ag.algebraic geometry - Groebner basis with group action

At one point my advisor, Mark Haiman, mentioned that it would be nice if there was a way to compute Groebner bases that takes into account a group action.

Does anyone know of any work done along these lines?

For example, suppose a general linear group $G$ acts on a polynomial ring $R$ and we have an ideal $I$ invariant under the group action. Suppose we have a Groebner basis $B$ of $I$. Then we can form the set $G(B) := { G(b) : b in B }$. Perhaps we also wish to form the set
$$IG(B) := { V : V text{ is an irreducible summand of } W, text{ for some }W in G(B) }$$
(note that $G(b)$ cyclic implies it has a multiplicity-free decomposition into irreducibles).

Can we find a condition on a set of $G$-modules (resp. $G$-irreducibles), analogous to Buchberger's S-pair criterion, that guarantees that this set is of the form $G(B)$ (resp. $IG(B)$) for some Groebner basis $B$?

Can the character of $R /I$ be determined from the set $IG(B)$ in a similar way to how the Hilbert series of $R /I$ can be determined from $B$?

soft question - books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is that a lot of math books lack the motivating examples or only provide very contrived ones stuck in between pages of definitions and theorems. Reading such books becomes a huge chore for me, even in areas in which I am interested. Besides I am certain no mathematical field was invented by someone coming up with a definition out of thin air and proving theorems with it (that is to say I know the good motivating examples are out there).

Can anyone recommend some graduate level books where the presentation is well-motivated with explicit examples. Any area will do, but the more abstract the field is, the better. I am sure there are tons of combinatorics books that match my description, but I am curious about the "heavier" fields. I don't want this to turn into discussion about the merits of this approach to math (i know Grothendieck would disapprove), just want to learn the names of some more books to take a look at them.

Please post one book per answer so other people can vote on it alone. I will start:

Fourier Analysis on Finite Groups and Applications by Terras

PS. this is a similar thread, but the main question is different.
How to sufficiently motivate organization of proofs in math books

co.combinatorics - The Matrix Tree Theorem for Weighted Graphs.

For signed graphs we have an interesting matrix-tree theorem. A signed graph is a graph with the additional structure of edge signs (weights) being either +1 or -1. We say that a cycle in a signed graph is balanced if the product of the edges in the cycle is +1. A signed graph is balanced if all of its cycles are balanced. Otherise, we say a signed graph is unbalanced.

We say a subgraph $H$ of a connected signed graph $G$ is an essential spanning tree of $Gamma$ if either

1) $Gamma$ is balanced and $H$ is a spanning tree of $G$, or

2) $Gamma$ is unbalanced, $H$ is a spanning subgraph and every component of $H$ is a unicyclic graph with the unique cycle having sign -1.

The matrix-tree theorem for signed graphs is stated as follows:

Let $G$ be a connected signed graph with $N$ vertices and let $b_k$ be the number of essential spanning subgraphs which contain $k$ negative cycles. Then

$$ det(L(G))=sum_{k=0}^n 4^k b_k.$$

For some references see:

T. Zaslavsky, Signed Graphs, Discrete Appl. Math, 4 (1982), 47-74.

S. Chaiken, A combinatorial proof of the all minors matrix tree theorem. SIAM J. Algebraic Discrete Methods, 3 (1982), 319-329.

Signed graphs are used in spin glass theory and network applications.

Considering mixed graphs, which are directed graphs that have some edges as undirected, we actually get the same theory. This is immediate since the matrix definitions are identical for signed graphs and mixed graphs. This seems to escape many of the people studying matrix properties of mixed graphs however. See:

R. Bapat, J. Grossman and D. Kilkarni, Generalized Matrix Tree Theorem for Mixed Graphs, Lin. and Mult. Lin. Algebra, 46 (1999), 299-312.

co.combinatorics - Tantrix from combinatorial viewpoint

This question is about the popular logic game called Tantrix. I would like to collect combinatorial theorems about it, eg. necessary conditions for making a cycle of one color from a given set of tiles that passes through all the tiles and the union of tiles has no holes in it. On the web I could only find theorems about the complexity of the game which is a completely different question. To show what kind of theorems I want, here are three easy observations.

Theorem Trivial. If there is a red cycle using all the tiles, then red must appear on all the tiles.

Theorem Crossing Parity. If there is a red cycle using all the tiles, then the number of red-blue crossings must be even.

Theorem Winding. Count straight red tiles 0 (can be denoted by I), big turns (which are almost straight, can be denoted by L) as 1 and small turns (when red touches adjacent sides, can be denoted by V) as 2. If there is a red cycle using all the tiles, then it must be possible to assign a sign to each tile such that the sum is 6.

I would be also interested in configurations that satisfy these theorems but it is still impossible to make a cycle, like 3 Vs and a non-zero number of Is.

triangulated categories - Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories.

The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large commutative diagram is particularly quizzical. While I am ok with understanding these axioms I was hoping to ask two questions about them.

1) What was the classical motivation for these axioms? Was there a particular example in mind to conform to?

and

2) Is there a modern motivating example for these axioms that differs from the classical?

I understand these things much better when I have specific examples to keep in mind, and since I am learning these in a general context, right now that is lacking. I was hoping you all could fill me in.

Thanks in advance!

soft question - Fundamental Examples

to understand curves, first study the abel map. and then the torelli map. [perhaps I should expand this rather succinct answer.]

8320 Spring 2010, day one Introduction to Riemann Surfaces

We will describe how Riemann used topology and complex analysis to study algebraic
curves over the complex numbers. [The main tools and results have analogs in
arithmetic, which I hope are more easily understood after seeing the original versions.]

The idea is that an algebraic curve C, say in the plane, is the image by a holomorphic
map, of an abstract complex manifold, the Riemann surface X of the curve, where X has
an intrinsic complex structure independent of its representation in the plane. Then Riemann's approach to classifying all complex curves, is to classify such Riemann surfaces, and then for each such surface to classify all maps from it to projective space. Briefly, the Torelli maps classifies complex surfaces, and the abel maps classify all projective models of a given complex surface.

More precisely, we will construct two fundamental functors of an algebraic curve:

i) the Riemann surface X, and

ii) the Jacobian variety J(X), and

natural transformations X^(d)--->J(X), the Abel maps, from the “symmetric powers” X^(d) of X, to J(X).

The Riemann surface X

The first construction is the Riemann surface of a plane curve:
{irreducible plane curves C: f(x,y)=0} ---> {compact Riemann surfaces X}.

The first step is to compactify the affine curve C: f(x,y) =0 in A^2, the affine complex
plane, by taking its closure in the complex projective plane P^2. Then one separates
intersection points of C to obtain a smooth compact surface X. X inherits a complex
structure from the coordinate functions of the plane. If f is an irreducible polynomial, X
will be connected. Then X will have a topological genus g, and a complex structure, and
will be equipped with a holomorphic map ƒ:X--->C of degree one, i.e. ƒ will be an
isomorphism except over points where the curve C is not smooth, e.g. where C crosses
itself or has a pinch.

This analytic version X of the curve C retains algebraic information about C, e.g. the field
M(X) of meromorphic functions on X is isomorphic to the field Rat(C) of rational
functions on C, the quotient field k[x,y]/(f), where k = complex number field. It turns out
that two curves have isomorphic Riemann surfaces if and only if their fields of rational
functions are isomorphic, if and only if the curves are equivalent under maps defined by
mutually inverse pairs of rational functions.

Since the map X--->C is determined by the functions (x,y) on X, which generate the field Rat(C), classifying algebraic curves up to “birational equivalence” becomes the question of classifying these function fields, and classifying pairs of generators for each field, but Riemann’s approach to this algebraic problem will be topological/analytic.

We already can deduce that two curves cannot be birationally equivalent unless their Riemann surfaces have the same genus. This solves the problem that interested the Bernoullis as to why most integrals of form dx/sqrt(cubic in x) cannot be “rationalized” by rational substitutions. I.e. only curves of genus zero can be so rationalized and y^2 = (cubic in x) usually has positive genus.

The symmetric powers X^(d)

To recover C, we seek to encode the map ƒ:X--->C, i.e. ƒ:X--->P^2, by intrinsic
geometric data on X. If the polynomial f defining C has degree d, then each line L in the
plane P^2 meets C in d points, counted properly. Thus we get an unordered d tuple of
points L.C, possibly with repetitions, on C, hence when pulled back via ƒ, we get such a d
tuple called a positive “divisor” D = ƒ^(-1)(L) of degree d on X. (D = n1p1+...nk pk,
where nj are positive integers, n1+...nk = d.)

Since lines L in the plane move in a linear space dual to the plane, and (if d ≥ 2) each line is spanned by the points where it meets C, we get an injection P^2*--->{unordered d tuples of points of X}, taking L to ƒ^(-1)(L).

If X^d is the d - fold Cartesian product of X, and Sym(d) is the symmetric group of
permutations of d objects, and we define X^(d) = X^d/Sym(d) = the “symmetric product”
of X, d times, then the symmetric product X^(d) parametrizes unordered d tuples, and
inherits a complex structure as well.

Thus the map ƒ:X--->C yields a holomorphic injection P^2*--->Π of the projective plane into X^(d). I.e. the map ƒ determines a complex subvariety of X^(d) isomorphic to a linear space Π ≈ P^2*. Now conversely, this “linear system” Π of divisors of degree d on X determines the map ƒ back again as follows:

Define ƒ:X--->Π* = P^2** =P^2, by setting ƒ(p) = the line in Π consisting of those
divisors D that contain p. Then this determines the point ƒ(p) on C in P^2, because a
point in the plane is determined by the lines through that point. [draw picture]
Thus the problem becomes one of determining when the product X^(d) contains a
holomorphic copy of P^2, or copies of P^n for models of X in other projective spaces.

The Jacobian variety J(X) and the Abel map X^(d)--->J(X).

For this problem, Riemann introduced a second functor the “Jacobian” variety J(X) =
k^g/lattice, where k^g complex g -dimensional space. J(X) is a compact g dimensional
complex group, and there is a natural holomorphic map Abel:X^(d)--->J(X), defined by
integrating a basis of the holomorphic differential forms on X over paths in X.

Abel collapses each linear system Π ≈ P^n* to a point by the maximum principle, since the
coordinate functions of k^g have a local maximum on the compact simply connected
variety Π. Conversely, each fiber of the Abel map is a linear system in X^(d).
Existence of linear systems Π on X: the Riemann - Roch theorem.

By dimension theory of holomorphic maps, every fiber of the abel map X^(d)--->J(X)
has dimension ≥ d-g. Hence every positive divisor D of degree d on X is contained in a
maximal linear system |D| , where dim|D| ≥ d-g. This is called Riemann’s inequality, or
the “weak” Riemann Roch theorem.

The Roch part analyzes the relation between D and the divisor of a differential form to
compute dim|D| more precisely. Note if D is the divisor cut by one line in the plane of C,
and E is cut by another line, then E belongs to |D|, and the difference E-D is the divisor of the meromorphic function defined by the quotient of the linear equations for the two
lines.

If D is a not necessarily positive divisor, we define |D| to consist of those positive divisors E such that E-D is the divisor of a meromorphic function on X. If there are no such positive divisors, |D| is empty and has “dimension” equal to -1. Then if K is the divisor of zeroes of a holomorphic differential form on X, the full Riemann Roch
theorem says:

dim|D| = d-g +1+dim|K-D|, where the right side = d-g when d > deg(K).

This sketch describes the abel maps and their relation to the RRT. The assignment X-->J(X) is the Torelli map, and classifies X by the numerical data in the lattice defining J(X), i.e. periods of integrals of the first kind on X. This assignment gives birth to the whole subject of "moduli" as numerical invariants of complex or geometric structure.a

nt.number theory - Detecting almost-primes quickly

this is my answer, I don't know if this can complete your answer or not:

(similar to falagar's answer )

Let $w_i$ be the product $p_{2i}p_{2i+1}$, where $p_{i}$ is ${i}$th prime, each $w_i$ is product of two
distinct primes and also $w_k$ and $w_t$ are coprime, now take $t^2$ such that $t^{2}equiv- kpmod{w_i}$for $k=-Ncdots 0,1,2cdots N$,and $i=1,2,cdots ,2N+1$ such that for $k=-N$,$i=1$ and,...,by chinese remainder theorem, we have such $t^2$.

Now $t^{2}+i$ has at least two primes because $t^{2}+i$ is multiple of $w_i$,since there are a prime between $t$ and $2t$ and also $t$ and $t/2$,so $t+h_1$
and $t-h_2$ are primes,so at least one of the numbers $t^{2}-N,cdots
t^{2}+1,t^{2}+2cdots ,t^{2}+N$ is exactly product
of two distinct primes.

co.combinatorics - Algebraic Kneser conjecture?

Recall that Kneser conjecture (now Lovasz theorem) claims that if the family of $k$-subsets (subsets of cardinality $k$) of given $(2k+d)$-set $M$, $dgeq 1$ are colored into $d+1$ colors, then there are two disjoint $k$-subsets of the same color (easy to see that for $d+2$ colors it does not hold, say, if $M={1,2,dots,2k+d}$, then color $Ksubset M$ to color $iin {1,2,dots,d+1}$ if $min K=i$ and color $K$ to color $d+2$ if $min K>d+1$).

This may be reformulated as follows (mad on first glance) way. Consider the (say, complex) non-commutative algebra $A$ with generators $g_1$, $g_2$, $dots$, $g_{2k+d}$ and relations $g_ixg_i=0$ for each $i=1,2,dots,2k+d$ and each $xin A$. This algebra is naturally graded by degree of monomials in $g_i$'s. Denote by $A_k$ homogeneous component of degree $k$. Note that if $A_kni x=sum_{K} c(K)g(K)$ (here $K$ runs over $k$-subsets of index set ${1,2,dots,2k+d}$, $g(K):=g_{a}g_{b}g_{c}dots $ for $K={ a < b < cdots } $, $C(K)$ are some complex coefficients), then $x^2=0$ iff no two $K$'s with $c(K)ne 0$ are disjoint. So, the Kneser conjecture is equivalent to the following statement:

not any element $A_k$ is a some of $d+1$ square roots of 0.

This is nothing but tautology, but the question is whether this statement holds aswell for some factors of $G$, which have nicer algebraic structure? For example, for commutative algebra with relations $g_i^2=0$, or for exterior algebra? (This last may happen only for even $k$, of course, else each element is a square root of 0).

st.statistics - Building a multi-variable regression model

The input dimension being 18 is a little problematic, so I have a few suggestions. I'm putting the statistical approach first due to your choice of tags and terminology..

Linear regression can be made to fit polynomials by simply explicitly creating all the monomial terms; ie for an input with two dimensions $x= (x_1,x_2)$, for a quadratic you'd instead use $x' = (1,x_1,x_2,x_1x_2,x_1^2,x_2^2)$. Notice that this means, to add all $d$th order terms, you would create $O(18^d)$ dimensions! Notice furthermore that your regression model has equivalently many parameters! Therefore, it may be beneficial to try to simplify the model a little with some regularization, maybe use lasso (l1-regularization). Note that, as specified, the degree of the polynomial is not being minimized. The regularization I mentioned only minimizes the l1 length, and even putting a sparsity constraint on the weights alone (which is simpler than minimizing degree) is nonconvex. To find the degree, you could binary search; ie try degrees $1,2,4,8,ldots$ until you get zero error, and then binary search within the last interval to find the exact order.

Another approach is to directly use polynomial interpolation. Simply grow $d$, building an interpolating polynomial at each iteration, and stopping when the one built on the provided points fits all other points. For univariate data, the Lagrange Polynomial[1] is the way to go. I don't know your case offhand but just noticed wikipedia has a "polynomial interpolation" page which should be helpful.

anything you try will be slow due to the dimension, your stipulation that you necessarily find the absolute smallest degree, and the vast number of parameters in any such polynomial.

[1] lagrange interpolation

fa.functional analysis - On the failure of the infinite dimensional Brouwer Theorem

Let $K$ be the closed unit ball of some infinite dimensional Banach
space, and let $H$ be an autohomeomorphism of $K$, having fixed
points. Can $H/2$ be fixed point free ?

Also, let ${mathcal{F}}$ := { $Sinmbox{C}(K,K), mbox{Fix}(S)neqtextrm{Ø } $}.

Let $T$ in $mbox{C}(K,K)$ such that $TSinmathcal{F}$ for all $Sinmathcal{F}$
. Must $T$ be necessarily compact ?

fa.functional analysis - When is a finite matrix a "good" approximate representation of an operator?

I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $rho(r), ; rin mathbb{R}^3$ by a finite linear combination of basis functions

$rho(r) = sum_{i=1}^N q_i phi_i (r) $

where $phi_i (r)$ is normalized to $int_{mathbb{R}^3} phi_i (r) dr = 1$ and has the interpretation of being the shape of some charge distribution (shape) of a unit charge. $rho$ and the $phi_i$s are real-valued but may be positive in some regions and negative in others. The basis functions are nonorthogonal and local in space but not strictly compact. Let's say for now that we use spherical Gaussians of the form $phi_i (r) propto exp (-alpha_i |r-R_i|^2)$, where $R_i$ is where the basis function is centered around. The number of basis functions chosen scales approximately as the number of atoms, as we expect charge to concentrate around atomic nuclei. (We may add additional basis function per atom of different shapes until we achieve a reasonable approximation to the desired shape of the charge distribution around an atom.)

The energy of the system can then be given by

$E = frac 1 2 sum_{i,j=1}^N q_i q_j J_{ij}$

where the matrix $J$ has elements

$J_{ij} = int_{mathbb{R}^{3times 2}} frac{phi_i(r_1) phi_j(r_2)}{|r_1 - r_2|} dr_1 dr_2$

and represents the Coulomb interaction between the unit charge distributions $phi_i$ and $phi_j$.

---

One way to look at the matrix $J$ is as a finite dimensional (approximate) representation of the Coulomb operator $hat J = 1 / {|r_1 -r_2|}$. We know that $hat J$ has certain nice properties such as positivity, so we expect a "good" representation of $hat J$ should be a symmetric positive definite matrix.

My question is this: are there conditions on the discrete representation (possibly expressible as conditions on the {$phi_i$} basis) to detect whether or not a given claimed representation $J$ is "good" in that it preserves such properties? Or asked another way, if I have some matrix $J$ which is claimed to represent $hat J$, what are necessary and sufficient conditions on its matrix elements for it to be a "good" representation of $hat J$?

I hope the question makes sense, and that I am not misusing too much terminology.

terminology - Compact and quasi-compact

The condition of quasi-compactness in the Zariski topology bears little resemblance to the condition of compactness in the classical analytic topology: e.g. any variety over a field is quasi-compact in the Zariski topology, but a complex variety is compact in the analytic topology iff it is complete, or better, proper over $operatorname{Spec} mathbb{C}$.

I think many algebraic geometers think to themselves that a variety is "compact" if it is proper over the spectrum of a field. I have heard this terminology used and occasionally it shows up in (somewhat informal) writing.

So a perhaps more accurate brief answer is that in algebraic geometry the distinction between quasi-compact and quasi-compact Hausdorff is very important, whereas in other branches of geometry non-Hausdorff spaces turn up more rarely.

Anyway, many mathematicians have been happy with the quasi-compact / compact distinction for about 50 years, so I don't think this usage is going away anytime soon.

To address the last question: when writing for a general mathematical audience, it is a good idea to give an unobtrusive heads up as to your stance on the quasi-compact / compact convention. (The same probably goes for other non-universal conventions in mathematics.) If I were speaking about profinite groups, I would say something like:

"A profinite group is a topological group which can be expressed as an inverse limit of finite discrete groups. Equivalently, a topological group is profinite if it is compact (Hausdorff!) and totally disconnected."

This should let people know what side I'm on, and thus be able to understand me. When writing for students, I might take pains to be more explicit, using a "By compact I mean..." construction as you have indicated above.

ct.category theory - Large cardinal axioms and Grothendieck universes

A Grothendieck universe is known in set theory as the set V_κ for a (strongly) inaccessible cardinal κ. They are exactly the same thing. Thus, the existence of a Grothendieck universe is exactly equivalent to the existence of one inaccessible cardinal. These cardinals and the corresponding universes have been studied in set theory for over a century.

The Grothendieck Universe axiom (AU) is the assertion that every set is an element of a universe in this sense. Thus, it is equivalent to the assertion that the inaccessible cardinals are unbounded in the cardinals. In other words, that there is a proper class of inaccessible cardinals. This is the axiom you sought, which is exactly equivalent to AU. In this sense, the axiom AU is a statement in set theory, having nothing necessarily to do with category theory.

The large cardinal axioms are fruitfully measured in strength not just by direct implication, but also by their consistency strength. One large cardinal property LC₁ is stronger than another LC₂ in consistency strength if the consistency of ZFC with an LC₁ large cardinal implies the consistency of ZFC with an LC₂ large cardinal.

Measured in this way, the AU axiom has a stronger consistency strength than the existence of any finite or accessible number of inaccessible cardinals, and so one might think it rather strong. But actually, it is much weaker than the existence of a single Mahlo cardinal, the traditional next-step-up in the large cardinal hierarchy. The reason is that if κ is Mahlo, then κ is a limit of inaccessible cardinals, and so V_κ will satisfy ZFC plus the AU axiom. The difference between AU and Mahloness has to do with the thickness of the class of inaccessible cardinals. For example, strictly stronger than AU and weaker than a Mahlo cardinal is the assertion that the inaccessible cardinals form a stationary proper class, an assertion known as the Levy Scheme (which is provably equivconsistent with some other interesting axioms of set theory, such as the boldface Maximality Principle, which I have studied a lot). Even Mahlo cardinals are regarded as rather low in the large cardinal hierarchy, far below the weakly compact cardinals, Ramsey cardinals, measurable cardinals, strong cardinals and supercompact cardinals. In particular, if δ is any of these large cardinals, then δ is a limit of Mahlo cardinals, and certainly a limit of strongly inaccessible cardinals. So in particular, V_δ will be a model of the AU axiom.

Rather few of the large cardinal axioms imnply AU directly, since most of them remain true if one were to cut off the universe at a given inaccessible cardinal, a process that kills AU. Nevertheless, implicit beteween levels of the large caridnal hiearchy are the axioms of the same form as AU, which assert an unbounded class of the given cardinal. For example, one might want to have unboundedly many Mahlo cardinals, or unboundedly many measurable cardinals, and so on. And the consistency strength of these axioms is still below the consistency strength of a single supercompact cardinal. The hierarchy is extremely fine and intensely studied. For example, the assertion that there are unboundedly many strong cardinals is equiconsistent with the impossiblity to affect projective truth by forcing. The existence of a proper class of Woodin cardinals is particularly robust set-theoretically, and all of these axioms are far stronger than AU.

There are natural weakenings of AU that still allow for almost all if not all of what category theorists do with these universes. Namely, with the universes, it would seem to suffice for almost all category-theoretic purposes, if a given universe U were merely a model of ZFC, rather than V_κ for an inaccessible cardinal κ. The difference is that U is merely a model of the Power set axiom, rather than actually being closed under the true power sets (and similarly using Replacement in place of regularity). The weakening of AU I have in mind is the axiom that asserts that every set is an element of a transitive model of ZFC. This assertion is strictly weaker in consistency strength thatn even a single inaccessible cardinal. One can get much lower, if one weakens the concept of universe to just a fragment of ZFC. Then one could arrive at a version of AU that was actually provable in ZFC, but which could be used for most all of the applications in cateogory theory to my knowledge. In this sense, ZFC itself is a kind of large cardinal axiom relative to the weaker fragments of ZFC.

soft question - Community experiences writing Lamport's structured proofs

From a proof-theoretic point of view, Lamport essentially suggests is writing proofs in natural deduction style, along with a system of conventions to structure proofs by the relevant level of detail. (It would be very interesting to study how to formalize this kind of convention -- it's something common in mathematical practice missing from proof theory.)

I have written proofs in this style, and once taught it to students. I find that this system -- or indeed any variant of natural deduction -- is extremely valuable for teaching proof to students, because it associates to each logical connective the exact mathematical language needed to use it and to construct it. This is particularly helpful when you are teaching students how to manipulate quantifiers, and how to use the axiom of induction.

When doing proofs myself, I find that this kind of structured proof works fantastically well, except when working with quotients -- i.e., modulo an equivalence relation. The reason for this is that the natural deduction rules for quotient types are rather awkward. Introducing elements of a set modulo an equivalence relation is quite natural:

$$
frac{Gamma vdash e in A qquad R ;mathrm{equivalence;relation}}
{Gamma vdash [e]_R in A/R}
$$

That is, we just need to produce an element of $A$, and then say we're talking about the equivalence class of which it is a member.

But using this fact is rather painful:

$$
frac{Gamma vdash e in A/R qquad Gamma, xin A vdash t in B qquad Gamma vdash forall y,z:A, (y,z) in R.;t[y/x] = t[z/x]}{Gamma vdash mbox{choose};[x]_R;mbox{from};e;mbox{in};t in B}
$$

This rule says that if you know that

• $e$ is an element of $A/R$, and


• $t$ is some element of $B$ with a free variable $x$ in set $A$, and


• if you can show that for any $x$ and $y$ in $R$, that $t(y) = t(z)$ (ie, $t$ doesn't distinguish between elements of the same equivalence class)

Then you can form an element of $B$ by picking an element of the equivalence class and substituting it for $x$. (This works because $t$ doesn't care about the specific choice of representative.)

What makes this rule so ungainly is the equality premise -- it requires proving something about the whole subderivation which uses the member of the quotient set. It's so painful that I tend to avoid structured proofs when working with quotients, even though this is when I need them the most (since it's so easy to forget to work mod the equivalence relation in one little corner of the proof).

I would pay money for a better elimination rule for quotients, and I'm not sure I mean this as a figure of speech, either.

soft question - What's your favorite equation, formula, identity or inequality?

My favorite equation is

$$frac{16}{64} = frac{1}{4}.$$

What makes this equation interesting is that canceling the $6$'s yields the correct answer. I realized this in, perhaps, third grade. This was the great rebellion of my youth. Sometime later I generalized this to finding solutions to

$$frac{pa +b}{pb + c} = frac{a}{c}.$$

where $p$ is an integer greater than $1$. We require that $a$, $b$, and $c$ are integers between $1$ and $p - 1$, inclusive. Say a solution is trivial if $a = b = c$. Then $p$ is prime if and only if all solutions are trivial. On can also prove that if $p$ is an even integer greater than $2$ then $p - 1$ is prime if and only if every nontrivial solution $(a,b,c)$ has $b = p - 1$.

The key to these results is that if $(a, b, c)$ is a nontrivial solution then the greatest common divisor of $c$ and $p$ is greater than $1$ and the greatest common divisor of $b$ and $p - 1$ is also greater than $1$.

Two other interesting facts are (i) if $(a, b, c)$ is a nontrivial solution then $2a leq c < b$ and (2) the number of nontrivial solutions is odd if and only if $p$ is the square of an even integer. To prove the latter item it is useful to note that if $(a, b, c)$ is a nontrivial solution then so is $(b - c, b, b - a)$.

For what it is worth I call this demented division.

gr.group theory - The inverse Galois problem and the Monster

Hi Charles,

the monster is a nice example of how the so-called rigidity method for the inverse Galois problem works. There is a lot of beautiful mathematics behind this, I will sketch the different steps.

A general remark: it is known that every profinite group, i.e. every group which could be a Galois group of some field extension, is indeed the Galois group of some Galois extension. This is still quite elementary (a result of Leptin, also proved by Waterhouse). To make the inverse Galois question more interesting, we should therefore consider a fixed base field $K$. We can't hope that any profinite group will still be a Galois group over $K$ - every Galois group over $K$ is a quotient of the absolute Galois group of $K$, and therefore the cardinality of a Galois group over $K$ is bounded from above, whereas it is easy to see that there are profinite groups which are "strictly bigger" in cardinality. So a very reasonable question is indeed to ask whether every finite group is a Galois group over some fixed base field $K$. The most natural case is to ask the question for $K = mathbb{Q}$, but also other base fields can be considered - for example, for $K = mathbb{C}(t)$ the inverse Galois conjecture is true. In fact, the inverse Galois problems for different base fields $K$ are sometimes closely linked; the method which I will sketch below is a perfect illustration for this, since we will have the consider four different base fields: $mathbb{C}(t)$, $overline{mathbb{Q}}(t)$, $mathbb{Q}(t)$, and of course $mathbb{Q}$.

(1) Start with the fact that each finite group $G$ can be realized as a Galois group over $mathbb{C}(t)$. This follows from the theory of coverings of Riemann-surfaces; if $G$ can be generated by $n - 1$ elements, then we can realize $G$ as a quotient of the fundamental group of the punctured Riemann sphere $pi_1^{text{top}}(mathbb{P}^1(mathbb{C}) setminus {P_1,P_2,,cdots,P_n})$, where we choose the points $P_1,P_2,,cdots,P_n$ to be rational.

(2) We use the theory of the étale fundamental group to get an isomorphism $pi_1(mathbb{P}^1(mathbb{C}) setminus {P_1,P_2,,cdots,P_n}) = pi_1(mathbb{P}^1(overline{mathbb{Q}}) setminus {P_1,P_2,,cdots,P_n})$, which allows us to realize $G$ as a Galois group over $overline{mathbb{Q}}(t)$. [$pi_1$ is the étale fundamental group - here the profinite completion of the topological version.]

[Of course, this is already advanced material; see the Wikipedia article for background. For a proper introduction to the theory, there is SGA 1 by Grothendieck; and the recent book "Galois groups and fundamental groups" by Tamas Szamuely is a very gentle introduction (and does all this in detail).]

(3) There exists an exact sequence

$1 to pi_1(mathbb{P}^1(overline{mathbb{Q}}) setminus {P_1,P_2,,cdots,P_n}) to pi_1(mathbb{P}^1(mathbb{Q}) setminus {P_1,P_2,,cdots,P_n}) to text{Gal}(overline{mathbb{Q}}|mathbb{Q}) to 1$

(this is a very fundamental result; see again the books I mentioned) and basically we now want to extend a surjective homomorphism from $\pi_1(mathbb{P}^1(overline{mathbb{Q}}) setminus {P_1,P_2,,cdots,P_n})$ to $G$ to a surjective homomorphism from $pi_1(mathbb{P}^1(mathbb{Q}) setminus {P_1,P_2,,cdots,P_n})$ to $G$. Of course this depends heavily on the structure of the group $G$. This works for many finite simple groups; the construction is quite general, but for particular groups there is always some technical work to do to show that the method applies. In particular it works for finite groups with a trivial centre, and a rigid system of rational conjugacy classes - these are quite technical conditions, of course, and I will just state the definitions. An $n$-tuple of conjugacy classes $C_1,C_2,,cdots,C_n$ of $G$ is rigid if there exists $(g_1,g_2,,cdots,g_n) in G^n$ such that the $g_i$ generate $G$, $g_1g_2cdots g_n = 1$ and $g_i in C_i$, and if moreover $G$ acts transitively on the set of all such $n$-tuples $(g_1,g_2,,cdots,g_n)$. A conjugacy class $C$ of $G$ is rational if $g in C$ implies $g^m in C$ for all $m$ coprime to the order of $G$. I won't explain why precisely these conditions give you what you want, since it is really technical. Szamuely explains this very clearly. The conditions can be generalized, but that doesn't make it more readable...

[References: section 4.8 in Szamuely's book I mentioned above, and also Serre's wonderful book "Topics in Galois theory", which should maybe be called "Topics in inverse Galois theory" :)]

(4) The previous step allows us to descend from $overline{mathbb{Q}}(t)$ to $mathbb{Q}(t)$, i.e. to realize $G$ as a Galois group of a regular extension - another technical notion which I won't explain, but it is not unimportant - of $mathbb{Q}(t)$. To descend from $mathbb{Q}(t)$ to $mathbb{Q}$, there is Hilbert's irreducibility theorem, or some slight generalization (I don't remember exactly).

According to Thompson, the Monster has a rigid system of three rational conjugacy classes of orders 2, 3 and 29. So the method will apply; of course, I guess that it will be very hard to construct these conjugacy classes, and it is clear that the classification of finite simple groups has played a very big role in these developments. (But I am not a group theorist, so anyone who knows how this works is welcome to give additional information about this construction :))

So I hope this gives you an idea; I wrote this up in a hurry, so suggestions to make this clearer or more coherent (or of course corrections of details which I got wrong) are always welcome.

soft question - Too old for advanced mathematics?

Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, several years ago I was affected by a serious illness and ended up temporarily disabled. I worked in the music business, and to help pass the time during my convalescence I picked up a book on musical acoustics.

That book reintroduced me to calculus with which I'd had a fleeting encounter with during high school, so to understand what I was reading I figured I needed to brush up, so I picked up a copy of Stewart's "Calculus". Eventually I spent more time working through that book than on the original text. I also got a copy of "Differential Equations" by Edwards and Penny after I had learned enough calculus to understand that. I've also been learning linear algebra - MIT's lectures and problem sets have really helped in this area. I'm fascinated with the mathematics of the Fourier transform, particularly its application to music in the form of the DFT and DSP - I've enjoyed the lectures that Stanford has available on the topic immensely. I just picked up a little book called "Introduction To Bessel Functions" by Frank Bowman that I'm looking forward to reading.

The difficulty is, I'm 30 years old, and I can tell that I'm a lot slower at this than I would have been if I had studied it at age 18. I'm also starting to feel that I'm getting into material that is going to be very difficult to learn without structure or some kind of instruction - like I've picked all the low-hanging fruit and that I'm at the point of diminishing returns. I am fortunate though, that after a lot of time and some great MDs my illness is mostly under control and I now have to decide what to do with "what comes after."

I feel a great deal of regret, though, that I didn't discover that I enjoyed this discipline until it was probably too late to make any difference. I am able, however, to return to college now if I so choose.

The questions I'd like opinions on are these: is returning to school at my age for science or mathematics possible? Is it worth it? I've had a lot of difficulty finding any examples of people who have gotten their first degrees in science or mathematics at my age. Do such people exist? Or is this avenue essentially forever closed beyond a certain point? If anyone is familiar with older first-time students in mathematics or science - how do they fare?

ca.analysis and odes - Truncated product of RiemannZeta(1)?

Hi, this is my first question. It appeared while solving a research problem in cryptography. I am computer science student, so I apologize for lack of mathematical rigor in this question. Thanks for any help.

Consider the RiemannZeta function at s = +1. It diverges, but the expression for the function is RiemannZeta(1) = $lim_{n rightarrow infty} sum_{i = 1}^{n} frac{1}{i}$ , the truncated sum of which are the $n$-th harmonic number, $mathcal{H}(n)$.

The question is,
How about the expression RiemannZeta(1) = $lim_{n rightarrow infty} prod_{textrm{primes} p_i leq n} frac{1}{1-p_i^{-1}}$. is the value of the truncated product $mathcal{H}(n)$ too?

My simulations for large values of $n$ tells me that it is some function of $log n$ (for example comparing the ratio of the function for $n$ and $n^2$ and $n^3$ etc) How do we prove this?

In summary, What is the value of $prod_{textrm{primes} p_i leq n} frac{1}{1-p_i^{-1}}$?
Thanks

gr.group theory - Combinatorial Techniques for Counting Conjugacy Classes

Since $A_n$ has index two in $S_n$, every conjugacy class in $S_n$ either is a conjugacy class in $A_n$, or it splits into two conjugacy classes, or it misses $A_n$ if it is an odd permutation. Which happens when is a nice undergraduate exercise in group theory. (And you are a nice undergraduate. :-) )

The pair $A_n subset S_n$ is typical for this question in finite group theory. You want the conjugacy classes of a finite simple group $G$, but the answer is a little simpler for a slightly larger group $G'$ that involves $G$. Another example is $text{GL}(n,q)$. It involves the finite simple group $L(n,q)$, but the conjugacy classes are easier to describe in $text{GL}(n,q)$ . They are described by their Jordan canonical form, with the twist that you may have to pass to a field extension of $mathbb{F}_q$ to obtain the eigenvalues.

The group $text{GL}(n,q)$ is even more typical. It is a Chevallay group, which means a finite group analogue of a Lie group. All of the infinite sequences of finite simple groups other than $A_n$ and $C_p$ are Chevallay groups. You expect a canonical form that looks something like Jordan canonical form, although it can be rather more complicated.

If $G$ is far from simple, i.e., if it has some interesting composition series, then one approach to its conjugacy classes is to chase them down from the conjugacy classes of its composition factors, together with the structure of the extensions. The answer doesn't have to be very tidy.

I suppose that finite Coxeter groups give you some exceptions where you do get a tidier answer, just because they all resemble $S_n$ to varying degrees. But I don't know a crisp answer to all cases of this side question. The infinite sequences of finite Coxeter groups consist only of permutation groups, signed permutation groups, and dihedral groups. (And Cartesian products of these.) In the case of signed permutation groups, the answer looks just like $S_n$, except that cycles can also have odd or even total sign. There is also the type $D_n$ Coxeter group of signed permutation matrices with an even number of minus signs; the answer is just slightly different from all of the signed permutation matrices, which is type $B_n$. The crisp answer that I don't have would be a uniform description that includes the exceptional finite Coxeter groups, such as $E_8$.

Connected components of the orthogonal group O(2n) in characteristic 2.

I am looking for a reference for the following fact:
The orthogonal group $O_{2n}$ over an algebraically closed field of characteristic 2
has exactly two connected components.

To be more precise, let $O_q$ denote the orthogonal group of the quadratic form $q(x)=x_1 x_2 +x_3 x_4+cdots +x_{2n-1}x_{2n}$
over an algebraically closed field $k$.
In characteristic $pneq 2$ the determinant takes two values on $O_q$, 1 and $-1$,
and therefore the subgroup $SO_q:=O_qcap SL_{2n}$ is of index 2 in $O_q$; it is known that $O_qcap SL_{2n}$ is connected.

In characteristic 2 the determinant takes only one value 1 on $O_q$ (because $-1=1$), and therefore $O_qcap SL_{2n}=O_q$.
Still there is a homomorphism $Dcolon O_qto mathbf{Z}/2mathbf{Z}$ given by a polynomial $D$ called the Dickson invariant,
see J.A.~Dieudonn'e, Pseudo-discriminant and Dickson invariant, Pacific. J. Math. 5 (1955), 907--910.
This homomorphism $D$ indeed takes both values 0 and 1 on $O_q$, and therefore its kernel ker $D$
is a closed subgroup of index 2 in $O_q$. I would like to know that ker $D$ is connected.
In other words, I am looking for a reference to the assertion that the orthogonal group $O_q$ has at most two connected components.
This is proved in Brian Conrad's handout "Properties of orthogonal groups" to his course Math 252 "Algebraic groups",
see http://math.stanford.edu/~conrad/252Page/handouts/O(q).pdf . Is there any other reference for this fact?

I will be grateful to any references, comments, etc.

Mikhail Borovoi

fa.functional analysis - What functorial topologies are there on the space of linear maps between LCTVS?

If $V$ is a vector space, there is a least topology $tau_V$ on $V$ which makes all linear maps from $V$ to any other vector space continuous (consider the initial topology from the collection of all linear maps from $V$ to all vector spaces which are quotients of $V$---this is a set of maps, which is nice, but it is not really needed) This topology is locally convex: it is the largest locally convex topology, in fact.

If you endow $L(V,W)$, the space of continuous linear maps between two lcvts $V$ and $W$, with the topology $tau_{L(V,W)}$, then you get a functor.

(You can play this game in many ways: pick a set $mathcal F$ of your favorite locally convex topological vector spaces — for example, let $mathcal F={L^{2/3}(mathbb R), mathcal E'}$, — and endow $L(V,W)$, for all lctvs $V$ and $W$, with the least topology $tau_mathcal F$ for which all linear maps from $L(V,W)$ to an element of $mathcal F$ are continuous. This gives again a functor.) (Moreover, one can do the same with final topologies, of course)

Remark that these topologies are probably useless :)

Later: There is another source of examples, generalizing the uniform topologies.
Suppose you have an assignment to each lctvs $V$ of a set $C(V)subseteqmathcal P(V)$ of subsets of $V$ which is functorial, in the sense that whenever $f:Vto W$ is a continuous linear map of lctvs then ${f(A):Ain C(V)}subseteq C(W)$. Then there is functorial topology on $L(V,W)$ given by uniform convergence on sets of $C(V)$.

If you take $C(V)$ to be the set of bounded subsets of $V$, of compact subsets of $V$, of relatively compact subsets of $V$, of finite subsets of $V$, then you get your examples 1 to 4. But there are other choices for $C(V)$: the set of star shaped subsets of $V$ (ie, subsets $Ssubseteq V$ such that there is a point $xin S$ such that for all $yin S$ the segment $[x,y]$ is contained in $S$; this over $mathbb R$---over $mathbb C$ there is a corresponding notion whose name escapes me now), say. There are others.

semigroups - References/literature for pushouts in category of commutative monoids? [ed. - amalgams]

This is more of a request for pointers to relevant literature than a question per se. I am, erm, looking at a paper which uses a kind of iterated pushout construction to obtain a commutative monoid with certain desired properties. The particular "gluing" construction the authors want to do is handled by quite direct means, and while that's probably fine for this particular problem, it would be nice to put it in the proper context.

My copy of Howie's "Introduction to Semigroup Theory" discusses the general notion of "semigroup amalgams" but doesn't seem to say anything about the pushouts in the smaller category of commutative monoids. The closest I can find via MathSciNet is a 1968 paper of Howie, which seems to be working in the category of commutative semigroups, but I can't get hold of a copy at the moment.

Anyway: I'm hoping that someone reading might know of a sensible place to look for a summary of some general results. (The issue is whether the constituent pieces "embed" into the pushout, not the mere existence of the pushout.)

EDIT: On further reflection, while pushouts are undoubtedly relevant, for the purposes of the paper I'm looking at, faithfulness of the embedding is more important than the universal property of pushouts. So I'm changing the title of the question to reflect this. I also think that while the suggestions below have been useful in a broad sense they don't quite address the case I need - which could just be an illustration that said case falls between the two extremes most commonly looked at by semigroup theorists.

at.algebraic topology - Are fundamental groups of aspherical manifolds Hopfian?

A group $G$ is Hopfian if every epimorphism $Gto G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth manifolds Hopfian?

Perhaps the manifold structure is irrelevant and makes examples harder to construct, so here is another variant that may be more sensible. Let $X$ be a finite CW-complex which is $K(pi,1)$. If it helps, assume that its top homology is nontrivial. Is $pi=pi_1(X)$ Hopfian?

Motivation. Long ago I proved a theorem which is completely useless but sounds very nice: if a manifold $M$ has certain homotopy property, then the Riemannian volume, as a function of a Riemannian metric on $M$, is lower semi-continuous in the Gromov-Hausdorff topology. (And before you laugh at this conclusion, let me mention that it fails for $M=S^3$.)

The required homotopy property is the following: every continuous map $f:Mto M$ which induces an epimorphism of the fundamental group has nonzero (geometric) degree.

This does not sound that nice, and I tried to prove that some known classes of manifolds satisfy it. My best hope was that all essential (as in Gromov's "Filling Riemannian manifolds") manifolds do. I could not neither prove nor disprove this and the best approximation was that having a nonzero-degree map $Mto T^n$ or $Mto RP^n$ is sufficient. I never returned to the problem again but it is still interesting to me.

An affirmative answer to the title question would solve the problem for aspherical manifolds. A negative one would not, and in this case the next question may help (although it is probably stupid because I know nothing about the area):

Question 2. Let $G$ be a finitely presented group and $f:Gto G$ an epimorphism. It it true that $f$ induces epimorphism in (co)homology (over $mathbb Z$, $mathbb Q$ or $mathbb Z/2$)?

examples - Applications of the Chinese remainder theorem

Here is a neat example that predates David Speyer's example of fast parallel arithmetic, but uses the same trick.

Richard J. Lipton, Yechezkel Zalcstein: Word Problems Solvable in Logspace. J. ACM 24(3): 522-526 (1977)

In this paper, the Chinese remainder theorem is used to prove that the word problem on several types of groups are solvable in logspace. (The Chinese remainder theorem is not explicitly invoked, but one can use it to justify the algorithms.) For instance, the paper states:

Corollary 6. The word problem for finitely generated free groups is solvable in logspace.

The word problem for a finite group is to determine if a given product of group elements equals the identity. The results are proved by embedding a group into a group of matrices over the integers, then computing the matrix product modulo all small primes.

co.combinatorics - Partition-Like Counting

Fix $k,ngeq 1$. I wish to know the number of ways of dividing $n$ up as follows. First find $c_1,c_2,ldots ,c_k$ such that $c_igeq 0$ and $sum_ic_i=n$. Then take each $c_i$ and find $c_{i,1},ldots ,c_{i,n}$ such that $c_{i,m}geq 0$ and $sum_mmc_{i,m}=c_i$. How many ways are there to do this?

ag.algebraic geometry - Deformations of hypersurfaces

Let's assume that we are working over $mathbb{C}$.

First of all, hypersurfaces in $mathbb{P}^n$ are unobstructed, so their first-order deformations always correspond to small deformations (deformations over a disk).

As a general fact, when you consider a smooth variety $X$ with a finite group $G$ acting $holomorphically$ on it, the invariant subspace $H^1(X, T_X)^G$, it parametrizes those first-order deformations that preserve the holomorphic $G$-action. This essentially comes from the fact that, being the action of $G$ holomorphic, if you take $sigma in G$, then $sigma_*$ commutes with $bar{partial}$ and the Green operator $boldsymbol{G}$, so if $varphi(t)$ solves the Kuranishi equation

$varphi(t)=t + frac{1}{2}bar{partial}^* boldsymbol{G}[varphi(t), varphi(t)]$

for $t$, then $sigma_*varphi(t)$ solves the Kuranishi equation for
$sigma_*t$, and $sigma_{*} varphi(t) = varphi(sigma_*t)$.

Example Let us consider a quintic Fermat surface $X subset mathbb{P}^3$ of equation

$x^5+y^5+z^5+w^5=0$.

It admits a free action of the cyclic group $mathbb{Z}_5$ given as follows: if $xi$ is a primitive $5$-th root of unity, then

$xi cdot (x,y,z,w)=(x, xi y, xi^2 z, xi^3 w) $.

The quotient $Y := X/mathbb{Z}_5$ is a Godeaux surface (i.e. a surface of general type with $p_g=q=0, K^2=1$ ) with fundamental group $mathbb{Z}_5$. M. Reid proved that, conversely, every Godeaux surface with fundamental group $mathbb{Z}_5$ arises in this way and that, moreover, the corresponding moduli space is generically smooth of dimension $8$. Then in this case we have

$dim H^1(X, T_X)=40$

$dim H^1(X, T_X)^G=H^1(Y, T_Y)=8$,

since the number of moduli of quintics keeping the free $G$-action equals the number of moduli of the Godeaux surface $Y$ (well, Horikawa showed that the deformations of quintic surfaces are complicated enough, anyway $40$ is the right number).

Actually, one can say more and check that for every irreducible character $chi$ of $G$ one has

$dim H^1(X, T_X)^{chi} = 8$,

but I do not know any easy interpretation of these eigenspaces in terms of the deformations of the quintic.

mg.metric geometry - Approximate solutions for trisecting the angle or squaring the circle

Hello all, it is well-known by transcendence results or Galois theory that famous geometric problems such as trisecting an angle or "squaring the circle" (i.e. given a disk of radius 1 construct a square of equal area) are not solvable by compasses-and-ruler only
constructions.
On the other side, it is equally well-known, by approximation results such as Weierstrass', that given any $varepsilon >0$
there is a definite construction process that yields an approximate solution which is correct up to $varepsilon$.
Of course, the obvious solution is to compute the coordinates of the points you need up to
the precision you need, and then place the points.
This solution however relies on some classical function tables (cosine, arc cosine or the decimal expansion of $sqrt{pi}$)
and I am wondering if there is a more "purely geometric solution" needing no calculator or tables.
Specifically, for angle trisection, one could ask the following :

Define explicitly a compasses-and-ruler only algorithm with the following properties :

Initial data : a circle with center O and radius 1 cm, two points I and J on that circle such that IOJ is a straight angle, and a point
M on the arc between I and J. Let us call N the point on that arc such that the angle ION is one third of IOM. The algorithm
must return a point N' which is undistinguishable from N to the naked eye, and must not rely on any calculator or tables.

Either that question is interesting or it isn't. If it isn't, the "shortest number of steps" solution has a large number of steps and is only a complicated reformulation of the compute-coordinates-with-calculator method.

soft question - Alternative Undergraduate Analysis Texts

Nobody has mentioned Folland's "Real Analysis with Applications"?? This was the textbook for my undergraduate real analysis course (measure theory, Banach spaces, Hilbert spaces), and I still go back to it all the time. I am not yet all that experienced (I just finished my third year of graduate school), but overall I have gotten more use out of this book than any other that I own.

It has the most comprehensive swath of applications of analysis of any introductory text I have ever encountered: basic functional analysis, Fourier theory, probability theory, distributions, Hausdorff measures, Haar measure, smooth measures, and more. The early material is covered with all the appropriate detail, while the later material quickly provides the essential definitions and results needed to come to grips with an unfamiliar idea in the literature. Also, the exercises are abundant and uniformly fantastic. My only complaints are that some of the later proofs are hard to read, and there is sadly no discussion of the spectral theorem.

mg.metric geometry - Estimating flat norm distance from a planar disc

Let $Dsubsetmathbb R^2subsetmathbb R^n$ be a unit planar disc in $mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $mathbb R^n$ such that $partial S=partial D$. Of course, we have $area(S)ge area(D)$. Assume that $area(S)< area(D)+delta$ where $delta>0$ is small.

Then $S$ is close to $D$ in the following sense: there is a 3-dimensional surface $F$ filling the gap between $S$ and $D$ such that $volume(F)<varepsilon(delta)$ where $varepsilon(delta)to 0$ as $deltato 0$ ($n$ is fixed). "Filling the gap" means that $partial F=S-D$.

This fact immediately follows from the compactness theorem for flat norms. But this proof is indirect and does not answer the following questions (I am especially interested in the second one):

1) Are there explicit upper bounds for $varepsilon(delta)$? How do they depend on $delta$ and $n$?

2) Can $varepsilon(delta)$ be independent of $n$? Or, equivalently, does the above fact hold true in the Hilbert space?

In the unlikely event that 2-dimensional surfaces are somehow special, what about the same questions about $m$-dimensional surfaces, for a fixed $m$?

Remarks: "Surfaces" here are Lipschitz surfaces or rectifiable currents or whatever you prefer to see in this context. Rather than talking about the filling surface $F$, one could equivalently say that the integral flat norm of $S-D$ is less than $varepsilon(delta)$.

set theory - Consistency Results Separating Three Cardinal Characteristics Simultaneously

Bumping an old question, because the situation has changed dramatically in the last year or so:

The most well-known cardinal characteristics of the continuum are those appearing in Cichoń's Diagram, which also presents all $ZFC$-provable relations between pairs of these characteristics: besides the dominating number $mathfrak{d}$ and the bounding number $mathfrak{b}$, these are all of the form $inv(mathcal{I})$, for $mathcal{I}$ the ideal of meager sets $(mathcal{M})$ or null sets $(mathcal{N})$, and $inv$ one of $add, cov, non, cof$:

• $add$ of an ideal $mathcal{I}$ is the least cardinality of a family of sets from $mathcal{I}$ whose union is not in $mathcal{I}$;




• $cov$ of an ideal $mathcal{I}$ is the least cardinality of a family of sets from $mathcal{I}$ covering all of $mathbb{R}$;




• $non$ of an ideal $mathcal{I}$ is the least cardinality of a set of reals not in $mathcal{I}$; and




• $cof$ of an ideal $mathcal{I}$ is the cofinality of the partial order $langlemathcal{I}, subseteqrangle$.

As far as I know, the first results separating more than two cardinal characteristics from Cichon's diagram came out in the last two years:

• In late 2012, Mejia http://arxiv.org/abs/1211.5209 constructed several examples of models separating multiple characteristics at once (see section 6).




• In a paper arxived today(!), Fischer/Goldstern/Kellner/Shelah http://arxiv.org/abs/1402.0367 produced a model of $aleph_1=mathfrak{d}=cov(mathcal{N})<non(mathcal{M})<non(mathcal{N})<cof(mathcal{N})<2^{aleph_0}$.

Now, I don't understand the proofs at all, but it seems the proofs by Mejia and by F/G/K/S are fundamentally different. The F/G/K/S paper has this to say about the two different approaches to separating multiple characteristics simultaneously:

"We cannot use the two best understood methods [to separate $ge 3$ characteristics simultaneously], countable support iterations of proper forcings (as it forces $2^{aleph_0}lealeph_2$) and, at least for the "right hand side" of the diagram, we cannot use finite support iterations of ccc forcings in the straightforward way (as it adds lots of Cohen reals, and thus increases $cov(mathcal{M})$ to $2^{aleph_0}$).

There are ways to overcome this obstacle. One way would be to first increase the continuum in a "long" finite support iteration, resulting in $cov(mathcal{M})=2^{aleph_0}$, and then "collapsing" $cov(mathcal{M})$ in another, "short" finite support iteration. In a much more sophisticated version of this idea, Mejia [Mej13] recently constructed several models with many simultaneously different cardinal characteristics in Cichon's Diagram (building on work of Brendle [Bre91], Blass-Shelah [BS89] and Brendle-Fischer [BF11]).

We take a different approach, completely avoiding finite support, and use something in between a countable and finite support product (or: a form of iteration with very "restricted memory").

This construction avoids Cohen reals, in fact it is $omega^omega$-bounding, resulting in $mathfrak{d}=aleph_1$. This way we get an independence result "orthogonal" to the ccc/finite-support results of Mejia.

The fact that our construction is $omega^omega$-bounding is not incidental, but rather a necessary consequence of the two features which, in our construction, are needed to guarantee properness: a "compact" or "finite splitting" version of pure decision, and fusion . . ."

- http://arxiv.org/pdf/1402.0367.pdf, pg. 2

mathematics education - Why is a topology made up of 'open' sets?

In this answer I will combine ideas of sigfpe's answer, sigfpe's blog, the book by Vickers, Kevin's questions and Neel's answers adding nothing really new until the last four paragraphs, in which I'll attempt to settle things about the open vs. closed ruler affair.

---

DISCLAIMER: I see that some of us are answering a question that is complementary of the original, since we are trying to motivate the structure of a topology, instead of adressing the question of which of the many equivalent ways to define a topology should be used, which is what the question literally asks for. In the topology course that I attended, it was given to us in the first class as an exercise to prove that a topology can be defined by its open sets, its neighbourhoods, its closure operator or its interior operator. We later saw that it can also be stated in terms of convergence of nets. Having made clear these equivalence of languages, its okay that anyone chooses for each exposition the language that seems more convinient without further discussion. However, I will mantain my non-answer since many readers have found the non-question interesting.

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Imagine there's a set X of things that have certain properties. For each subset of S there is the property of belonging to S, and in fact each property is the property of belonging to an adequate S. Also, there are ways to prove that things have properties.

Let T be the family of properties with the following trait: whenever a thing has the property, you can prove it. Let's call this properties affirmative (following Vickers).

For example, if you are a merchant, your products may have many properties but you only want to advertise exactly those properties that you can show. Or if you are a physicist, you may want to talk about properties that you can make evident by experiment. Or if you predicate mathematical properties about abstract objects, you may want to talk about things that you can prove.

It is clear that if an arbitrary family of properties is affirmative, the property of having at least one of the properties (think about the disjunction of the properties, or the union of the sets that satisfy them) is affirmative: if a thing has at least one of the properties, you can prove that it has at least one of the properties by proving that property that it has.

It is also clear that if there is a finite family of affirmative properties, the property of having all of them is affirmative. If a thing has all the properties, you produce proofs for each, one after the other (assuming that a finite concatenation of proofs is a proof).

For example, if we sell batteries, the property P(x)="x is rechargeable" can be proved by putting x in a charger until it is recharged, but the property Q(x)="x is ever-lasting" can't be proved. It's easy to see that the negation of an affirmative property is not necessarily an affirmative property.

Let's say that the open sets are the sets whose characteristic property is affirmative. We see that the family T of open sets satisfies the axioms of a topology on X. Let's confuse each property with the set of things that satisfy it (and open with affirmative, union with disjunction, etc.).

Interior, neighbourhood and closure: If a property P is not affirmative, we can derive an affirmative property in a canonical way: let Q(x)="x certainly satisfies P". That is, a thing will have the property Q if it can be proved that it has the property P. It is clear that Q is affirmative and implies P. Also, Q is the union of the open sets contained in P. Then, it is the interior of P, which is the set of points for which Q is a neighbourhood. A neighbourhood of a point x is a set such that it can be proved that x belongs to it. The closure of P is the set of things that can't be proved not to satisfy P.

Axioms of separation: If T is not T0, there are x, y that can't be distinguished by proofs and if it is not T1, there are x, y such that x can't be distinguished from y (we can think that they are apparently identical batteries, but x is built in such a way that it will never overheat. So if it overheats, then it's y, but if it doesn't, you can't tell).

Base of a topology: Consider a family of experiments performable over a set X of objects. For each experiment E we know a set S of objects of X over which it yields a positive result (nothing is assumed about the outcome over objects that do not belong to S). If you consider the properties that can be proved by a finite sequence of experiments, the sets S are affirmative and the topology generated by them is the family of all the affirmative properties.

Compactness: I don't know how to interpret it, but I think that some people know, and it would be nice if they posted it. (Searchable spaces?)

Measurements: A measurement in a set X is an experiment that can be performed on each element of X returning a result from a finite set of possible ones. It may be a function or not (it is not a function if there is at least one element for which the result is variable). The experiment is rendered useful if we know for each possible result r a set T_r of elements for which the experiment certainly renders r and/or a set F_r for which it certainly doesn't, so let's add this information to the definition of measurement. An example is the measurement of a length with a ruler. If the length corresponds exactly with a mark on the ruler, the experimenter will see it and inform it. If the length fits almost exactly, the experimenter may think that it fits a mark or may see that it doesn't. If the length clearly doesn't fit any mark (because he can see that it lies between two marks, or because the length is out of range), he will inform it. It is sufficient to study measurements that have only a positive outcome and a negative outcome, a set T for which the outcome is certainly positive and a set F for which the outcome is certainly negative.

Imprecise measurements on a metric space: If X is a metric space, we say that a measurement in X is imprecise if there isn't a sequence x_n contained in F that converges to a point x contained in T. Suppose that there is a set of imprecise measurements available to be performed on the metric space. Suppose that, at least, for each x in X we have experiments that reveal its identity with arbitrary precision, that is, for each e>0 there is an experiment that, when applied to a point y, yields positive if y=x and doesn't yield positive if d(y,x)>e. Combining these experiments we are allowed to prove things. What are the affirmative sets generated by this method of proof? Let S be a subset of X. If x is in the (metric) interior of S, then there is a ball of some radius e>0 centered at x and contained in S. It is easy to find an experiment that proves that x belongs to S. If x is in S but not in the interior (i.e, it is in the boundary), we don't have a procedure to prove that x is in S, since it would involve precise measurement. Therefore, the affirmative sets are those that coincide with its metric interior. So, the imprecise measurements of arbitrary precision induce the metric topology.

Experimental sciences: In an experimental science, you make a model that consists of a set of things that could conceivably happen, and then make a theory that states that the things that actually happen are the ones that have certain properties. Not all statements of this kind are completely meaningful, but only the refutative ones, that is, those that can be proved wrong if they are wrong. A statements is refutative iff its negation is affirmative. By applying the closure operator to a non refutative statement we obtain a statement that retains the same meaning of the original, and doesn't make any unmeaningful claim.

An example from classical physics: Assume that the space-time W is the product of Euclidean space and an affine real line (time). It can be given the structure of a four-dimensional real normed space. Newton's first law of motion states that all the events of the trajectory of a free particle are collinear in space-time. To prove it false, we must find a free particle that incides in three non-collinear events. This is an open condition predicated over the space W^3 of 3-uples of events, since a small perturbation of a counterexample is also a counterexample. Assuming that imprecise measurements of arbitrary precision can be made, it is an affirmative property. I think that classical physicists, by assuming that these kind of measurements can be done, give exact laws like Newton's an affirmative set of situations in which the law is proved false. I also suspect (but this has more philosophical/physical than mathematical sense) that the mathematical properties of space-time (i.e. that it is a normed space over an Archimedean field) are deduced from the kind of experiments that can be done on it, so there could be a vicious circle in this explanation.