you're right that PDEs come up with complex varieties, but the Weil conjectures are about finite fields, so it would be very surprising if PDEs are applicable in any way
I need to emphasis to my haters here : the work of Grothendieck is what I consider honost catagory theory . Am mainly arguing about the catagory theory stuff that is not directly related to good ol classical math
And to each his own , that’s the nice thing about math , you study what you like , but don’t come and try to take over my spot
I can tell you that the stuff I was discussing with Leaky is related to good ol classical math: Algebraic groups, Lie groups, group schemes, topological groups. It provides a different perspective (you probably don't like) on each of those
maybe it's more useful for algebraic groups and group schemes, but for that it can definitely be helpful
@LeakyNun let me just mention one thing: if you think of universal properties as statements about hom-sets, then the fully faithfulness of the Yoneda embedding tells you that two objects with the same universal property are isomorphic. And if you look closely e.g. at the proof for uniqueness of the tensor product you can see that it's related to the proof you gave for the fact how fully faithful functors relfect isomorphismic objects and the Yoneda argument
@LeakyNun Not really 'an application' - but I may have mentioned about functor of points before (or maybe you guys did in the above) - specifically about how one could view a scheme as a generalisation of solving equations. For example if I take the scheme $X=\operatorname{Spec} \mathbb{Z}[x,y]/(x^2+y^2-1)$, then I can consider the functor from the category of affine schemes to the category of sets mapping each scheme $Y$ to the set $\operatorname{Hom}_{Sch}(Y,X)$. Yoneda tells you that your scheme is determined by this functor -
sometimes the only thing we know about a scheme is that it represents a certain functor, and understanding the functor allows us to say some things about the scheme.
A $K$-point of a scheme $X$ (say defined over $k$) would just be a morphism of $k$-schemes $\mathrm{Spec}(K) \rightarrow X$.
There's no need to only consider fields etc. but anyway the idea is the same.
Now let's suppose I take the scheme $\operatorname{Spec} k[x,y]/(x^2+y^2+1)$, and say you take $K$ to be a field extension of $k$. Then a morphism of $k$-schemes $\operatorname{Spec}(K) \rightarrow \operatorname{Spec} k[x,y]/(x^2+y^2+1)$ corresponds to a ring map $k[x,y]/(x^2+y^2+1) \rightarrow K$ fixing the underlying field $k$.
@XanderHenderson fine. Reading up on No Game No Life light novels :)
@FamousMichaelWang I tend to be that person who makes very fast geowing functions for no particular reason. Seeing as no-one is occupying the chat, we can play a game involving such functions, if you're up for it.
Currently trying to show via cantor diagonal argument of decimals that the number of new element counting function grows quicker than any computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all...
Hello. I don't understand the difference between tensor product and Kronecker/outer product. Here (en.wikipedia.org/wiki/Outer_product#Tensor_multiplication) it is said that tensor product of two tensors of order 2 makes a new tensor of order 4. But how is Kronecker product still "tensor product" and it produces still a matrix?
The number of $\omega$ places increases in the sequence 1,1,2,2,5,5,12,12,...
The total number of entries per units of $\omega$ increases in the sequence:
1,2,3,5,7,12,17,29,...
Therefore given any $\alpha$th iteration there is at least $f(\alpha)$ elements where $f$ computable, and the next iteration will produce something such that $f(\alpha + 1) > f(\alpha)$ for all countable $\alpha$
Therefore the lower bound of the number of reals by "counting" is $\omega_1^{CK}$ the supremum of all computable ordinals
The proof is complete if the following conjecture is true:
Given $\omega$ uncomputable binary sequences that are distinct. Place them into a table. Then the elements obtained from reading each columns are always unique
@mercio:"that if $V_\sigma$ doesn't branch, it shouldn't be in an antisymmetric square of one of the $V_{\rho'_i}$ because it should be invariant by the "action" of $G$, and I suspect $G$ should switch around the antisymmetric squares at least if $G'$ is normal in $G$ there might be a bit of sense in that" :
I truly believe you can get very far in mathematics , research level far , without spending more than a couple hours a day , as long as you are consistent and you do it everyday
@Rudi_Birnbaum Right now what you seem to be "missing" is mainly familiarity with what everything is called in math. Plus some more exposure to the way mathematicians talk about things, since this differs from the way others do
My background is quite a bit of what we call "analysis" so its more than calculus. and lots of linear algebra, then complex calculus and a bit of complex analysis.
what we call "Funktionentheorie".
My main Problem is that I miss a bit interacting with other maths students
Well , something draws you to mathematics , find out what it is , is it visual ? Abstract ? Analytic ? Pretty much find your area of interest and dive in deep
For me personally , the best way to learn something specific is to read an introductory book , and whenever it mentions things I don’t understand , I go to other books and read the sections
@Rudi_Birnbaum There is certainly some merit to start trying to learn about the stuff you really want and then go back and learn the necessary ingredients as they come up
Another maybe my central interest since I was a child in maths was primes. the quest for an explicit formula. But I don't know if that is something I could approach and then how?
I fell kind of attracted by this proposed programme from the "birds and frogs" about classification of "1D quasi crystals" but I am not sure if this is still what people consider an interesting attack on primes and RH?
I read Dummit and Foote as a beginner. Years later, I wished I had read Rotman's advanced modern algebra first, because he presented stuff in a much more beautiful way...
I even tried to do some maths on it, but sadly got stuck...
I wanted to reproduce a paper, but came to point where I didn't understand it and contacted the author, he firstly responded but couldn't resolve the issue ...
It was some quite simple FT stuff on some "self-similar" sequence.
D&F is good. Some problems are really hard, but most are doable. Also, for an introduction to ring theory I'd highly recommend "Rings and Ideals" by McCoy. It's really old, but cleared a lot of confusion
well, besides algebra, I think you should have to learn about Fourier transforms and a bit of functional analysis (spectral theory). Perhaps it is a bit of a tall order
@Rudi_Birnbaum I see. I just remember hearing about them because they should be the geometric object that corresponds to those Coxeter groups which are not crystallographic but which are still finite (or affine)
The idea is that these should be reflections, so when you compose them, you should get something like a rotation, and due to geometry, these are the possible orders of those rotations
I think so, yes, because these rotations say something about the possible angles between suitable vectors, and if these are not right, then you can have translational symmetries
(this is all a very imprecise and vague description)
@TobiasKildetoft: In my understanding the central requirement for "normal" crystalls ist the 3D periodicity. And from that follows that one can only have what you said. And than you all of a sudden dicovere that you get diffraction patterns of 5-fold symmetry. Then you fight very stubborn for 20 years with ignorant colleagues and then you get the nobel price!!
But "really" the reason is coming from Lie algebras, since these force some integrality on the entries in the associated Cartan matrix, and the relation to the above means that the only possibilities are like that
So the thing about finite Coxeter groups is that almost all of them come from Lie algebras. But there are some that don't, and these are very annoying.
A Coxeter group is one that is generated by elements of order $2$, and where we only put relations on what the orders of products of two generators are
An example is one you already know, called $I_2(3)$ (or $A_2$), namely $\langle s, t\mid s^2 = 1, t^2 = 1, (st)^3 = 1\rangle$
@Rudi_Birnbaum We don't require the elements to commute, so if for example we left out the $(st)^3 = 1$ above, all products alternating between $s$ and $t$ would be different
Not sure about Coxeter's book. I tend to avoid books that are too old unless they are very well regarded. So much work has been done in the meantime improving our understanding and thus improving the exposition of the results
Whats then the intuition for the definition of the Coxeter group? ("A Coxeter group is one that is generated by elements of order $2$, and where we only put relations on what the orders of products of two generators are")
Translation invariance huh. Any operator bounded on L^2 which commutes with translations must be a Fourier multiplier. Translation invariance on a locally compact abelian group also immediately suggests we should study its Pontryagin dual (the space of frequencies). A very beautiful result is that the Pontryagin dual of a compact group must be discrete (e.g. a lattice) and vice versa. So I am inclined to believe the theory of quasicrystals is within the domains of harmonic analysis.
Part of the idea is that of a reflection group, meaning that one should start with some set of reflections in $n$-dimensional space and see what group they generate
Not completely sure how people ended up with the precise requirements for being a Coxeter group from that though, other than seeing that this gave some importat examples while still being manageable (apart from those annoying ones that are are not crystallographic)
@Rudi_Birnbaum So all of the dihedral groups are Coxeter groups ($D_n$ is called $I_2(n)$). And these tend to be annoying in some cases. For example, I have a paper with three others where we show something for all finite Coxeter groups except $D_{12}$, $D_{18}$ and $D_{30}$
@Rudi_Birnbaum Ohh, and it was not just that the thing we were showing did not hold for those three groups. It was that we were simply not able to figure out what did hold for them.
Like if you have a 2-D periodic structure and put a line at an irrational ascend in it. and then project the closest points onto the line, then you get a quasi-periodic sequence.
it is an interesting viewpoint. That the spatial side should be ignored in favor of the Fourier side. This is akin to the development of Fourier multipliers in harmonic analysis, where some people start to disregard whatever the kernel looks like on the spatial side, but only focus on the Fourier side.
wavelet theory is what happens when people want to look at a function in close details, by restricting both the spatial range and frequency range you are interested in. Think of the high frequencies as the treble, and the low frequencies as the bass of the song which is the function :). If you restrict only the frequency, you get Littlewood-Paley theory. If you restrict both the spatial location and the frequency, you get wavelet theory. OK, that is a dirty summary, but that is how I see it.
well, if it really is Yves working on this stuff, the obvious advice is to read his books and harmonic analysis books, though I can not in good conscience suggest this to all but the most determined learners
you can probably find Tao's epsilon of room book online, which should contain the analysis prerequisites (including Pontryagin duals) before going into harmonic analysis
@Iza_lazet: Noted! Guys it was nice to talking to you! I'll have to continue my work now (I hope later on I can motivate @mercio to help me a bit with my current project).
and of course the other part is that if $|F| = n$, then $n = 0 \in F$, so one of the prime factors of $n$, say $p$, satisfies $p=0 \in F$, and then characteristic is unique by Bezout, and then $F$ is a vector space over $\Bbb F_p$, so its order is a power of $p$
I have a math-dad-joke that I cannot tell properly :( Imagine working in a non-strict monoidal category with left duals, and you write something like $X\otimes (Y\otimes Y^*)\xrightarrow{1\otimes \operatorname{ev}_Y} X$ and someone interrupts you and says "Well strictly speaking you have $\to X\otimes 1 \to X $"
Does anyone understand what I'm getting at. I just said that to myself (by accident) and laughed like an idiot
Of course, my duals are mixed up. Whatever
It happened again. "Strictly speaking, we have to include the bracketing because the associator is not trivial"
We have the relation $f : \mathbb{Z} \rightarrow \mathbb{Z}$ with $a \mapsto 3-2a$. This is a map because every integer $a$ is mapped to unique integer, or not? What special properties does $f$ have?
Let $f(a_1)=f(a_2)\Rightarrow 3-2a_1=3-2a_2\Rightarrow a_1=a_2$. That means that f is injective. That is one special property, isn't it? Now we have to check the surjectivity.
That map is not surjective since for $2\in \mathbb{Z}$ there is no $a$ such that $f(a)=2$, right? SInce $3-2a=2\Rightarrow 2a=1\Rightarrow a=\frac{1}{2}\notin \mathbb{Z}$.
And when you play a difficult level puzzle, sometimes one must guess a number or infill. In the best case you have 50% chance of guessing correctly. I was thinking if there are scenarios in similar games where the chance would be above 50%. Should I ask such a guestion?
@mercio:"that if $V_\sigma$ doesn't branch, it shouldn't be in an antisymmetric square of one of the $V_{\rho'_i}$ because it should be invariant by the "action" of $G$, and I suspect $G$ should switch around the antisymmetric squares at least if $G'$ is normal in $G$ there might be a bit of sense in that" : The cases where sigma doesn't ever branch, are exactly the cases where this $V_\sigma$ is identical to the *peculiar* irrep (lets call it $\Gamma_a$) I am after. The other cases are with respect to $\Gamma_a$ exceptions and in these cases anything is possible. That is in detail: a) $\Ga…
The inverse relation I(f) is $3-2a\mapsto a$, also $-2a\mapsto a-3$, also $a\mapsto \frac{3-a}{2}$, right? I want to fins the biggest possible subset of $\mathbb{Z}$ such that $I(f)$ is a function. Is the inverse a function when the function $f$ is surjective? We have that f is surjective if we consider the function $f:\mathbb{Z}\rightarrow f(\mathbb{Z})=\{2k+1\mid k\in \mathbb{Z}\}$, right?
Does it mean that the inverse relation of f is a function is we consider the domain $\{2k+1\mid k\in \mathbb{Z}\}$, i.e. $I(f):\{2k+1\mid k\in \mathbb{Z}\}\rightarrow \{2k+1\mid k\in \mathbb{Z}\}$ ?