As per your second question, the following algorithm allows one to compute the identity element.
Let $c$ denote the maximal stable configuration; i.e. $c = sum_{vin V}(d(v)-1) v$ This is always recurrent. Let $a^{circ}$ denote the stabilization of a configuration $a$. Then this will give you the identity $e$:
$e =(2c - (2c)^circ)^circ$
If you are interested, check out this applet for doing a lot of this stuff (and it's pretty, too): http://people.reed.edu/~davidp/sand/program/program.html
The answer to the first part is indeed true. In fact, something more general is true. Let $mathcal{A}$ be a small category and let $mathcal{C}$ be a cocomplete category (which is locally small, i.e., there is just a set of morphism between any two objects). Then any cocontinuous functor $L colon mathrm{Set}^{mathcal{A}^mathrm{op}} rightarrow mathcal{C}$ has a right adjoint, given by $C mapsto mathcal{C}(K-,C)$, where $K colon mathcal{A} rightarrow mathcal{C}$ is the composite of the Yoneda embedding and $L$.
This is for example proved in Kelly's "Basic concepts of enriched category theory", Theorem 4.51. He proves the enriched version of this result, where $mathrm{Set}$ is replaced by any complete and cocomplete category $mathcal{V}$. I must say I don't know of a reference that just treats the $mathrm{Set}$-case.
If the target is the category of presheaves on some large category, then this might fail. Take for example $mathcal{D}$ the large discrete category whose objects are sets, and let $F colon mathcal{D} rightarrow mathrm{Set}$ be the canonical inclusion functor. Then the functor $mathrm{Set}rightarrow mathrm{Set}^{mathcal{D}}$ which sends a set $X$ to the functor $Ftimes X$ (i.e., the functor which sends a set $A$ to $Atimes X$) is cocontinuous, because $Atimes -$ preserves colimits. However, there is a proper class of natural transformations $F rightarrow F$ (a natural transformation just amounts to choosing an endomorphism of each set with no compatibility conditions), so if this functor had a right adjoint $R$, then we would have a bijection $mathrm{Set}(ast,RF) cong mathrm{Set}^{mathcal{D}}(F,F)$, i.e., $RF$ would have to be a proper class. The reason for this failure is of course that $mathrm{Set}^mathcal{D}$ is not locally small. Note that this problem doesn't go away when we use universes: the above example would give you an isomorphism between a small set and a large set, i.e., a set outside of the universe.
There are also simple examples of ideals of commutative rings that have no minimal generating sets, e.g., any nonzero ideal $I$ with $I = J(R)I$. (Adapt the proof of Nakayama's Lemma.) For example, let $K$ be a field, $M$ be the maximal ideal of $K[{x^s mid s in mathbb{Q}^+}]$ consisting of the elements with zero constant term, and note that $M_M = (M_M)^2$.
Since I don't have enough points to comment yet, I would also like to point out here that there is an erratum for the paper Martin referenced where the author points out that his proof is flawed. As far as I know, it is known that that condition implies perfect, but whether the converse is true is still open.
With all this unanimous enthusiasm, I can't help but add a cautionary note. I will say, however, that what I'm about to say applies to anyone of any age trying to get a Ph.D. and pursue a career as an academic mathematician.
If you think you want a Ph.D. in mathematics, you should first try your best to talk yourself out of it. It's a little like aspiring to be a pro athlete. Even under the best of circumstances, the chances are too high that you'll end up in a not-very-well-paying job in a not-very-attractive geographic location. Or, if you insist on living in, say, New York, you may end up teaching as an adjunct at several different places.
Someone with your mathematical talents and skills can often find much more rewarding careers elsewhere.
You should pursue the Ph.D. only if you love learning, doing, and teaching mathematics so much that you can't bear the thought of doing anything else, so you're willing to live with the consequences of trying to make a living with one. Or you have an exit strategy should things not work out.
Having said all that, I have a story. When I was at Rice in the mid 80's, a guy in his 40's or 50's came to the math department and told us he really wanted to become a college math teacher. He had always loved math but went into sales(!) and had a very successful career. With enough money stashed away, he wanted to switch to a career in math. To put it mildly, we were really skeptical, mostly because he had the overly cheery outgoing personality of a salesman and therefore was completely unlike anyone else in the math department. It was unthinkable that someone like that could be serious about math. Anyway, we warned him that his goal was probably unrealistic but he was welcome to try taking our undergraduate math curriculum to prepare. Not surprisingly, he found this rather painful, but eventually to our amazement he started to do well in our courses, including all the proofs in analysis. By the end, we told the guy that we thought he really had a shot at getting a Ph.D. and have a modest career as a college math teacher. He thanked us but told us that he had changed his mind. As much as he loved doing the math, it was a solitary struggle and took too much of his time away from his family and friends. In the end, he chose them over a career in math (which of course was a rather shocking choice to us).
So if you really want to do math and can afford to live with the consequences, by all means go for it.
I have data set X = {x_1, x_2, ldots, x_N}, each x_i
is a d-dimensional vector, where scalars are from some finite field
(In practice they are categories, represented by integers from 1...C).
If my data would be 1-d, then multinomial density would be ok to model it.
How do I extend it to multidimensional case? If scalars in vector are iid, then
I guess, I could just model it by d-product of multinomial densities? Did I miss something?
There is a general strategy, which unfortunately is a failure because the information needed for success is scarcely ever available.
Let $A$ denote a set of primes, and $B$ the set of all products of powers of the primes in $A$. Then
$$
prod_{p in A}(1 - p^{-s})^{-1} = sum_{n in B}n^{-s}
$$
by the Euler product formula. The Dirichlet series on the right hand side has nonnegative coefficients, so by a theorem of Landau, the point where the line of convergence crosses the real axis is a singularity of the sum function. If $A$ is finite, this singularity is at the origin. Thus if it can be established that the line of convergence is to the right of the imaginary axis, $A$ must be infinite (The only reason to invoke the theorem of Landau is to guarantee that there always is a singularity on the line of convergence). Of course, existence of a singularity of the sum function somewhere in the open right hand half plane would also work, even if we do not know the line of convergence.
This tends to fail because we can't get a good grip on $B$. Suppose $A$ is the set of twin primes (primes $p$ such that $p+2$ is also prime). Nothing really useful is known about the set $B$ of integers that are products of powers of twin primes. But if the accepted conjecture about the distribution of twin primes holds, there will be a singularity at $s = 1$, so $B$ cannot be really sparse.
As an example, there are infinitely many primes $p equiv 1pmod{4}$. Letting these primes, together with $2$, constitute $A$, we see that $B$ contains the
values of the polynomial $n^2 + 1$, since the latter is never divisible by any prime $q equiv 3pmod{4}$. Then
$$
sum_{n in B}n^{-sigma} geq sum_{m = 1}^{infty}(m^2 + 1)^{-sigma}
$$
and thus the Dirichlet series over $B$ has a singularity in the half plane $sigma geq 1/2$. So $A$ has to be infinite.
Admittedly, this example is not that interesting.
On reflection, SAS tells me that Euclidean geometry has a strong semi-local homogeneity, in that every neighborhood of every point is isotropically isomorphic with some neighborhood of any other point --- once you find a good way to say "neighborhood", that is.
The parallel postulate, on the other hand, can be used to construct *canonical* isomorphisms of point(ed)-neighborhoods --- by parallel translation of course; but since the constructed isomorphisms are all parallel in a good sense, we don't get the isotropy structure without SAS. (edit/add:ed): in the other direction, SAS doesn't give any canonical isomorphisms, which is just as well because hyperbolic and elliptical space both have SAS, but not the parallel postulate. (end edit)
The related postulate that Euclid states properly --- that all right angles are equal --- only gives a pointwise isotropy; it doesn't help much for segments subtended by respectively equal segments at equal angles.
The Wikipedia article is more technical than it should be, and for the reader in a hurry not all that well written. Here is a summary of the main points as best I understand them:
Complex manifolds are analogous to smooth complex algebraic varieties, not to the singular ones. But that discrepancy is surmountable, because you can also have complex analytic varieties which can have similar singularities. Then the first and most important relation is that every complex algebraic variety is a complex analytic variety. Every Zariski open set is analytically open; analytic gluing maps are more general than algebraic ones; and the allowed analytic charts are more general than the allowed algebraic charts. Also every algebraic morphism is an analytic morphism, so you get a morphism between categories.
But the connection is better than that because of the GAGA principle (globally analytic implies globally algebraic). My understanding of GAGA is very sketchy, but I think that the following is correct. Among other consequences of GAGA, a closed analytic subvariety of a proper (equivalent to compact) algebraic variety is algebraic. An analytic isomorphism between two proper algebraic varieties is algebraic. I would suppose that there is a similar principle for compact fibrations as well.
So, if you make compact analytic varieties algebraically, you can't escape from the algebraic class some of the main constructions of complex manifolds do not escape from the algebraic class. (But not all: deformations and infinite group actions can escape.) All projective analytic varieties are algebraic, and in dimension 1 all compact curves are projective. Moreover, there are limited ways for a compact analytic manifold to avoid being projective, by Moishezon's theorem and Kodaira's theorem. In practice, then, most of the complex manifolds that people make are algebraic. Also, most of the analytic calculations on a proper algebraic variety are algebraic: Many global calculations are algebraic by GAGA, and many local calculations are algebraic just by truncating Taylor series.
Contrast all this with real algebraic vs real analytic. It is still true that (the real points of) a smooth real algebraic variety is a real analytic manifold. More strongly than in the complex case, although it is highly non-trivial, every compact real analytic manifold is real algebraic. But the real algebraic structure is massively not unique, even for a circle, and that makes all the difference.
The other answerers in this thread, who are more expert in this topic than I am, had more information about why a compact complex manifold might not be a smooth projective variety. Just for clarity, I will restrict attention to the compact, smooth case. Also, you say "proper" rather than "compact" in the algebraic category because every algebraic variety is "compact" in the extremely coarse Zariski topology. An algebraic variety is proper if and only if it is analytically compact. The main use of the word proper is to emphasize that it is more general than projective, which means given by polynomial equations in complex projective space.
There are two very different initial reasons that an analytic complex manifold might not be projective. It might not be Moishezon: A complex $n$-manifold is Moishezon if it has $n$ algebraically independent meromorphic functions. (The number of algebraically independent elements or the transcendence degree of a field is called the Krull dimension. The meromorphic Krull dimension of a compact complex $n$-manifold is at most $n$.) Or it might not be Kähler: A complex $n$-manifold is Kähler if it has a Riemannian metric such that the covariant derivative of the complex structure vanishes. So to summarize what people said about compact complex manifolds (much of which is in the back of Hartshorne's book):
projective ⇒ algebraic ⇒ Moishezon ⟺ bimeromorphically projective
projective ⇒ Kähler ⇒ symplectic ⇒ non-zero $H^2$
algebraic ⇒ non-zero $H^2$ (exposited by David Speyer)
Moishezon and Kähler ⟺ projective (Moishezon)
Kähler and integrally symplectic ⟺ projective (Kodaira)
In addition, projective and algebraic structure and the Moishezon property are all unstable with respect to analytic deformation. And bimeromorphic equivalence preserves $pi_1$. Taubes found compact complex manifolds that have the wrong $pi_1$ to be Kähler; indeed they can have any $pi_1$. Voisin found compact Kähler manifolds with the wrong homotopy type to be projective, disproving Kodaira's conjecture that every compact Kähler manifold can be deformed to projective. And, way out in left field, a left-invariant complex structure (LICS) on a compact simple Lie group is a compact complex manifold that has no $H^2$ and can be simply connected too.
Still, despite these beautiful non-projective compact complex manifolds, it's generally easier to study projective examples. It's generally easier to sidestep analysis and do algebra instead.
Take a smooth closed curve in the plane. At each self-intersection, randomly choose one of the two pieces and lift it up just out of the plane. (Perturb the curve so there are no triple intersections.) I don't really know anything about knot theory, so I don't even know if I'm asking the right questions here, but I'm wondering: What is the probability that this is the trivial knot? What can we say about how knotted this knot might be, and with what probabilities? (Measure "knottedness" in whatever way you like.) More generally, can we say anything about the probability of the various possible values in the usual invariants that people use to study knots?
I only have an idea of how to approach the first question, and even then it's only by brute force. I was just playing around with the easiest cases, and I think that with 0, 1, or 2 intersections, all knots are trivial, and with 3 intersections the knot is trivial with probability 75%.
A general analysis should presumably involve calculating the probability that we can simplify using various Reidemeister moves, but I don't know how to incorporate this. I'd imagine a computer could brute-force the first few cases pretty easily (I'm not so bold as to venture an order-of-magnitude guess on whether it's the first few hundred or the first few million)...
I'm not an expert, but, here is my understanding. Right-fibrations are important because they are the infinity-version of a category fibered in groupoids (that is an infinity-category fibered in infinity-groupoids). In particular, given an infinity-category $C$,there is a model structure on $sSet/C$, called the contravariant model structure, such that the fibrant and cofibrant objects are precisely the right-fibrations over $S$, and this model structure is Quillen-equivalent (through a generalization of the Grothendieck construction) to the projective model-structure of simplicial presheaves over $w(C)$, where $w(C)$ is a simplicial category and $w$ is the left-adjoint to the homotopy-coherent nerve. Both of these (simplicial) model categories model $Fun(C^{op},infty-Gpd)$- the infinity-categeory of "weak presheaves in infinity groupoids". So, the upshot is, right fibrations are the infinity-analogue of Grothendieck fibrations in groupoids and provide a model for weak presheaves. This presheaf infinity-topos is the starting point for higher topos theory; infinity topoi are just left-exact (accessible) localizations of such presheaf-infinity categories.
Now, dually, left-fibrations are a model for "infinity-categories COfibered in infinity-groupoids".
The next step, is Cartesian-fibrations. Cartesian-fibrations are the infinity-version of categories fibered in categories (not necessarily groupoids), i.e. they are "infinity categories fibered in infinity-categories". Nearly everything above goes through again, except we need to work with marked-simplicial sets, where we "mark the cartesian-edges".
Again, dually, CoCartesian fibrations model "infinity categories cofibered in infinity-categories". You may wonder why we need both notions? In fact, we need both notions TOGETHER in order to define adjunctions. An adjunction between two infinity-categories $C$ and $D$ is a functor $K to Delta[1]$ which is simultaneously a Cartesian-fibration and a CoCartesian fibration, together with Joyal-equivalences $K_{0} cong C$ and $K_{1} cong D$. This definition is a generalization to the infinity-world of a characterization of adjunctions using cographs.
Now, Kan-fibrations are a relic of homotopy theory. They are the fibrations on the Quillen-model structure on simplicial sets. In a similar spirit, categorical fibrations are the fibrations in the Joyal-model structure on simplicial sets. Other than that, they are not that well behaved; they don't really play a role in infinity-category theory.
Finally, inner fibrations, as far as I know, are only used in defining Cartesian fibrations. That is, a Cartesian fibration is defined to be an inner fibration satisfying extra properties.
I hope this helps.
Given a finite CW complex X, there is a filtration of the topological K-theory of X given by setting $K_n(X) = ker left(K(X) to K(X^{(n-1)})right)$, where $X^{(n-1)}$ is the (n-1)-skeleton of X. (The choice of indexing here is from Atiyah-Hirzebruch.)
My question is:
How does this filtration interact with the external product $K(X)times K(Y)to K(X times Y)$? I believe the answer should be that $K_n (X) cdot K_m (Y) subset K_{n+m} (Xtimes Y)$.
Just to be clear, and to set notation, this external product is the one induced by sending a pair of vector bundles $Vto X$ and $Wto Y$ to the external tensor product, which I'll write $Vwidetilde{otimes} W = pi_1^* V otimes pi_2^* W to Xtimes Y$.
Of course, if $Vin K_n (X)$ and $W in K_m (Y)$, then $Vwidetilde{otimes} W$ restricts to zero in both $K(X^{(n-1)} times Y)$ and $K(X times Y^{(m-1)})$, and $(Xtimes Y)^{(n+m-1)}$ is contained in the union of these two subsets. Is there some way to deduce from this information that the class $Vwidetilde{otimes} W$ is actually trivial in $K((Xtimes Y)^{(n+m-1)})$?
Here's the reason I'm asking (which is really a second question, I guess).
In Characters and Cohomology Theories, Atiyah states (without comment) that for the internal product $K(X)times K(X)to K(X)$, one has $K_n (X) cdot K_m (X) subset K_{n+m} (X)$. In Atiyah-Hirzebruch, they state this formula and say that it "admits a straighforward proof."
I thought I remembered that the straighforward proof was the following:
Show that the external product satisfies $K_n (X) cdot K_m (Y) subset K_{n+m} (Xtimes Y)$
Observe that if $f:Xto Xtimes X$ is a cellular approximation to the diagonal $Xto Xtimes X$, then $f(X^{(n+m-1)}) subset (Xtimes X)^{(n+m-1)}$. So for any $V, Win K(X)$, we have $Votimes W = f^*(Vwidetilde{otimes} W)$, and if $Vin K_n (X)$ and $Win K_m (X)$, it then follows from 1. that $Votimes Win K_{n+m} (X)$.
Am I barking up the wrong tree here?
Presumably these questions will turn out to have an easy answer, but I've been thinking about them for a while now and haven't gotten any further. Any suggestions or references would be great! I haven't found any sources other than the two mentioned above that talk about the relation between skeleta and products, and neither of these sources mentions case of external products.
There is a projective model structure on the category of (pre)sheaves with value in any reasonnable model category (e.g. simplicial sets, complexes of abelian groups, commutative k-dg-algebras, where k is some field of char. 0, or k-dg-algebras for any commutative ring k); see for instance def. 4.4.33 and 4.4.40 and cor. 4.4.42 in Ayoub's book (Astérisque 315), whose online version is here (there is also a paper of Barwick which does the job if you want descent à la Lurie (to appear in HHA soon, I think)).
If you have a site C and a left Quillen functor F:M->M', you get a left Quillen functor Sh(C,M)->Sh(C,M') between model categories of sheaves. The fibrant objects will always have the good taste of being exactly the termwise fibrant sheaves which satisfy (hyper)descent, and, if you have enough points, the weak equivalences are defined stalk-wise.
If M=SSet and M'=Complexes of R-vector spaces, the usual adjunction SSet<->Comp(R) gives a Quillen adjunction:
Sh(C,Sset) <-> Sh(C,Comp(R))
If you consider the projective model structure on Sh(C,Sset), any simplicial sheaf X such that, for each n, X_n is a sum of representables, is cofibrant (and any cofibrant object is weakly equivalent to such a thing). Hence, if C={differential real manifolds}, if you allow your manifolds to be stable under small sums, the simplicial manifolds are just your favourite cofibrant objects. You can thus apply all the machinery of model categories (e.g. the homotopy category of cofibrant objects is equivalent to the whole Ho(M) for any model category M). As the de Rham complex satisfies descent on manifolds, then it is a fibrant object (for the projective model structure on sheaves on complexes), and using the above Quillen adjunction, you will get that the derived sections of the de Rham complex over a simplicial manifold behaves like maps from a cofibrant object to a fibrant object in any model category: quite well. From there, you should be able to prove that de Rham cohomology of simplicial manifolds is in fact the explicit description of the total left derived functor of the left Quillen functor DR:Sh(manifolds,Sset)->cdga^op.
[Added comments] You can also consider the left Bousfield localization L_R of the projective model category of simplicial sheaves on the category of manifolds which consists to invert the maps XxR->X for any manifold X (where R denotes the real line). The result is a model category which is Quillen equivalent to the model category of simplicial sets. Using the Poincaré lemma (which gives you the homotopy invariance of de Rham cohomology), the functor
DR:Sh(manifolds,Sset)->cdga^op induces a functor of shape
Ho(Sset)=Ho(L_R Sh(manifolds,Sset))->Ho(cdga)^op
which is simply (isomorphic to) the functor of Quillen-Sullivan.
Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $widetilde{X}$ is Kahler, it is an irreducible symplectic variety, although it may fail to be projective.
Are there any criteria/known cases where one can guarantee the projectivity of $widetilde{X}$?
I know this is a little late for Christmas, but nevertheless, I have a few (some of which have already been mentioned) books I've read that I've quite enjoyed. For the sake of brevity, I'll let you search the titles on Amazon for reviews and better descriptions.
Title: Everything & More: A Compact History of Infinity
Author: David Foster Wallace
Title: The Mathematical Experience
Author(s): Philip J Davis & Reuben Hersh
Title: One, Two, Three...Infinity
Author: George Gamow
Title: Pi in the Sky
Author: John D. Barrow
Title: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Author: John Derbyshire
Title: Strength in Numbers
Author: Sherman Stein
Title: e: The Story of a Number
Author: Eli Maor
Title: A History of Pi
Author: Petr Beckmann
Title: Nature's Numbers
Author: Ian Stewart
Title: Mathematics: The Science of Patterns
Author: Keith Devlin
Title: Zero: The Biography of a Dangerous Idea
Author: Charles Seife
Title: How to Enjoy Calculus
Author: Eli S. Pine
(Not really a "popular" book, per se', but still pretty good)
Title: How to Think About Weird Things
Author(s): Theodore Schick & Lewis Vaughn
(Not really about mathematics, but not so far out of the way that you wouldn't enjoy it if you also enjoy mathematics)
We need to check $eta(u,v)+eta(v,w)geeta(u,w)$. Introduce coordinates $x,y,z$ so that the form is $x^2+y^2-z^2$.
First, verify that there is a Lorentz map sending $v$ to $(0,0,1)$. Since it is an isometry, we may now assume that $v=(0,0,1)$. This is the main idea. For added convenience, you may also rotate the $xy$-plane so that the $y$-coordinate of $u$ equals 0.
Next, observe that the formula yields equality in the case when the projections of $u$ and $w$ to the $xy$-plane are endpoints of a segment containing (0,0). This is straigtforward if you write $u=(sinh a,0,cosh a)$ and $w=(-sinh b,0,cosh b)$ where $a,bge 0$.
Finally, rotate $w$ around the $z$-axis until it comes to a position as above. The product $ucdot w$ grows down (it equals contant plus the scalar product of the $xy$-parts, since $z$-coordinates are fixed). Hence $eta(u,w)$ grows up while the two other distances stay, q.e.d.
Of course, for writing purposes the last step is just an application of Cauchy-Schwartz for the scalar product in $mathbb R^2$.
This was about two-dimensional hyperbolic plane, in higher dimensions just insert more coordinates (they will not actually show up in formulae).
A natural (iso)morphism is one that is "canonical", or defined without making "choices", or that is defined "in the same way" for all objects.
This is a heuristic I found in every introductory text on category theory I can remember reading (and usually followed with the single/double dual of a vector space as an example) and it took me quite a while to realize that this is not only inaccurate, but just plainly wrong.
Explanation of "wrongness": A natural morphism is a morphism between two functors. That is, a morphism in the category of functors between two categories. And as such, should be thought as usual as mapping the "data" in a way that preserves the "structure" and choices have really nothing to do with it.
For example, thinking of a group $G$ as a one object category, functors from it to the category of sets form the category of $G$-sets. A morphism of $G$-sets is a map of sets preserving the action of $G$ and not a map of sets that "does not involve choices". Same goes for other familiar categories of functors (representations, sheaves etc.)
Another example is the category of functors from the one object category $G$ again to itself. To give a natural map (isomorphism) from the identity functor of $G$ to itself is just to pick an element of the center of $G$. I don't imagine anyone describing it as doing something that "doesn't involve choices".
Moreover, every category $C$ is the category of functors from the terminal one-object-one-morphism category to $C$. Hence, every morphism in any category is a "natural morphism between functors" so there is really no point in specifying a heuristic for when a morphism is "natural". This is utterly meaningless.
In the other direction, it is easy to write down "canonical" object-wise maps between two functors that fail to be natural in the technical sense. Conisder the category of infinite well ordered sets with weakly monotone functions. The "successor function" is definitely defined "in the same way" for all objects, but is not a natural endomorphism of the identity functor in the technical sense.
Explanation of harmfulness": Well I guess it is clear that a completely wrong heuristic is a bad one, but I'll just point out one specific example that is perhaps not so important, but shows clearly the problem. When showing that every category is equivalent to a skeletal category there is a very "non-canonical" construction of the natural isomorphisms. I saw several people get seriously confused about this.
Some thought: One might argue that this heuristic was advanced by the very people who invented category theory (like Maclane) and thus, it is perhaps a bit presumptuous to declare it as "plainly wrong". My guess is that at the time people where considering mainly large categories (like all sets, all spaces, all groups etc.) as both domain and codomain of functors and were focusing on natural isomorphisms. In such situations it is unlikely that the functor will have non trivial automorphisms (or have very few and "uninteresting" ones) and therefore a natural isomorphism will be in fact unique so maybe this is the origin of the heuristic (It is just a guess, I am not an expert on the history of category theory).
This relates to the point that by definition, if specifying an object does not involve choices, then it is unique (this is a tautology). So when we say that an isomorphism is "canonical" we usually mean that given enough restrictions, it is unique (and not just natural in the technical sense). For example, the reason we identify the set $Atimes (B times C)$ with the set $(Atimes B)times C$ is not because there is a natural isomorphism between them, but because if we consider the product sets with the projections to $A,B$ and $C$, then there is a unique isomorphism between them. And this is in line with the general philosophy of identifying objects when (and only when) they are isomorphic in a unique way. In contrast, we don't identify two elements of a group $G$, just because they are conjugate (This is "naturally isomorphic" viewed as functors of one object categories $mathbb{Z}to G$) precisely because this natural isomorphism is not unique.
Well, I did not intend this to get so lengthy... I was just anticipating some "hostile" responses defending this heuristic, so I tried to be as convincing as possible!
Hello, this is my first post here. I am no mathematician and English is not my first language, so please excuse me if my question is too stupid, it is poorly phrased, or both.
I am developing a program that creates timetables. My timetable-creating algorithm, besides creating the timetable, also creates a graph whose nodes represent each class I have already programmed, and whose arcs represent which pairs of classes should not be programmed at the same time, even if they have to be reprogrammed. The more "heavily linked" a node is, the more inflexible its associated class is with respect to being reprogrammed.
Sometimes, in the middle of the process, there will be no option but to reprogram a class that has already been programmed. I want my program to be able to choose a class that, if reprogrammed, affects the least possible number of other already-programmed classes. That would mean choosing a node in the graph that is "not very heavily linked", subject to some constraints with respect to which nodes can be chosen.
Do you know any algorithms that solve this problem?
it really depends on how you're doing the hierarchical clustering, but if (for example) you're using a sum-of-squares measure, then the centroidal merging that Mikael recommends is in fact the exact representative.
There are entire families of clustering algorithms that rely on the ability to store representatives of lower parts of the clustering and merge them. While these are not hierarchical per se, they use a hierarchical approach, and so the concept is the same. For reference, check out the BIRCH algorithm:
Let $W = M + D$, where $M$ is the original $n times n$ matrix and $D$ is the added diagonal matrix that we want to determine.
$W$ is symmetric, thus diagonalizable by an adjoint action of the orthogonal group.
Larger multiplicities in the eigenvalues of $W$ imply smaller dimensions of the adjoint orbits.
For example, if we have an eigenvalue of multiplicity $n-1$, then the adjoint orbit will be
$O(n)/(O(n-1) times O(1)) = S^{n-1}/Z_2 = RP^{n-1}$ of dimension $n-1$;
while, if all the eigenvalues are distinct, the adjoint orbit is the real flag manifold $Fl_{mathbb{R}}^n = O(n)/(Z_2)^n$ of dimension $ frac{n(n-1)}{2}$. (Of course, in the case of scalar multiple of the unit matrix, the adjoint orbit is just a single point).
Thus in order to achieve large multiplicities, we need to minimize the dimension of the adjoint orbit.
This problem can be reduced to a problem of matrix rank minimization as follows:
Let ${l_i }_{i=1,...,n(n-1)/2}$, be a set of generators of the Lie algebra of $O(n)$ normalized according to:
$textrm{tr}(l_i l_j) = delta_{ij}$. The dimension of the adjoint orbit equals the rank of the Gram matrix $C$ whose elements are given by, $C_{ij} = textrm{tr}([l_i, W][l_j, W])$. The problem is thus reduced to the minimization of the rank of the Gram matrix whose elements depend linearly on the added diagonal matrix elements.
One of the possible methods to solve this problem is through a convex programming
heuristic approach for the solution of matrix rank minimization,
based on replacing the rank by the nuclear norm (the sum of the singular values), as explained in the following lecture notes by: P.A. Parillo. The nuclear norm is a convex envelope of the rank which may explain why this method works well in practice in general.
Here is a nice cyclic ordering of the eight geometries:
HxR, SxR, E^3, Sol, Nil, S^3, PSL, H^3
derived from staring at Peter Scott's table of Seifert fibered geometries. The table is organized by Euler characteristic of the base 2-orbifold and Euler class of the bundle. (See his BAMS article.) The cyclic ordering also has a bit of antipodal symmetry.
I didn't come up with geometric pictures of the eight geometrics but I have thought about "icons" to represent them. Here are my suggestions - I'm interested to hear what other people think/suggest.
I think it is also reasonable to ask for a "prototypical" three-manifold for each of the eight geometries. Here is an attempt:
Notice that all of the examples are either surface bundles over circles or circle bundles over surfaces, or both (ie products).
Yes you can get a $phi$ that is a homomorphism. Here is a quick sketch.
First let $p=sup$ {$p_alpha,$ projections in $Ker theta$}. So $pin Ker theta$. Furthermore $pin Z(M)$, the center of M.
To see this note that if this were not true then we could find a unitary $uin M$ with $pneq upu^star$. So then $pwedge upu^star$ would be a projection in $Ker theta$ bigger than $p$.
From here you can get that $Ker theta=pMp$, and so we can decompose $M=Ker theta oplus M_1$ and $theta|_{M_1}$ is injective and thus an isomorphism, thus $phi$ can be chosen to just be the inverse of $theta$ on $M_1$.
Note that if we demand that $phi$ be unital, this doesn't work and I don't think it can be done in general. I will have to give it more thought.
I'm not sure this is an answer, but it got too long to be a comment! The projectivity of $G/B$ seems to me to follow from two facts: a) a homogeneous space $G/H$ is always a quasi-projective variety (which is due to Chevalley), and b) the variety $G/B$ must be complete.
The first fact clearly has nothing to do with Borel subgroups but it would be maybe interesting to ask about how many proofs we have of it. The idea of the construction is clear: find a representation of $G$ which contains a line $L$ which has $H$ as its stabilizer and then take the orbit of $L$ in $mathbb P(V)$, but to see that you get a categorical quotient this way takes some more care: you need to use some infinitesimal properties, (which you can tidily say using the Lie algebra of course).
The second fact perhaps depends on what you're willing to assume, but it must come down to the Borel fixed point theorem in some form or other, since this tells you that if $G/H$ is complete, then any solvable subgroup will be contained in a conjugate of $H$, thus Borel subgroups are the only solvable subgroups with a chance of having an associated homogeneous space which is complete.
The argument from Humphreys' book uses the strategy of picking first a maximal solvable subgroup of largest dimension so as to show the orbit has to be closed (essentially because the stabilizer is largest so the orbit has smallest, but even then you use the Borel fixed point and the theorem for $GL_n$ if I remember correctly). I don't know how Onishchik and Vinberg get around this (if they do). Then, as the question says, you get the general result from the Borel fixed point theorem.
The proof for $GL_n$ shows that $G/B$ is projective if you take $B$ to be the subgroup of upper triangular matrices, but one still needs to show that this subgroup is a Borel, which again I only know how to do using the Borel fixed point theorem in some form. (And of course you can use the same strategy for other classical groups if you can eyeball a candidate Borel subgroup).
Given that, it's maybe worth pointing out that in all of this you can get away with a weak version of the Borel fixed point theorem: namely if $V$ is a representation of a solvable group $H$, and $X$ is an $H$-stable closed subvariety of $mathbb P(V)$ then $X$ has an $H$-fixed point. This can be shown just using the Lie-Kolchin theorem (that is, that a representation of a solvable group contains a one-dimensional subrepresentation), which can be proved directly. Since the standard proof of the general Borel fixed point requires you to use something like Zariski's main theorem, this is maybe a noticeable saving.
All of this leads me to wonde how many proofs do we know for i) the fact that for any closed subgroup $H$ the homogeneous space $G/H$ has to be quasi-projective and ii) the Borel fixed point theorem (or some variant)?
I don't know who first observed this (maybe Archimedes?) but it is true because $C({0,1 }^Gamma)$ is a quotient of $ell_1^Gamma$ and hence $ell_1(2^Gamma)$ embeds into $ell_infty(Gamma)$.
@Ady
Here is a more serious answer to your question. Take a quotient map $Q$ from $ell_1(2^Gamma)$ onto $C([0,1]^{2^Gamma})$ and extend to a norm one mapping $T$ from $ell_infty(Gamma)$ into some injective space $Z$ that contains $C([0,1]^{2^Gamma})$ (you cannot extend $Q$ to an operator from $ell_infty(Gamma)$ into $C([0,1]^{2^Gamma})$ because, e.g., $C([0,1]$ is not a quotient of $ell_infty$). Use partitions of unity to get a net $(P_a)$ of norm one finite rank projections on $Z$ taking values in $C([0,1]^{2^Gamma})$ and whose restrictions to $C([0,1]^{2^Gamma})$ converge strongly to the identity. A weak$^*$ cluster point of $(P_a^* T^*)$ gives an isometric embedding of the dual of $C([0,1]^{2^Gamma})$ (which contains $L_1([0,1]^{2^Gamma})$) into the dual of $ell_infty(Gamma)$. Thus if $Y^*$ is any reflexive subspace of $L_1([0,1]^{2^Gamma})$, such as $ell_2(2^Gamma)$, then $Y$ is isometric to a quotient of $ell_infty(Gamma)$.
Even when viewed as an additive category, Rep(G) is not semisimple, so it's not really clear to me what such a description would entail... But simple objects in it are (in a different language) irreducible representations of a semidirect product of groups and Mackey theory was invented precisely with the goal of determining them. The answer, in short, is that each simple G-module has the form $operatorname{Ind}_{L}^{G}(sigma^{prime})$, where $sigma$ is a simple $H$-module, $K_sigma$ is the stabilizer of $sigma$ in $K$ (i.e. consists of all elements $kin K$ s.t. the action of $k$ on $H$ conjugates $sigma$ into an isomorphic module), $L=K_{sigma}H<G$ and $sigma^{prime}$ is the natural extension of $sigma$ to $L$. The catch is that the usual Mackey theory applies to unitary representations (which can be infinite-dimensional) of topological groups. Nevertheless, induction functor is defined in the algebraic setting and I think that the "Mackey machine" works for formal reasons (this must be described in Jantzen, but I don't have it close at hand to check).
Returning to the full category Rep(G), there are various filtrations of finite-length representations with simple quotients, and one can take tensor products of filtered objects. However, in general one doesn't expect a manageable description of Rep(G) even in the special situation of a semidirect product of a unipotent group and a torus: this already includes the case when G is the Borel subgroup of a semisimple algebraic group, which has been studied but is not completely understood. Good luck!
Addendum Of course, if G is a solvable connected algebraic group, by Lie – Kolchin every irreducible representation is one-dimensional, so simple G-modules are the same as simple G/[G,G]-modules, which have an easy description.
More on the artistic side I appreciated
The Invention of Infinity: Mathematics and Art in the Renaissance by J.V. Field
Its theme is the interaction of mathematical and artistic inquiries as characteristic of Western art in the Renaissance, with perspective and precise description of geometrical forms (such as polyhedra) as turning point, and embodied in several key artists such as Piero della Francesca, Leonardo da Vinci, Albrecht Dürer.
The same author has written a book dedicated to Piero della Francesca:
Piero Della Francesca: A Mathematician's Art
Another source are books about the camera obscura and pinhole photography.
More contemporary: the techniques used to enhance digital images and their perspective with mathematical models of camera lens deformation have given birth to relatively sophisticated applied mathematics. There are a few private companies such as DxO selling software to correct (among other things) perspective in image files by reversing the nonlinear effects of multiple lens systems found in camera.
Let $mathcal{C}$ be a Waldhausen category. Suppose that $mathcal{B}$ is a subcategory of $mathcal{C}$, and that $mathcal{B}$ is closed under extensions. If $mathcal{B}$ is strictly cofinal in $mathcal{C}$ (in the sense that given any $Cin mathcal{C}$ there exists a $Bin mathcal{B}$ such that $Camalg Bin mathcal{B}$), can we say anything about $K(mathcal{B}) rightarrow K(mathcal{C})$?
In Waldhausen's paper "Algebraic K-theory of spaces" Waldhausen claims that the inclusion $mathcal{B}rightarrow mathcal{C}$ induces a weak equivalence $wS_bullet mathcal{B}rightarrow wS_bulletmathcal{C}$ (and thus an equivalence on K-theories), but I'm not sure that this is right, as $mathcal{B}$ does not need to be a full subcategory. In particular, if there are objects $C,C'$ which are in $mathcal{B}$ but are not isomorphic in $mathcal{B}$ they may well be isomorphic (or at least weakly equivalent) in $mathcal{C}$.
Consider the following example. Let $mathcal{C}$ be the category of pairs of pointed finite sets, whose morphisms $(A,B)rightarrow (A',B')$ are pointed maps $Avee Brightarrow A'vee B'$, and let $mathcal{B}$ be the category of pairs of pointed finite sets whose morphisms $(A,B)rightarrow (A',B')$ are pairs of pointed maps $Arightarrow B$ and $A'rightarrow B'$. We make $mathcal{C}$ a Waldhausen category by defining the weak equivalences to be the isomorphisms, and the cofibrations to be the injective maps. $mathcal{B}$ is clearly cofinal in $mathcal{C}$, but $K_0(mathcal{B}) = mathbf{Z}times mathbf{Z}$, while $K_0(mathcal{C}) = mathbf{Z}$. Going even further, the Barratt-Priddy-Quillen theorem should tell us that $K(mathcal{B}) = QS^0times QS^0$, while $K(mathcal{C}) = QS^0$.
If we add the condition that $mathcal{B}$ needs to be a full subcategory of $mathcal{C}$, then I believe that Waldhausen's paper is correct. But even without that, it is possible to say anything about the map $K(mathcal{B})rightarrow K(mathcal{C})$?
Most of the results/conjectures I'm aware of in asymptotic group theory (involving whether "most" groups satisfy a certain property) assign an equal weight to all groups of the same size. For instance, the conjecture that "most groups are nilpotent" claims that the quotient of the number of nilpotent groups of order at most $n$ to the total number of groups of order $n$ approaches 1. (See this paper for a survey).
But it may also be beneficial to assign different "weights" to different p-groups, similar to the Cohen-Lenstra measure assigned on the set of all abelian p-groups, which weights them by the reciprocal of the size of their automorphism group (see this MathOverflow discussion and this blog post by Terence Tao).
What are the typical or desirable ways of doing this for finite p-groups? In other words, what are the typical measures that we can assign to various sets of finite p-groups, such as:
This question is motivated by this (highly recommended) comment by Emerton on Terry Tao "Learn and relearn your field" post. In particular, the following paragraphs:
In particular, the first couple of sections of Ribet’s famous Inventiones 100 article give a great example of how hundreds of pages of theory (including a lot of the theory of Neron models, and a quite a bit of SGA 7) can be summarized in ten or so pages of “working knowledge”.
and
For example, p-adic Hodge theory is another tool which plays a big role in a lot of arithmetic algebriac geometry, and which has a technically formidable underpinning. But, just like etale cohomology, it has a very nice formalism which one can learn to use comfortably without having to know all the foundations and proofs.
My question is: Do you know of other sources that provide useful manual for your favorite technical topic? (the kind of paper or section of a book which you use time and time again and would recommend it for new researchers.)
As an example, where can I find the "nice formalism" of p-adic Hodge theory Emerton was referring to? (I was trying to learn more about almost ring theory and such a reference might be useful for bigger picture)
Here is another example, just to fix ideas, since the question is a bit vague. At some point in graduate school I was trying to learn about "Riemann-Roch theorem for local rings". I found Fulton's intersection theory book, Chapter 18 and this paper by Kurano to be quite helpful.
This is obviously community wiki, so one topic per answer please. Answers with more description of the sources and how to use them would be very appreciated. Thanks a lot.
Since $mathbb{R}$ and any 3-manifold $N$ must be non-exotic, their product $mathbb{R}times N$ cannot possibly be diffeomorphic to exotic $mathbb{R}^4$, correct?
Update: Andy Putman already answered this question in a different thread, as pointed out by Steven Sivek below. The answer is yes, but not for the reasoning I implied above, because, I gather, the product could in principle be taken in a nontrivial way that alters the differentiable structure.
The proof outlined by Andy relies on $mathbb{R}times N$ being piecewise linearly isomorphic to $mathbb{R}^4$, which is said to be proved in "Cartesian products of contractible open manifolds" by McMillan, which happens to be freely available here:
http://www.ams.org/journals/bull/1961-67-05/S0002-9904-1961-10662-9/S0002-9904-1961-10662-9.pdf .
The relevant part of that paper is as follows:
"A recent result of M. Brown asserts that a space is topologically $E^n$ if it is the sum of an ascending sequence of open subsets each homeomorphic to $E^n$.
THEOREM 2. Let $U$ be a $W$-space. Then $Utimes E^1$ is topologically $E^4$
Proof. Let $U=sum_{i=1}^{infty}H_i$ where $H_i$ is a cube with handles and $H_isubseteq text{Int } H_{i+1}$. By the above result of Brown, it suffices to show that if $i$ is a positive integer and $[a,b]$ an interval of real numbers ($alt b$), then there is a 4-cell $C$ such that
$H_itimes[a,b]subseteqtext{Int }Csubseteq Csubseteq Utimes E^1$.
There is a finite graph $G$ in $(text{Int }H_i)times{(a+b)/2}$ such that if $V$ is an open set in $Utimes E^1$ containing $G$ then there is a homeomorphism $h$ of $Utimes E^1$ onto itself such that $h(H_itimes[a,b])subseteq V$. But $G$ is contractible to a point in $Utimes E^1$. Hence, by Lemma 8 of [Bull. Amer. Math. Soc. 66, 485 (1960)], a 4-cell in $Utimes E^1$ contains $G$, and the result follows."
A $W$-space was earlier defined as a contractible open 3-manifold, each compact subset of which is embeddable in a 3-sphere.
I'm not sure what it means for a simply connected manifold such as $mathbb{R}^3$ to be equal to an infinite sum of cubes with handles, but given that, can we say that the above machination qualifies as a piecewise linear isomorphism because each $H_itimes[a,b]$ can be covered with a chart, and each $C$ can be covered with a chart, such that there is a linear mapping between the two?
I don't have much time, but maybe the following can lead you to a solution. I'm sloppy too, writing $G$ both for the algebraic group (over some finite field $k$) and for the set of points $G(k)$.
This is semilar to a special case of a formula of Humphreys on Deligne-Lusztig characters.
("Deligne-Lusztig characters and principal indecomposable modules". J. Algebra 62 (1980), no. 2, 299--303.) The special case says that
$sum_{w in W} R_{T_w}(1) = |W| St_G$,
where $W$ is the Weyl group and $T_w$ the rational maximal torus defined by $w$ (by twisting a fixed rational maximal torus $T$), and $St_G$ is the Steinberg character. In
the simplest case, $T$ is split, and then the assignment $w mapsto T_w$ induces a bijection between conjugacy classes in the Weyl group and rational conjugacy classes
of rational maximal tori. This is like the multiplicites you found for $GL_3$. Also
signs come up, since the sign of the "dimension" of a DL representation is related to the parity of the split rank of the
corresponding torus.
[Note that $R_T(1) = Ind_B^G(1)$ if $T$ is split and lies in the rational Borel $B$.]
(Humphreys proved it for $G$ simply connected, semisimple, split algebraic groups over finite fields and Jantzen (unpublished)
generalised it to arbitrary reductive groups over finite fields.)
EDIT: $R_T(theta)$ is the virtual representation of DL defined by the rational maximal torus $T$ and the character $theta$ of the finite group $T(k)$.
For $GL_2$ the formula boils down to $(St_G+1)+(St_G-1) = 2St_G$.
Let $fin mathbb{C}[x_1,dots, x_n]$ be a reduced homogeneous polynomial of degree n.
Let $mathfrak{g}={ delta in Der_{mathbb{C}^n}|delta(f)in (f)mathcal{O}_{mathbb{C}^n} text{ and weight}(delta) =0 }$ be a (reductive) complex Lie algebra with minimal system of generators $langle sigma_1, dots, sigma_s, delta_1, dots, delta_rrangle$ such that:
Is it true that the centre of $mathfrak{g}$ is made only of diagonalizable elements?
Take $A$ local (you already reduced to it), with $m$ the max. ideal. I claim that $A/m$ is a finite field. Suppose first that it has char. 0. Then we get injections $mathbb Z to mathbb Q to A/m$. By Zariski's lemma, $mathbb Q to A/m$ is finite, since it is of finite type.
Now (unfortunately I don't have it on me), Atiyah-Macdonald have a beautiful lemma which says that if $A subset B subset C$ are (comm.) rings, $A$ noetherian, $A subset C$ of finite type, $B subset C$ finite, then $A subset B$ is of finite type.
In our case, $mathbb Z to mathbb Q$ is of finite type, contradiction. Thus $mathbb Z/p to A/m$ is of finite type, hence finite for some prime number $p$. So $A/m$ is a finite field. Also $m^n = 0$ for some $n$ since $A$ is artin local. Finally, $m^i/m^{i+1}$ is a f.d. $A/m$-vector space (since $A$ is noetherian), so it is finite as well. And $|A| = sum |m^i/m^{i+1}|$.
From A Mathematician’s Apology, G. H. Hardy, 1940:
"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. ... I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself."
Have matters improved for the elderly mathematician? Please answer with major discoveries made by mathematicians past 50.
For algebraic curves $C$ over a closed field, a correspondence on $C$ is a the same thing as a divisor, and so, a line bundle on $C times C$. Can I assume that this simplification does not extend to the case of general varieties? Does there exist a nice characterization of those correspondences 'expressible' as vector bundles?
In the case of finite fields, is there a bundle formation of the Frobenius correspondence $(v,$Fr$(v)) in V times V$ in terms of bundles?
I am particularly interested in the case of projective and Grassmannian spaces.
I can share what I did having a similar concern in mind, but it was for point-set topology, not linear algebra. I am not sure how much of this can be translated to linear algebra, since student's minds are already full of preconceptions about what a vector space, but not about what a topological space is.
After many years of tutoring point-set topology, I observed that students systematically thought of all topological spaces as $mathbb{R}^n$, and that they always wanted to use balls, even if the topology was non-metrizable. Hence, when I got to teach my own point-set topology course, I tried something a bit radical: I did not talk about metric spaces at all until later in the course.
I started with motivation. On the second day, I defined the notions of topology, homeomorphism (but not continuous function), and convergence of a sequence. Then I did only small finite examples first. I gave the students the following exercise: 1) How many topologies can you define in {0,1,2}; 2) How many of them produce homeomorphic topological spaces?; and 3) In how many of them does the sequence $0,1,0,1,0,1, ldots$ converge to $2$? Then I made sure to give students enough time (and guidance) to solve this exercise before moving to anything else.
I wanted to force the students to accept the abstract notion of topology and to not be scared by it (and to realize that everything we do in point-set topology is logical). Also, in this example, there is no way a student is going to attempt to use balls (particularly when I have not talked about balls). I think it worked well.
A quick counter-example to the question as stated is $a_0=0$, $a_1=1$ $a_2=-1$. Since $2arccos(0)-arccos(b)-arccos(-b)=0$ for all $b$, we have $2M_1-M_2-M_3=0$ where $M_1, M_2, M_3$ are the rows of the matrix.
So it is better to assume that the $a_i$ are nonnegative. In this case, the answer is yes. More generally, consider this problem for a function $f$. You want to choose $n$ linearly independent vectors from the set
$$
X := { (f(a_1x), f(a_2x),dots, f(a_nx)) mid xinmathbb R } subset mathbb R^n
$$
This is not possible if and only if $X$ lies in an $(n-1)$-dimensional subspace. This means that there are constants $c_1,dots,c_n$ (not all zero) such that $c_1v_1+dots+c_nv_n=0$ for all $vin X$. Or, equivalently,
$$
c_1 f(a_1x) + dots c_n f(a_nx) = 0
$$
for all $xinmathbb R$ (such that all $a_ix$ belong to the domain of $f$). In other words, the functions $xmapsto f(a_ix)$ are linearly dependent over $mathbb R$.
This cannot happen if $f=arccos$. Indeed, assuming that $a_n$ is the maximum of $a_i$ such that $c_ine 0$, the above sum is well-defined and analytic on $[0,1/a_n)$ but its derivative goes to infinity as $x$ approaches $1/a_n$. Hence it is not constant on $[0,1/a_n)$, and hence non-constant in any neighborhood of 0.
UPDATE. A similar argument shows that the answer is the same for any non-polynomial function analytic near the origin (assuming $a_i>0$). Indeed, if $f(x)=sum_{j=1}^infty q_j x^{k_j}$ where $q_jne 0$, then Taylor expansion of the identity
$$
c_1 f(a_1x) + dots c_n f(a_nx) = 0
$$
implies that $sum_i c_i a_i^{k_j}=0$ for all $j$. This cannot happen because, if $a_n$ is the maximum of $a_i$, the term $a_i^{k_j}$ grows faster (or decays slower) than all other terms as $k_jtoinfty$.
The short answer is that they are very different, but become quite similar if you 1) stabilize, i.e invert smash product by $mathbb{P}^1_k$ on the homotopy side and invert tensor product by the Tate motive of the motivic side and 2) pass to rational coefficients. This is the analogue of the similar result in topology : the rationalized stable homotopy category is equivalent to the derived category of $mathbb{Q}$-vector spaces.
The precise comparison result if you do those two operations was announced by Morel in http://www.mathematik.uni-muenchen.de/~morel/Splittinggrassman.pdf and a proof was written down recently by Deglise and Cisinski in the preprint http://www.math.univ-paris13.fr/~deglise/docs/2009/DM.pdf paragraph 15.2
Even the stable, integral versions are quite different : one way to quantify this is to say that spectra in $SH_{mathbb{A}^1}(k)$ represent generalized cohomology theories - oriented ones like motivic cohomology, algebraic K-theory, algebraic cobordism but also non-oriented like Balmer-Witt groups, Hermitian K-theory - while objects in $DM_{k}$ represent only "oriented cohomology theories with additive group law" (in the sense of Quillen) : see e.g the Déglise-Cisinski preprint above, paragraph 10.3
For the different between unstable versions, a good simple example is the case of curves of genus greater that 1 : their effective motives are non-trivial (weight one effective motivic cohomology detects Pic ) while their unstable homotopy type is in a sense completely disconnected. This is essentially the reason why unstable $mathbb{A}^1$-homotopy seems most interesting for "nearly rational" varieties, see the papers of Asok and Morel.
Does there exist a two variable analogue of the Weil conjecture?
What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points of a smooth algebraic variety over finite fields of characteristic $p$. Is it possible to have a sensible two-parameter family of finite rings instead? Any references?
For instance, one can consider finite quotients of Witt vectors, and form a two-parameter family of numbers $V_{n,m} = V ( Witt(GF(p^n))/I^m)$ (where $I$ is the maximal ideal of the Witt vectors) from a variety $V$ (smooth, projective over $mathbb Z$). Is there a sensible two variable zeta-function cooked with these numbers?
For $d=3$ the homotopy groups can be pretty elaborate. Consider the connect-sum of some lens spaces. The universal cover embeds in $S^3$ as the complement of a cantor set (except for a few degenerate cases where you have $mathbb RP^3$ summands). So the homotopy-groups are pretty complicated ($pi_2$ is finitely generated over $pi_1$). You could probably make an argument that this is about the worst thing that can happen for the homotopy-groups of 3-manifolds.
You might want to phrase your question as a question about the Postnikov towers of manifolds. Eilenberg-Maclane spaces are rarely compact boundaryless manifolds.
edit: I guess another spin on your question could go like this. We know the fundamental groups of compact manifolds are all possible finitely presentable groups provided $n geq 4$. So is there a sense in which the homotopy-algebras of manifolds can be anything finitely presentable? Say, for example, $pi_2$. As a module over the group-ring of $pi_1$, are there any restrictions beyond being finitely generated? I suppose you could construct a compact $6$-manifold with $pi_2$ (almost) any finitely-presented thing over any finitely-presented $pi_1$ pretty much the exact same way $4$-manifolds with any finitely presented $pi_1$ are constructed. I think if $H_2(pi_1)$ is non-trivial you might run into problems following the analogous construction, in that $pi_2$ might strictly contain the $pi_2$ you're trying to create.
2nd edit: So regarding 3-manifolds I think your question has something of an answer now, right? $pi_n M$ is $pi_n$ of the universal cover provided $n > 1$. The universal cover of a geometric 3-manfold is homeomorphic to $mathbb R^3$ or $S^3$. So by climbing up the JSJ and connect sum decomposition of a 3-manifold, the universal cover is diffeomorphic to a punctured $S^3$ -- the number of punctures is either $0$, $1$, $2$ or a Cantor set's worth of punctures. In the Cantor set case we're giving this complement the compactly generated topology induced from the Cantor set complement's subspace topology. So among other things, $pi_2 M$ is a direct sum of copies of $mathbb Z$, similarly $pi_3 M$, torsion first appears in $pi_4 M$. The complement you think of as a directed system of wedges of $S^2$'s so the Hilton-Milnor theorem tells you the homotopy groups.
The method of choosing a solution Ansatz to an equation and then actually deriving an exact solution is quite common in soliton theory, which is a sub-field of the study of hyperbolic equations. All methods described below, to my knowledge, only work on hyperbolic equations. Sorry, diffusion folks.
You must know properties of your equations to know which Ansatz will yield reasonable or good results. If you know that the tails of the solution die off quickly, you may choose a Gaussian $$ A exp^{-b x^2} $$, or if they die off very quickly, a super Gaussian $$ A exp^{-b a(x)^2} $$, where a(x) can be any polynomial. Also, based on the properties of your equation, you may want to multiply these 'basic' Ansatzen by other functions, to represent behavior that is known to be present. For example, if you know that solutions to the equation are not monotonic and/or 'wiggly', then you might want
$$ A exp^{-b x^2 } sin(k x) $$ The latter Ansatz is a two-parameter Ansatz and is the most likely to have a chance of working on a real equation. You may think that $ k $ is a third parameter, but actually, it is determined, usually algebraicly, by $A$ and $b$. Single parameter Ansatzen usually only work on very specific coefficients of equations and are too simple to model real equations.
There are obviously many, many other good Ansatzen, such as soliton solutions
$$ A {sech}^n{left(k x - omega tright)} $$ (where $n$ is a positive even integer, and $omega=omegaleft(kright)$ is the dispersion relation)
if your equations has symmetry properties. There is a large theory, mostly derived from the work of R. Hirota, of how to derive exact solutions to systems of nonlinear PDE's which have certain symmetry properties or invariants, using the properties of bilinear operators.
Note: Directly translated, the word der Ansatz in German has many meanings, but it most usually is translated as 'approach' or 'basic approach', but it really just means: an educated guess of a solution, with enough degrees of freedom (in the form of parameters) such that the Ansatz is able to solve the equation.
Also, in the above equations, $A$ can be constant, or only a function of $ t $ or a function of both $x$ and $t$, depending on which behavior is being modeled.
There is a straightforward encoding of 3-SAT into this problem, which means that unless P=NP, no, there cannot be a polynomial-time algorithm that solves it. The encoding can be constructed as follows. Given a formula $phi=psi_1wedgedotswedgepsi_k$ in conjunctive normal form over propositional variables $v_1,dots,v_n$, you have the following boxes:
This can be done in time linear in the size of $phi$.
Then a set of circles satisfying your constraints must contain at least one of $A_i,B_i$ for all $i$, and has weight $n$ if and only if it corresponds to a valid valuation of the $v_i$ which satisfies $phi$. In particular, satisfiability of $phi$ can be decided by checking if the minimum weight solution has weight $n$.
Theres graded local duality which works just like local duality; however, it requires that $A_0$ is a field. I've had some luck making things work when A_0 is not a field, but then the local duality becomes more derived. Specifically, Matlis duality, which is an exact functor in the graded local case, gets replaced by $operatorname{RHom}_{A_0}(-,A_0)$, the derived graded hom over $A_0$ into $A_0$. Then, at least in some cases I've looked at, the local cohomology $Rtau$ satifies the equation
$$Rtau(M) = operatorname{RHom}_{A_0}(operatorname{RHom}_A(M,A),A_0)[d](l)$$
This should be true when the ring $A$ is 'relatively Gorenstein over $A_0$', which means that $operatorname{Ext}^i_A(A_0,A)$ is non-zero for a single $i=d$, and $operatorname{Ext}^d_A(A_0,A)=A(l)$.
As for translating this to Serre duality, there should be an exact triangle $Rtau(M) to M to RGamma(M)to$ relating the local cohomology to the derived global sections.
This is true! I assume $M$ compact.
Method 1. Real algebraic geometry. Cf. this article. By a version of the Nash-Tognoli embedding theorem, one can realise $M$ as an real affine algebraic variety $V_mathbb{R}$, cut out by polynomials $f_i in mathbb{R}[x_1,dots,x_N]$. The complex variety $V_mathbb{C}$ will then be smooth in a small neighbourhood $U$ of $V_mathbb{R}$, hence Kaehler in that region, with $V_{mathbb{R}}$ as a Lagrangian submanifold. But $U$ is diffeomorphic to $T^ast M$. The resulting symplectic structure on $T^ast M$ may be non-standard; via the Lagrangian neighbourhood theorem, you can take the symplectic form to be the canonical one if you'll settle for a Kaehler structure only near the zero-section.
Method 2. Eliashberg's existence theorem for Stein structures. See Cieliebak-Eliashberg's unfinished book, Symplectic geometry of Stein manifolds, Theorem 9.5. We observe that $T^ast M$ has an almost complex structure $J$ (one compatible with the canonical 2-form, for instance) and a bounded-below, proper Morse function $phi$ whose critical points have at most the middle index (namely, the norm-squared plus a small multiple of a Morse function pulled back from $M$). In this situation Eliashberg, via an amazing chain of deformations, finds an integrable complex structure $I$ homotopic to $J$ such that $dd^c phi$ is non-degenerate. This makes $T^ast M$ Stein! His theorem only applies in dimensions $geq 6$ (this paper of Gompf explains what you have to check in dimension 4), so without doing those checks or appealing to other methods, the case of $M$ a surface is left out.
I think that the more precise version of Eliashberg's theorem, which may not yet be in the book, would tell us that the Stein structure is homotopic to an easy-to-write-down Weinstein structure on $T^ast M$ involving its canonical symplectic structure $omega_{can}$, hence that $dd^cphi$ is symplectomorphic to $omega_{can}$.
Dear Steve Huntsman, I would refer you to the version for hamiltonian systems of a result known as Poincarè-Lyapunov theorem that describes the periodic orbits around a known one when a certain condition is satisfied.
Let $(M,omega)$ be a $2n$-dimensional symplectic manifold, an $H$ a smooth regular function on $M$.
Let $Lambda$ be a $1$-dimensional compact connected submanifold of $M$ which is invariant under the flow of $X_H$, i.e. $Lambda$ is the image of a periodic integral curve of $X_H$.
If $1$ is not an eigenvalue for the derivative of the first recurrence map for $X_H$ in a point of $Lambda$ then there exists a $2$-dimensional symplectic submanifold $N$ of $(M,omega)$ containing $Lambda$ such that $H|_N$ is a summersion whose fibers are compact connected and invariant under the flow of $X_H$.
So under the stated non-degeneracy condition a periodic trajectory of $X_H$ is included in a family of periodic orbits forming a symplectic submanifold and parametrized by $H$.
For a reference and a generalization which joins together the Poincarè-Lyapunov theorem with the Liouville-Arnol'd theorem, I would suggest N.N. Nekhoroshev: The Poincare'-Lyapunov-Liouville-Arnold theorem. Funct. Anal. Appl. 28 (1994), no. 2, 128--129
A few questions were asked, so a few answers will be given. (main point: likelihood is not necessarily a product density, though this is the common interpretation.)
Frequently, the likelihood is the product of densities over some provided set of examples. The examples are drawn i.i.d., and therefore this product density is the density for the corresponding product measure over the product space. What I'm saying is that yes, from this perspective, you have constructed a product density.
Since you are dealing with densities, not probabilities, values are not constrained to [0,1], and your density can easily be greater than one. In fact, if you are dealing with
dirac measure (which puts all mass on one point on the real line), you essentially have "infinite" density. I put that in quotes since this is not a continuous probability measure, ie it does not have a density wrt to Lebesgue measure, let alone one with infinite mass on a point. (A quick fact check: the corresponding integral wrt lebesgue measure would have value zero since it is off zero only on a set of lebesgue measure zero, which means it is not a probability distribution; but it was, which contradicts this being its density.) perhaps a more apt example: any (continuous) distribution on [0,0.5] will have to have density greater than one on a set of nonzero lebesgue measure. (you can try
to construct a sequence of these which convergence to something which violates what i said, but that will be the density of something which is not continuous!)
things can get a little confusing because you can write discrete probability distributions
as densities wrt a measure putting 1 on each point in the support set of the probability (ie it is counting measure wrt that set). NOTE that this is a density wrt a measure which is NOT a probability measure. But anyway, the density values at each point are exactly the probability values. This allows an interchanging probability masses and densities, which can be confusing.
I'll close with some further reading. A good book on machine learning is "A probabilistic Theory of Pattern Recognition" by Devroye, Gyorfi, Lugosi. Chapter 15 is on maximum likelihood and you'll notice they do NOT define likelihood as being a product probability or density, but rather as a product of functions. This is because they are careful to encompass the differing interpretations; rather, they ignore the interpretations there and work out the math.
A geometry question that I thought about more seriously a few years ago... thought it'd be a good first question for MO.
I'm aware that there are a number of Torelli type theorems now proven for compact HyperKahler manifolds. Also, I think that Y. Andre has considered some families of HyperKahler (or holomorphic symplectic) manifolds in some paper.
But, when I see such a moduli problem studied, the data of a HyperKahler manifold seems to include a preferred complex structure. For example, a HyperKahler manifold is instead viewed as a holomorphic symplectic manifold. I'm aware of various equivalences, but there are certainly different amounts of data one could choose as part of a moduli problem.
I have never seen families of HyperKahler manifolds, in which the distinction between hyperKahler rotations and other variation is suitably distinguished. Here is what I have in mind, for a "quaternionic-Kahler family of HyperKahler manifolds:
Fix a quaternionic-Kahler base space $X$, with twistor bundle $Z rightarrow X$. Thus the fibres $Z_x$ of $Z$ over $X$ are just Riemann spheres $P^1(C)$, and $Z$ has an integrable complex structure.
A family of hyperKahler manifolds over $X$ should be (I think) a fibration of complex manifolds $pi: E rightarrow Z$, such that:
In other words, the actual hyperKahler manifold should only depend on a point in the quaternionic Kahler base space $X$, but the complex structure should "rotate" in the twistor cover $Z$.
This sort of family seems very natural to me. Can any professional geometers make my definition precise, give a reference, or some reason why such families are a bad idea? I'd be happy to see such families, even for hyperKahler tori (which I was originally interested in!)
The following paper:
Bhattacharyya, P., and B. K. Chakrabarti. The mean distance to the nth neighbour in a uniform distribution of random points: an application of probability theory. Eur J. Phys. 29, pp. 639-645.
Claims to provide exact, approximate, and handwaving estimates for the mean 'k'th nearest neighbor distance in a uniform distribution of points over a D-dimensional Euclidean space (or a D-dimensional hypersphere of unit volume) when one ignores certain boundary conditions.
However, Wadim's response is making me feel some concern that the exact problem is much more complex. Please see the paper for the full derivation (and approximate methods), but I'll write the exact expression they converge on using two different method of absolute probability and conditional probability.
Let $D$ be the dimension of the Euclidean space, let $N$ be the number of points randomly and uniformly distributed over the space, and let $MeanDist(D, N, k)$ be the mean distance to a given points $kth$ nearest-neighbor. This yields:
$MeanDist(D, N, k) = frac{(Gamma(frac{D}{2}+1))^{frac{1}{D}}}{pi^{frac{1}{2}}} frac{(Gamma(k + frac{1}{D}))}{Gamma(k)} frac{Gamma(N)}{Gamma(N + frac{1}{D})}$
Where $Gamma(...)$ is the complete Gamma function.
Wadim - might it be possible for you to provide some feedback about the derivations here vs. the method of box integrals you described in your comment?
Let $z$ be a complex variable and $f(z)$ be a formal power series with rational coefficients (an element in $mathbb Q[[z]]$), with a finite radius of convergence, and assume $f(z)$ has a meromorphic continuation to the whole complex plane (so it has at most countably many poles). What do we know about the number-theoretic property of the roots and poles? Are they algebraic numbers? If they are, are they stable under the action of the Galois group of the rationals?
More generally, if the coefficients of $f(z)$ are algebraic numbers, and let $sigma$ be an automorphism of the algebraic closure of the rationals, then what is the relation between roots of $f(z)$ and roots of $f^{sigma}(z),$ where $f^{sigma}(z)$ is the power series obtained by applying sigma to the coefficients of $f(z)?$
Without assuming the finiteness of radius of convergence, $sin(z)$ gives a counter-example.
Edit: Let me give a second try, by imposing more requirements on $f(z).$ I'm thinking about the case where $f(z)$ is the zeta function of an algebraic stack over a finite field, so let's assume $f(z)$ has an infinite product expansion over $ell$-adic numbers, like $prod P_{odd}(z)/prod P_{even}(z),$ where each $P_i(z)$ is a polynomial over $mathbb Q_{ell}$ with constant term 1. Assume they have distinct weights, e.g. reciprocal roots of $P_i(z)$ have weights $i.$ Then can we conclude that all coefficients of $P_i(z)$ are rational numbers? Thanks.
Hello. After flipping through a few textbooks on birth-death processes, I can't seem to find anything about genealogical distribution of survivors (conditioned on non-extinction). What I am looking for is a statement roughly of the type, "in generation n, there is probability at least P(j,k,n) that descendants have survived from at least k distinct members of generation j".
I'm interested in the critical case of the Galton-Watson process, where the number of descendents is i.i.d. with mean 1. If necessary, assume the distribution is Bin(r,1/r).
Also, my sample question about P(j,k,n) is just the easiest thing I could think to write down. But I would like to know how likely it is for there to exist a very uniformly distributed subpopulation of size roughly n in generation n.
The answer is no for $n=2$. It is sufficient to construct 3 surfaces with common boundary (say $Sigma_i$, $iin{1,2,3}$)
such that there is no choice of points $p_iinSigma_i$ with pairwise parallel tangent planes.
Let us take a smooth function $f:S^1to mathbb R$, $f(t)approxsin(2cdot t)$ with one little bump near zero
so it has 3 local minima and maxima.
We want to construct three functions $h_1,h_2,h_3$ from unit disc $D$ to $mathbb R$ such that
each has $f$ as boundary values and
if $nabla h_1(x)=nabla h_2(y)$ then $nabla h_1(x)=0$
$nabla h_3not=0$ anywhere in the disc.
Then graphs of functions give the needed surfaces.
The graphs of $h_1$ and $h_2$ are parts of boundary of
convex hull of graph of $f:partial Dtomathbb R$; it is easy to check (1).
The graph of $h_3$ is a ruled surface which formed by lines passing
through points $(u,f(u))$, $(phi(u),f(phi(u))inmathbb R^3$, $uin S^1$ for some involution diffeomorphism $phi: S^1to S^1$.
To have the property one has to choose $phi$ with two fixed points (say at global minima of $f$)
so that if $f(phi(x))=f(x)$ for some $x$ then $f'(phi(x)cdot f'(x)<0$.
The later is easy to arrange, that is the place we need the bump of $f$.
P.S. Hopefully it is correct now :)
Let $S$ be a locally noetherian scheme, $C$ the category of proper smooth $S$-schemes with geometrical connected fibres and $C_*$ the category of pointed objects of $S$, i.e. objects of $C$ together with a morphism $S to C$. Also denote $A$ the category of abelian schemes over $S$. There is a well-known rigidity result stating that a pointed morphism between $X,Y in A$ is already a group morphism. In other words, the inclusion functor
$A to C_*$
is fully faithful. Is there a nice description for the image? In other words, which purely scheme-theoretic properties do abelian schemes have and are there enough to characterize them? For example, $X in A$ is "homogeneous".
Since Quinn's article is a long opinion piece which he says is 90% complete and welcomes comments, it seems entirely appropriate to contact him for clarification on this point. He would probably be happy to tell you more.
One example that springs immediately to my mind is the classification of finite simple groups. This was, by a safe margin, the largest scale collaborative activity in the history of mathematics, taking place over a decade or so. The accounts I have read describe Aschbacher, Thompson and (especially) Gorenstein as acting like army generals overseeing a war: they had the most insight into the global structure of the argument and they used it to apportion and subcontract various pieces of the proof. So far as I can think of at the moment, it is much more usual for a visionary mathematician (e.g. Langlands, Thurston, Hamilton) to lay out a program which other mathematicians are then inspired to work on as they see fit than to have this kind of explicit top-down organization.
The rest of the story is well-known: in the early 80's Aschbacher, Thompson and Gorenstein were photographed on an aircraft carrier in front of a victory banner (figuratively speaking of course) and all the other group theorists shouted hurrah and cleared out. But certain key parts of the argument had never been published in any form, as a small number of mathematicians (e.g. Serre) spent the next 20 years reminding the community. It seems fair to say that the finite group theorists cleared out a little too early. I don't really know why or exactly what motivated the recent moderate resurgence of interest in the classification, including the 2004 (!) publication of a two-volume work completing the quasi-thin case (a mere 1300 additional pages were required). In the last few years it seems that there has been "the right amount" of tidying up these massive argument by those involved in the "second generation" and "third generation" classification efforts.
See
http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
and the references therein for more details. Especially highly recommended is Aschbacher's 2004 Notices article
http://www.ams.org/notices/200407/fea-aschbacher.pdf
which, in addition to being gracefully written and informative, is admirably forthright.
It's said that most groups arise through their actions. For instance, Galois groups arise in Galois theory as automorphisms of field extensions. Linear groups arise as automorphisms of vector spaces, permutation groups arise as automorphisms of sets, and so on.
On the other hand, abelian groups often arise without any natural (or at least obvious) action -- the "class groups" such as the ideal class group and Picard group, as well as the various homology groups and higher homotopy groups in topology are examples. [ADDED: One (sloppy?) way of putting it is that abelian groups arise quite often for "bookkeeping" purposes, where we think of them simply as more efficient ways to store invariants, and their actions are not obvious and not necessary for most of their basic applications.]
What are some good examples of non-abelian groups that arise without any natural action? Or, where the way the group is defined doesn't seem to indicate any natural action at all, even though there may be an action lurking somewhere? The only prima facie example I could think of was the fundamental group of a topological space, but as we know from covering space theory, for nice enough spaces (locally path-connected and semilocally simply connected), the fundamental group is the group of deck transformations on the universal covering space.
This might be somewhat related to the question raised here: Why do groups and abelian groups feel so different?.
To clarify: There are surely a lot of ways of constructing groups within group theory (or using the tools of group theory, which includes various kinds of semidirect and free products, presentations, etc.) where there is no natural action. These examples are of interest, but what I'm most interested in is cases where such groups seem to arise fully formed from something that's not group theory, and there is at least no immediate way of seeing an action of the group that illuminates what's happening.
This inequality is essentially equivalent to the Classical isoperimetric inequality. If you have a measurable body $X$ in $mathbb{R}^n$ and a ball $Bsubset mathbb R^n$ of same volume then you have the following:
$$Area(X)=lim_{epsilon to 0} frac{operatorname{Vol}(X_{epsilon})-operatorname{Vol}(X)}{epsilon}$$
$$Area(B)=lim_{epsilon to 0} frac{operatorname{Vol}(B_{epsilon})-operatorname{Vol}(B)}{epsilon}$$
Proving that $Area(X)geq Area(B)$ follows from $operatorname{Vol}(X_{epsilon})geq operatorname{Vol}(B_{epsilon})$, which is your inequality. ($n=2$)
Now this follows from the Brunn Minkowski inequality because
$$operatorname{Vol}(X_{epsilon})=left(operatorname{Vol}(X+epsilon B)^{1/n}right)^n geq left(operatorname{Vol}(X)^{1/n}+epsilon operatorname{Vol}(B)^{1/n}right)^n=operatorname{Vol}(B_{epsilon})$$
EDIT: My first answer was confusing and not quite accurate. Let me try again.
In arbitrary characteristic, the structure sheaf of any homogeneous space $G/P$ (for $G$ a semisimple group) has no higher cohomology. This is an instance of Kempf's vanishing theorem. The space of sections is 1-dimensional, so this implies that the arithmetic genus is 0.
Now using Serre duality, we conclude that only the top cohomology of the canonical sheaf is nonzero, which means that the geometric genus is also 0.
Sci-Hub is pretty helpful in accessing articles, even for those researchers who already have access to several journals. The interface is great, the site is pretty fast, and the database is huge. See this article and other linked articles there for a nice overview of who all are downloading pirated papers.
Edit: as pointed out in the comments, it should be noted that there is an ongoing lawsuit against the website.
I heard this from Haskell Rosenthal many years ago.
If V is a complex vector space, say the opposite of V is the complex vector space with the same elements, the same operations except switch scalar multiplication to scalar multiplication by the complex conjugate scalar. Of course this definition applies in particular to a complex Banach space. Question: Is every complex Banach space isomorphic to its opposite? (An isomorphism is a complex-linear homeomorphism.)
[ prompted by question Silly question about opposite groups ]