to weight all natural numbers equally, then only the trivial `zero measure' can assign a finite measure to an infinite set.