The blog has been pretty quiet the last few weeks with the usual end-of-term business, research, and A-exams (mine is coming up quite soon). I was looking through some of my notes recently and came upon two very short Fourier analysis proofs of the isoperimetric inequality. Both proofs are among my all-time favorites; the result is of general interest (though it is subsumed in more general and useful facts), and the proofs are quick and elegant. The proofs are similar, but the second generates a Poincare inequality which is one of the fundamental tools of analysis — basically, the inequality says that for a function with a derivative, the norm of the function minus its average value (this is known as a BMO norm) is controlled by the norm of its derivative.

Anyway, here goes.

Theorem. Suppose that is a simple (sufficiently smooth) closed curve in with length and enclosing an area . Then .

Proof #1. Suppose that is a parametrization for . By scaling, we may assume that — scaling points to changes arclength by a factor of (by the arclength formula) and the area by a factor of (by linear algebra). Furthermore, we shall parametrize by arclength so that for every .

Let and denote the coordinate functions for , i.e. Observe that both are periodic, and so we may think of them as living on the circle. We can then compute some Fourier series:

Since both and are real-valued functions, we know, for instance, that , and so Parseval’s identity says that

We know the arclength formula is given by

Since , we have that

Hence

Green’s Theorem tells us that

Thinking of as the inner product between and , using the fact that and are real-valued and hence , and the fact that integrates to zero unless , we see that

But we now have the foolish inequalities

This gives

and hence

The fact that the circle is the only curve for which equality between the two is possible follows by the above with the facts that when , some high school algebra, and knowing what a circle is.

Proof #2. Another fairly easy way to prove the isoperimetric inequality is to use a Poincare inequality:

where denotes . Since gives the constant term for the Fourier series of , this Poincare inequality follows immediately by the Fourier series formula for the derivative, since

In fact, since the terms coincide exactly, one sees that

Hence the only way one can get equality is if for all . To get the isoperimetric inequality, we use Green’s Theorem again, to get a slightly different (but more familiar) area formula:

Assume that the orientation of our parametrization makes the inside of the absolute value positive. Use this formula and the fact that to deduce that

Using the fact that , we get that

The last inequality follows since we’re dropping a negative term. Applying the Poincare inequality, we get that

Thus, again, The argument that equality only holds for circles is just as easy as before.

    I tend to think of homotopy theory a little bit like ‘The One That Got Away’ from mathematics as a whole.  Its full of wistful fantasies about how awesome it would have been if things could only have worked out.  Imagine if homotopy groups of spaces and homotopy classes of maps were as easy to compute as homology groups… we’d be teaching undergrads about Postnikov towers and topology might very well end up a subset of group theory.

   Instead, homotopy theory is a hopeless, incalculable mess in all but the trivial cases… that bitch.  The canonical example here is the Toda’s Table of the Homotopy Groups of Spheres.  That’s right, even the simplest imaginable case - homotopy classes of maps between spheres - is a wildly unpredictable mess with only a handful of a structure theorems.

   So homology and cohomology theories rule the day; not as powerful as homotopy groups, but infinitely more tractable.  However, recently I’ve become somewhat enamored of a weaker form of homotopy which is just weak enough where you can actually say things: Rational Homotopy Theory.  The general idea is to simply ignore any information coming from torsion homotopy groups.  After all, all the hideousness in Toda’s table is finite groups; we know the infinite homotopy groups, and they represent reasonably interesting phenomena.

    The main upshot of this is that all the information of a space (up to rational homotopy) can be packaged in a differential graded algebra.  Rational homotopy equivalence becomes quasi-isomorphisms, and so the question of whether two spaces are rational homotopic is very reasonable.  With some mild restrictions, it can be shown that spaces up to rational homotopy are the same as DGAs up to quasi-isomorphism.  This opens the door for recasting much of topology as purely algebraic constructions.

    For the sake of concreteness, whenever I say ’space’ in this post, I mean connected CW complex. 

    We want to think about passing from the category of spaces to the homotopy category of spaces, but we would like to be as lazy as possible.  We could identify any pair of maps which are homotopic, but it turns out we can get away with only identifying pairs of maps which are homotopic and are homotopy equivalences.  To see this, notice that any homotopic maps factor through the maps , and and are homotopic and homotopy equivalences, so and also get identified.

    This may seem like a rather silly point to make, but the advantage is that the latter class of maps is easier to characterize algebraically.

Whitehead’s Theorem. A map between spaces and is a homotopy equivalence if it induces an isomorphism on all homotopy groups.  Furthermore, two such maps are homotopic if and only if they induce the same isomorphism.

Hence, to form the homotopy category, we can identify maps which induce the same isomorphism on homotopy groups.  This is usually done by adding arrows to the category which are formal inverses to such classes of maps.

    This leads naturally to the definition of rational homotopy.  Given a space , we can throw away all the torsion information in by tensoring with ; ie, looking at the rational homotopy group .  Inspired by Whitehead’s theorem, we say a map is a rational homotopy equivalence if it induces an isomorphism on all rational homotopy groups, and two such maps are rationally homotopic if they induce the same isomorphism.  Thus, we can form the rational homotopy category by adding formal inverses to rational homotopy equivalences.

    This is a coarser equivalence relation on spaces than you might think.  To get a sense for what is going on here, think about any finite-sheeted covering map .  This induces an isomorphism on all , and a surjection on with finite order kernel.  Therefore, it is an isomorphism on all rational homotopy groups and hence a rational homotopy equivalence.

    So rational homotopy is blind to quotienting by the free action of a finite group.  This may seem weird, but in some ways it is desirable.  Take, for instance, the study of hyperbolic Riemann surfaces.  By the Riemann mapping theorem, these are all quotients of the hyperbolic disc by some subgroup of (a Fuchsian group).  A useful relationship between two Fuchsian groups is that of commensurablity, that is, having intersection which is cofinite in each group.  By the same argument as the previous paragraph, two Riemann surfaces with commensurable Fuchsian groups are rational homotopy equivalent (and the converse is also true).

   Ok, so this is all well and good, except that homotopy groups are hard to calculate, and we don’t really know how much easier will be to compute.  Thankfully, there is a theorem of Whitehead and Serre’s which allows us to avoid this entirely:

Theorem (Whitehead-Serre).  A map induces an isomorphism on all rational homotopy groups (ie, is a rational homotopy equivalence) if and only if the map induced on all rational cohomology groups is an isomorphism.

Therefore, we can go back and redefine the rational homotopy category by inverting morphisms which induce an isomorphism on all rational cohomology groups.

    Here’s where the algebra comes in.  To any space  we can assign its cochain complex .  This determines a functor from the category of spaces to the category of differential graded algebras; specifically DGAs which are commutative and have no negative-degree terms.  Let’s call these DGAs topological, and we can endow them with an equivalence relation by inverting maps which induce isomorphisms on cohomology (quasi-isomorphisms). 

    Then the rational homotopy category of spaces maps into the category of topological DGAs modulo quasi-isomorphism.  What is even better is that if we restrict to simply connected, finite dimensional spaces , and topological DGAs which are finite dimensional in each degree and have zero first cohomology, then this functor is an equivalence of categories.

    This allows us to abandon spaces entirely and work with DGAs.  I find this personally very satisfying, since I already know several circumstances where it is useful to think of DGAs as almost like spaces, and this theorem lets me know exactly what is lost when thinking like this (its the information inaccessible to rational homotopy theory).

    This also transitions into one of the most fun aspects of math (at least for me personally), which is taking an equivalence and seeing what natural constructions on one side look like on the other side.  Topology is full of natural constructions like various topological products, suspensions, loop spaces, classifying spaces, fiber bundles, etc…  Each of these becomes an interesting construction on DGAs which might otherwise seem bizarre and unmotivated.  If the mood strikes me, I might write a follow-up post to these outlining some of these constructions.

    The blog’s been rather quiet lately, due in a large part to me being in research mode right now.  That also explains why my posts, when they occur, are mostly advanced.  I just don’t have much general-consumption math on the brain at the moment.

    Today, I’d like to talk about some of the more basic things in the subjects I tend to work (as a compromise).  As you might recall, an abelian category is a category where the set of morphisms between any two objects, , is an abelian group; with some additional properties that make it nice enough to do things like homological algebra.  The classic example of an abelian category is , the category of finitely generated left modules of some ring .

    One can ask the question: how far is an arbitrary abelian category from being a module category?   One result in this direction worth knowing is the Freyd-Mitchell Embedding Theorem, which says that any abelian category has a fully faithful, exact embedding into some module category.  The principle use of this theorem is to make homological algebra proofs which assume the existance of ‘elements’ work.

    But, how can we tell if an abelian category is equivalent to a module category?  As we will see, finding such an equivalence is the same as finding a sufficiently nice element in the category, called a compact progenerator.  It can also be interesting to find multiple progenerators, giving us non-trivial equivalences , where we call and ‘Morita equivalent’.

    We should start by considering a module category, and thinking about what we know aside from that it is an abelian category.  The first thing that comes to mind is that has an object , thought of as a left module over itself.  One thing we can do with is to look at maps from it into any other object, .  Of course, this is isomorphic to itself, with the left module structure induced by the right module structure on (being the first argument in a is dualizing, and so it swaps left and right actions). 

    Ok, so this is just the identity functor on , but now let’s pretend we are ignorant of the fact that our category is .  Picking this object still determines a functor , but where is this functor landing?  Well, always has a left action by (which is the ring ), given by composition.  Therefore, this functor maps to a concretely realized module category , and we know by construction it is an equivalence.

    Next, let’s take any abelian category .  If it is a module category, then there is some special object inside such that the “Hom out of” functor is an equivalence.  Therefore, let’s think about for an arbitrary object .  It determines a functor to , and we want to know if there is an inverse equivalence going the other way.  However, our functor does have a left adjoint , and we know that if a functor has both an adjoint and an inverse, they must coincide.  Therefore, it suffices to check if is an inverse to .

    Now, let’s assume that and are equivalences of categories.  Since they must be exact functors, is projective as both an object in and a right  module. 

    We also know that every object in is isomorphic to one of the form , for some module .  If we replace with a free resolution , then we get that is a resolution of .  Hey, then every object in has a resolution by sums of copies of !  We say that generates the category .  A projective generator is often called a progenerator.

    Technically, we have made the mild assumption that even for infinite , but this only needs that is a finitely presented module.  This requirement on is called being compact.

    So what have we found so far?  A compact object for which is an equivalence of categories is a progenerator.  Ah, but it’s not too hard to see that this is a sharp characterization.  This fact is usually called ‘Morita’s Theorem’:

Morita’s Theorem.  Let  be an object in an abelian category , and let .  Then is an equivalence of categories between and with inverse functor if and only if is a compact projective generator (’progenerator’) in .  Furthermore, every equivalence between and a category of modules arises in this way.

    We haven’t completely justified every statement here, but we have given the rough outline.  To see why the last statement is true, that every equivalence arises in this way, assume the existance of the equivalence and let be the image of the ring .

    We can reap some immediate fruits from this theorem.  Let’s say we have a module category , and we want to know what other module categories it is equivalent to.  This is a strong relationship between and called Morita equivalence.  We now know that and are Morita equivalent if and only if there exists a bimodule such that it is a compact progenerator of and .  This first condition is the same as being finitely presented, projective and being a direct summand of for some .

    Well, we can think of some examples of these right away.  , for instance, most certainly satisfies all the above requirements.  is , the ring of matrices with coefficients in .  Thus, we have established the equivalence of categories between and .   and are the classic example of Morita equivalent rings, so this is comforting.

    This idea, in particular the way I’ve presented it, beg to be generalized to derived categories.  After all, we replaced an element in with a free resolution to see where it went in .  If we work with derived categories, we no longer need resolutions to go to resolutions, they can go to complexes; therefore, we no longer need projectivity.

    In the context of derived categories, this process of finding an object which defines an equivalence to a derived module category is called ’tilting’, with the object called a ’tilting sheaf’.  The biggest success story in this direction is perhaps the Beilinson equivalence, which reveals the surprising fact that is equivalent to , for a relatively simple quiver .

   I recently came across an excellent survey article, “Morse Theory Indomitable“, by Raoul Bott.  It starts with the basic history of Morse functions, and covers the additions of Smale and Witten, and the connections to symplectic reduction.  Though, even moreso than being a clear and concise overview of some beautiful mathematics, it is all liberally dosed with personal anecdotes from the life of someone who lived throughout virtually the entire story.  It was a throughly enjoyable read, even though I had seen almost all the contained math before.

    Something has been bothering me about Koszul duality lately.  Well, technically many things have been bothering me about it, but here’s a particular thing that has been bothering me.   Usual (homological) Koszul duality assigns to an augmented -dga the algebra .  Since is central, is again an augmented -dga, and so makes sense.  In many (all?) cases, (hence, duality).

    I am interested in the case when is no longer central in .  Then an action of on no longer is the same as an augmentation map, so we only assume we have a left -module structure on (where the contained copy of acts by mutliplication).  We can still form in the same way; however, no longer contains , so it seems impossible to form .

    However, I have reason to suspect that one should be able to form , and that it should often be quasi-isomorphic to .  My evidence in this direction is a paper of Positsel’skii’s, “Nonhomogeneous Quadratic Duality and Curvature”, wherein he shows that for a very narrow class of algebras which both contain and act on , that can be recovered from .

     An important case of this duality is for the ring of differential operators on a smooth curve.  This has a canonical left action on , and so exists, and is in fact the deRham complex.  So, a specific case of my question is: is there a sense in which the deRham complex of a space acts on , such that something like ?

    More generally, one can replace with the universal enveloping algebra  of any Lie algebroid over .  can still be defined, and is the complex which computes the Lie algebroid cohomology.

   Both of these examples do fall under Positsel’skii’s explicit duality.  The problem is that the framework of this duality isn’t self-contained.  It takes filtered algebras which aren’t too far from being quadratic algebras, and sends them to commutative dgas whose underlying graded algebra is quadratic.  Then there is a different construction which takes quadratic commutative dgas to filtered algebras which are almost quadratic, and these two functors induce an equivalence of categories.

    What I want, though, is for these two dualities to be facets of the same duality.  Maybe its not possible, but I imagine that people have thought about this, at least in the specific case of Lie algebroids.

    Its now been several days since the conference at IAS, and I might as well do a quick wrap up.  Overall, the conference was great.  Almost all the talks were understandable, and contained new mathematics of a truly humbling quality.  There was a surprising collection of internet mathies: Ben Webster, Joel Kamnitzer, Peter Woit, Charles Siegel, Aaron Bergman and David Ben-Zvi (and maybe even some I’m forgetting or didn’t notice).  Also, I didn’t actually play Bad Talk Bingo; I had more than enough of my own math to do during downtime and bad talks.  It was standing room only for virtually every talk (though that was at least in part because we were in a small room when a big auditorium was being unused).  I even sat in the aisle next to Deligne for the first talk.

    One downside is that it was a pretty intimidating atmosphere for asking questions.  Usually, I’m pretty good about asking potentially stupid questions, but I wasn’t confident in my knowledge of the ‘basics’.  I asked some of the speakers my questions after the talks, and I tried to write them all down for further contemplation.  I figured I’d put ‘em here, both for my own reference and in case anyone out there knows the answer.  I’ve included the name of the talk the question is from; but in some cases, these are unrelated questions I was thinking about.

• Vezzosi - One theme he emphasized was that, in derived algebraic geometric, the information carried in the higher homotopy groups is like a ‘nilpotent’ part of the scheme.  My general question is: how far can one take this analogy?  Specifically, given a (non-derived) scheme , can one construct the category of derived schemes whose is (similar to Artinian schemes over )?  Can one define a derived version of ‘pro-representability’ for this category?  Presumably, this requires looking at the ind-completion of this category, which would be the derived analog of a formal scheme?  Since moduli spaces of schemes are often more naturally derived schemes than schemes, I’m wondering if ‘derived pro-representability’ is somehow nicer than pro-representability.  I asked Vezzosi about it after his talk, and he seemed to affirm most of these questions, saying that Jacob Lurie has addressed them.
• Costello - He mentioned in passing that any time we have a complex/dga (whose differential is of degree +1), and we equip it with a square-zero derivation which commutes with and is of degree -1, that this amounts to giving the complex/dga a circle action.  I understand why giving something a grading gives it a circle action, but not for a mixing differential.
• Okounkov - He mentioned the quantum Calogero-Moser equation (I think).  I know and care enough about the classical CM equation that I am curious as to what it looks like.  Anybody know a resource?
• Seidel - He took with marked points, and the branched double cover over those points.  By connecting the marked points into a tree, and looking at the pre-image, he got a sub-locus whose Floer homology generated the Floer homology of the total space.  The multiplication here corresponded to thinking of this doubled tree as a quiver and giving is some relations.  This whole process was eerily reminiscent of ‘doubling a quiver’, even though the relations seemed a bit different.  Is there any relation? (Warning: my comprehension of this talk wasn’t superb, so there might have been some incorrect facts above)
• Keller - What do Quiver Mutations correspond to?  He showed us neat facts about them, and told us that other people do care.  Why do they care?  What does a mutation correspond to?  Also, he wrote down an algebra of Ginzburg’s associated to a quiver with superpotential.  It was very close to the preprojective algebra of the quiver (I believe they coincide when the superpotential is zero).  Is there anyway to deform this algebra, so that it corresponds to deformed preprojective algebras?
• Lurie - I would have liked to see some examples of Koszul duality for -algebras.  In particular, he never mentioned what the Koszul dual of the motivating example, , was.  Well, presumably we would need to make it an algebra first, but is almost certainly an -coalgebra, and so has a dual.  I believe someone asked a question to this effect, and Lurie answered ‘its just the de-looping’, which doesn’t make much sense to me.
• Me - If I have a Lie groupoid which is sufficiently nice, how do I construct a classifying space for it?  I’m interested in equivariant cohomology for Lie groupoids.  Is there a Weil algebra for a nice Lie groupoid?
• Me - Is there a good resource for the meta-theory of the ‘field with 1 element’?  I’ve read several posts of John Baez on the subject, and I think its neat enough that I’d like to give a graduate colloquium on it.  However, I’d like a more complete resource before I talk, and I haven’t found one.

I’m sure I had more questions, but I didn’t think to start writing them down til the fourth day.  Anyway, thanks again to the speakers for a delightful conference.

    A fun little math problem today.  It came about when I misheard a problem asked by a friend, and solved a slightly different problem.  It goes as follows:

    A child watches television everyday.  He always watches at least one hour a day, and he only watches tv in whole integer amounts of hours.  Concerned for his well-being, his parents impose a restriction: he can never watch more than 11 hours of tv in any 7 day period.  Show that there is some consecutive string of days in which the child watches exactly 20 hours of tv.

The number 20 is a relic of the problem that I misheard (show that this happens in any 11 week period).  Of course, why stop there?  Show that for any positive integer , there is some string of consecutive days in which he watches exact hours of tv.

    This begs an interest question: for what other ‘restrictions’ is this property true?  That is, what other numbers and have the property that if the child can only watch hours of tv in a day span, then he achieves every positive integer as a consectutive total?

    I’ll review my proof of the original problem for any number first.  We have the list of hours the child watches each day, and we want to find a consecutive substring whose total in . First, note that the child can never watch more than 5 hours of tv in a day.  Otherwise, he couldn’t watch 1 hour each of the next 6 days and still stay under 11.

    Next, let us proceed in the most naive possible way.  We pick some starting day, and we keep a running total of the amount of tv watched since that day.  We stop once the total is bigger than or equal to .  However, since the previous day’s total was less than , and the child can’t watch more than 5 hours in any day, we know that this final total is between and .  We could try to fix this overage by removing some of the first days, but in general we know nothing about them.

    So now lets suppose that there is some string of 4 days in which the child watches only 1 hour each day.  If we start counting on the first of these days, then no matter what the final total is, we can cut off some of the first days and get a total of exactly . 

    Of course, theres no reason to suspect that we can find four 1’s in a row.  We can still use arguments of this type in the following way.  If we have a string of 1’s of some length, we can start counting at the first one.  Either the total is close enough to that we can fix it by cutting off some days, or the last day counted had a large number on it.  However, if there is a single day with alot of tv, then all the surrounding days must have very small amounts in them; in particular, there will be a chunk of 1’s in a row.  We can put all this together into a kind of ‘reverse’ induction.

• If there are 4 ones in a row, its solved (by the above paragraph).
• If there are 3 ones in a row, either the final total is less than or equal to (in which case some of the first days can be removed) or the final total is .  This means that the child watched 5 hours of tv on the last day.  However, if he watched 5 hours of tv on some day, then the next 6 days must all be ones.
• If there are 2 ones in a row, either its solvable or the last day is at least 4 hours.  This means the next 6 days can have at most 7 hours of tv.  Therefore, there is at least one string of 3 ones in a row.
• If there a single one, then either is solvable or the last day is at least 3 hours.  This means the next 6 days can have at most 8 hours of tv.  Therefore, there is at least one string of 2 ones in a row.
• There must be at least one day the child watches exact one hour of tv.

Therefore, we can wind through the induction and always find some string of total exactly . 

    One annoying thing about this proof is that it seems pretty inefficient, in terms of bounding the number of days one needs to check.  A quick estimate shows that one needs to check days before finding a sum up to .  This is because for each starting day, we need to figure out the maximum number of days before the total is above .  However, if this takes a long time, it means lots of the intervening days were ones, and we don’t need to work so hard to force there to be lots of ones in a row.  Therefore, its clear that we can bring down this bound alot, but it would require alot of special arguments.  In particular, I doubt a method of this form could be effective for computing sharp bounds.

    So the next logical question is: what other pairs of numbers have this property?  Certainly, I have no idea how to answer the question in general, but we can see for which numbers does the above proof work.

    Let be the number of hours of tv the kid is allowed to watch in any day period, for fixed integers and .  Also, let , the maximum number of hours the kid can watch in any day.

    So now lets assume we can find ones in a row.  Then starting with the first one, we get some total which we can fix as long as it is less than or equal to .  If the total is more than , then the final day counted must be at least .  This means that the next days have only hours maximum.  After assigning the minimum one hour to each day, there are only hours left to play with.  This forces there to be at least ones in a row somewhere (where is rounds the fraction up).

     As long as , then the reverse induction from before will still work (because it will always force increasingly large strings of ones).  Moving terms around gives .  Since the coefficient of is positive and the vertex of the parabola is at , this will only be true for all positive if the discriminant is negative.  This means that .  Heuristically, this means that as the number of days in the restriction gets larger, the number of ‘extra’ hours can only increase like .

    Of course, I’m sure that this property is true for all sorts of that this method doesn’t show.  Can anyone think of examples?

    Next week I am off to IAS for the conference on Algebro-Geometric Derived Categories and their Applications.  I am starting to get worried about it, since the list of talk titles is up and only a few of them appear to be welcoming to non-experts in their respective fields.  Also, when registering, there was no option under ‘Occupation’ for ‘Grad Student’, which struck me as a subtle hint that I wasn’t welcome.  Perhaps just being at IAS will compel speakers to turn the difficulty up to eleven.

    In any event, I am preparing for there to be a fair share of bad talks.  This, and other recent experiences with bad talks has given me the following idea of something to do: Bad Talk Bingo.  The idea is to create a 5×5 grid of events and signs that a talk is going badly, so that I can check off which ones happen and try to get 5 in a row.  My plan is that if I ever succeed in getting 5 in a row, I am allowed to pretend to get a phone call and rush hurriedly from the talk.

    However, to do this, I need at least 25 possible signs that a talk is going badly.  Ideally, I want many more, so that I can write a program to randomly generate a 5×5 grid, to make the game fun and exciting each time.  This is where you, the reader come in!  I want more ideas, and I’m sure everyone with any experience in math has seen their share of doomed talks.

    Below is the list so far of possible topics, separated into Keepers, Not Bad, and Would Like Better ideas.  Also, when a word or phrase appears in quotation marks, it means the speaker uses that word or phrase.

Keepers:

• Poor English speaker
• “quantum”
• Slides
• “Ricci”
• Trouble with laptop
• Went over allotted time
• Ignored a reasonable complaint
• Name dropping
• Gratuitous use of indices
• Screwed up a computation

Not Bad:

• Important theorem has hidden assumptions
• Erased the last thing written
• “trivial”
• “string”
• Wrote out unnecessary technical details
• Didn’t write down important theorem
• Speaker was late
• Referenced a ‘folklore result’ that is ‘known to experts’
• Topic was TBA
• Inaudible
• Talked facing the blackboard
• Construction that wasn’t functorial

Would Like Better:

• Annoying noises with chalk
• Wrote out long list of collaborators
• Wrote subjects of math and drew arrows between them
• Hand gestures instead of explanation
• Various hygiene concerns
• “categorify”

If possible, I’d like five general categories of topic, so that each column could have a theme.  These might be ‘Buzzwords’, ‘Apathy’, ‘Intentional Obfuscation’, ‘Technical Shortcomings’, etc.  I want more than 25 good ones, so that I can randomly pick from them to make many sheets.  Also, this is all in good fun, so please try to stay away from pointed criticisms of specific people and things the speaker might not have any control over (like speech impediments). 

I was motivated by a comment on Jim Pivarski’s recent post to speak about the Heisenberg Uncertainty Principle. Someone asked,

If uncertainty in quantum mechanics comes from (or is inseparable from) quantization, then where does it come from in its mathematical formulation i.e in terms of a space and its Fourier transform?

The Heisenberg Uncertainty Principle is a curious fact: it requires no physical intuition whatsoever and yet has profound physical ramifications. It is also interesting because it is among a small group of facts which are both physically and mathematically interesting. It is an important (to harmonic analysis) and commonly known fact that a function and its Fourier transform cannot both be compactly supported. There are stronger statements than that, though, of the following flavor: if a function is a narrow spike near zero, then its Fourier transform will be a shorter and fatter bump around zero and vice versa. The Heisenberg Uncertainty Principle is a quantitative statement about this kind of fact.

The Principle follows from several simple but fundamentally powerful aspects of the Fourier transform. First, polynomials in derivatives acting on a function can be pulled outside the Fourier transform into corresponding polynomials in frequency variables. More specifically,

where is a polynomial and means derivative with respect to Second, translation in space variables leads to modulation in frequency variables:

Third, modulation in space variables leads to translation in frequency variables:

Now, let’s derive the Uncertainty Principle. Let denote the functions which are square integrable — that is, We will write

On , define the operators and by

— position.

— momentum.

Here is Planck’s constant which is on the order of in the macroscopic units of Joules-seconds. The presence of is unimportant mathematically, so let’s assume we’re using units so that is 1. That corresponds to the position operator of a particle comes from the interpretation of the wave function: if is a solution to the Schrödinger equation, then it is a probability distribution — or rather, is — so that gives the expected value for position. That corresponds to the momentum operator basically comes from the fact that it’s like the velocity of the wave function, and momentum is mass times velocity. More precisely, the Schrödinger equation,

is nothing more than a statement about conservation of energy. The kinetic energy is which corresponds to the first term, and so our choice for the momentum operator reflects that.

One should note here that, as indicated above, taking the Fourier transform of momentum gives the position operator in the frequency variable:

The Fourier transform of position is basically the same thing as momentum, but it’s off by a constant multiple.

Observe that and do not commute; specifically,

We know that has as an inner product . Also, it’s pretty easy to see that both and are self-adjoint, meaning that and likewise for . Using these two facts, we find that for arbitrary real constants and ,

The first quantity is non-negative which means the last quantity is also, and so:

Picking and gives

For any fixed pair of numbers and , we can apply the above relation to and deduce that

The second term in the product on the left side of above comes from a fact called Plancherel’s formula:

For any wave function , we must have that since must be a probability distribution and thus have a total mass of 1. By choosing

the expected position, and

the expected momentum, we get that

where denotes variance — the variance is the square of the standard deviation and so for an arbitrary distribution the square root of variance is the standard way to measure deviation from the mean.

More commonly, the above relation is written as

By carefully retracing how Planck’s constant fits in, one can see that the physically relevant equation is

From a physical point of view, this means that our confidence in a measurement of position is at best inversely proportional to our confidence in a measurement of momentum. This gives us the usual qualitative interpretation of the Uncertainty Principle: one cannot simultaneously know with perfect certainty both position and momentum, or, as Heisenberg himself said, “The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.”

    I have always been fascinated with the study of how people make decisions.  Its a complicated process that involves as much ‘gut instinct’ as rational evaluation, and is rife with systematic errors in judgement.  For a long list of common mistakes, check out Wikipedia’s list of cognitive biases, particularly the decision-making section.  The majority of them are things that most people are probably aware of, like picking a option just because other people picked it.  However, a couple of them are a bit surprising, especially the one I want to talk about today: the phenomenon of hyperbolic discounting.

    The core question is how people penalize various options for having a ‘delayed payoff’.  Would you rather have $30 now and $50 in five years?  Implicit in the decision-making process is that later payoffs aren’t worth as much.  You might need the money now more than later, or there’s a risk you won’t get the money later, due to death/bankrupcty of the source/etc.  The drop in value of a payoff due to the delay involved is called the ‘delay discount’.

     How might one rationally evaluate this discount?  Well, virtually every reason you can name for de-valuing later payoffs is due to some source of roughly constant risk.  Therefore, the value of an object should decay exponentially with time, at a rate determined by the amount of risk.  However, this is not what people usually do!  Studies have shown that people discount delayed payoffs hyperbolically; that is, roughly proportional to the inverse of the delay.  Specifically, a payoff of value  becomes one of value if it is delayed for time , where is a constant that determines roughly how ‘risky’ the delay is.

    This has two main consequences.  The first is that we tend