@BalarkaSen Suppose I have a smooth function $f$ on a compact Riemannian manifold $M$. Can I find a smooth diffeomorphism $\phi$ of $M$ such that $f\circ\phi$ is close to $\inf f$ for a set of arbitrarily small comeasure?
Give $M$ a cell decomposition $M = e^n \cup M^{n-1}$ where $e^n$ is the top cell and $M^{n-1} \subset M$ is the codimension 1 skeleton. $M^{n-1}$ should be measure zero inside $M$. Given any ball $B$ around $p \in e^n$, you should be able to find a sequence of diffeomorphisms $f_k : M \to M$ such that "$f_k(B) \to e^n$"
I think it can be done without too much Riemannian geometry though. Define a vector field on $e^n$ with source at $p$, radially extending out but compactly supported on a ball $B_k \subset e^n$ of radius $1 - 1/k$ (I am identifying $e^n$ with the unit open disk).
This in turn gives a vector field on $M$ (just set it to be zero on the rest)
@BalarkaSen I'm almost done with one of the Kazdan-Warner theorems. I just have to write down this ball argument...they didn't actually explain how to do it at all in the original paper
instead they refer to an older paper for "details" but it actually has zero details
and the "modern" proof of that result completely avoids this argument
okay so I fill my open set with ball bearings, each of which is, WLOG, the largest open ball the fits entirely in each closed non-overlapping cubes. These ball bearings need not be uncountable because $\mathbb R^n$ is completely seperable. But they can't be finite because ...
I mean we loved Rudin but doing every question might've been... nah actually if we just had to do every Rudin problem that would've way better than Sally tbh
lol @Daminark "you are the only one of your group who expressed interest. For us this is not a problem since any of the three of the faculty can give you more advanced material."
Knuth rates the problems in "The Art of Computer Programming" on a difficulty scale from 1 to 50. In the earlier editions, Fermat's last theorem was given as a 50 exercise . After Wiles found a proof, Knuth demoted it to a 45 problem.
Consider the three statements:
(i) If $X$ is a set and $\mathcal{A}$ is a collection of subsets of $X$ having the countable intersection property, then there is collection $\mathcal{D}$ of subsets of $X$ such that $\mathcal{D} \supset \mathcal{A}$ is maximal with respect to the counta...
All of my exams are done. I'm going to retake complex analysis and numerical analysis (only the exams, not the courses, I could've passed them, but not with grades I'd like), I've got an A in algebraic number theory and I'm still waiting for the grades in elementary number theory, but I guess it's either an A or A-, because I've got one silly mistake and the rest of the exam was really straightforward
So I'm doing fine, thanks :)
I'm currently wondering what I should do in the semester break (or more precisely, which books I should read)
@MatheinBoulomenos Congrats :) Currently reading Cox "Primes of the form $x^2 + ny^2$" for a classical intro to class field theory, which might interest you if you're starting hilbert class fields next?
@Twink @Secret @abcd: I don't know if you all figured it out, but you need to use power of a point. If you take a point $P$ outside a circle and draw a segment from it tangent to the circle at $T$ and a segment from it that gives a diameter $AB$, then $(PT)^2 = (PA)(PB)$.
@ÍgjøgnumMeg thanks! I already have that book in my shelf and I've read parts of it. The course on class field theory will focus more on the proofs than the applications, but Cox' book is a great motivation for sure
@MatheinBoulomenos Fair! I'm reading it for the motivation atm, most other treatments I've seen use a lot of local theory etc. that I'm not familiar with so the classical approach is nice. I also have some notes from Franz Lemmermeyer with the same classical approach, but none of the theorems have proofs, which I find strange.
@ÍgjøgnumMeg I think the only proofs which don't use any local theory use a lot of complex analysis. (These analytical proofs are for example in Janusz "Algebraic Number Fields" and Langs "Algebraic Number Theory")
But I've heard that no matter which approach you take, the proofs are really hard
@MatheinBoulomenos I've heard this too :( I'm a lot more familiar with complex analysis than I am with any local theory so this approach is fine for what I need
I'm actually taking two different courses on class field theory next semester. The main lecture on algebraic number theory will do local first and then use local CFT to prove global CFT, we will use group cohomology as the main tool in the proofs. The other course is a seminar on Lubin-Tate theory which is a purely local theory based on formal groups, but it's a lot more explicit than the abstract proofs (so you can actually compute the equations that define the class fields)
That's cool :) I'm just self-teaching global cft as an "additional exercise" for my dissertation, I don't think it should be too heavy, just to establish an isomorphism between the galois group of the hilbert class field and the ideal class group, which I think can be done without anything too "dark" lol
Definitely, but I'm leaving it until my masters for now, I don't know any topology and I don't know if I have the time to learn any since my dissertation needs to be finished by april 16th
The weirdness I’m running into is that, at the level of the a matrix representation, I seem to have two operators commuting. But at the level of the operator algebra I don’t
@TedShifrin do you know of any book that can help with things like how to write negations? i was stuck on a question all day, and the thing was i thought of its negation wrong.
@Shobhit: This is basic intro to higher math stuff. My favorite book, which I've recommended numerous times on MSE, is Houston's How to Think Like a Mathematician.
If I work in the basis I think is appropriate, I find that a certain operator is conjugate to itself under an appropriate transformation. But if I work at the level of the operators as I understand them, no such conjugacy seems to occur
@ÍgjøgnumMeg do you know roughly what metric spaces and normed vector spaces are? You don't really need topology for local fields, sure authors will use words like "topological fields", but all the topologies under consideration are induced by metrics
@MatheinBoulomenos Yeah I do roughly, most of my mathematics is quite patchy because everything is self-taught, so often I need to retrace my steps to learn some theory before I continue!
@Semiclassic: If you're thinking of matrices, you're already choosing a "ground" basis in terms of which you represent the operators. That formula says that if you change basis by $A$ you get the same matrix representation. If you think of $A$ as an operator (rather than as a change of basis), it's still in that same basis.
Well, you're wrong one of the two times.
Hi, demonic @Alessandro. How was the driving lesson?
@MatheinBoulomenos When you say induced by metrics, do you mean that all of this comes out of valuations on fields? I've read a little bit about valuations and this seems to be what leads to defining local fields
@ÍgjøgnumMeg yes, a valuation gives you a metric and a metric gives you a topology. But not every topology comes from a metric and not every metric comes from a valuation. But it's not really important to pass to the lower levels, all the properties will be mostly checked using valuations
@MatheinBoulomenos That's cool, perhaps I will take a look at this in the summer between my bachelor and master degrees as preparation lol, I still need to get my Reifezeugnis sent to me before I can apply for masters in Germany though -.-
@TedShifrin My groups are typically rank 20-50 and my matrices are in the 200 by 200 to the 600 by 600 range, so there's proably a much easier way of doing what I did.
@shobhit 24^2 is the rank of H_2 of \{(x,y,z) \in \Bbb C^3| x^2+y^25+z^25= \epsilon \} if you are curious where this number is coming from. The matrix is the intersection form.
Idk if SmithDecomposition is really optimized for sparseness.
There's probably a topology way of generating much smaller matrices, but I think it'd be much more complicated to program in mathematica
entries in the matrix would have to be defined by some continued fraction expansions and the length of the matrices would be dependent on the lengths of these expansions.
im ok w mathematical drawing bc i like to make technical drawings of plants (cuz plants are cute and i like them) and it's made me a better math drawer
Consider the three statements:
(i) If $X$ is a set and $\mathcal{A}$ is a collection of subsets of $X$ having the countable intersection property, then there is collection $\mathcal{D}$ of subsets of $X$ such that $\mathcal{D} \supset \mathcal{A}$ is maximal with respect to the counta...
