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3
votes
2answers
133 views

Specific (algebraic) directions in NCG

Since my initial question (How much algebra is there in Noncommutative Geometry?), I got to study the basics of NCG and I now consider starting a PhD. program in NCG. I read Khalkhali's Basic NCG and ...
0
votes
1answer
263 views

Manifold learning/nonlinear dimensionality reduction for beginners

I'm a computer science graduate student. I recently discovered manifold learning. I think I understand the very basic, high-level concept of nonlinear dimensionality reduction, but I'd like a ...
4
votes
1answer
108 views

Trustworthiness of foundational systems

Naively, we might think that if a foundations of mathematics is consistent, then its fair game. Then we learn a bit more, and we realize that even if a foundations of mathematics is consistent, it may ...
4
votes
1answer
229 views

Hydra game and quantum superposition

Goodstein's theorem is not provable in Peano Arithmetic showed by Kirby and Harrington in 1982 [Wolfram Mathworld]. Any reference of a "quantum" hydra game where a head can remain in a state of ...
0
votes
3answers
327 views

A list of all algebras?

Reading @Qiaochu Yuan's blog the other day, I came across a new term: Poisson algebra. I had never heard of it before and wondered what other algebras are out there that I'm not aware of. Neither ...
6
votes
2answers
139 views

Popular textbooks and current research.

I am sure that I will be finding out first hand as I am entering a PhD program, but I will ask my question anyway. Say, for instance, a student has worked through the majority of a textbook like ...
4
votes
2answers
181 views

How to determine the big questions in a field of mathematics?

During my self-study (and soon to continue at a university) of mathematics, one thing I have been interested in is how to to effectively learn the material. An answer to a question provided by ...
0
votes
1answer
37 views

What is the difference between “model” and “method”

I am not sure which forum to ask this question since the answer may change depending on the scientific area. I am analysing some time series using linear regression. I predict data using the linear ...
1
vote
3answers
91 views

The ubiquitous “helper function” $\frac{f(z) - f(a)}{z - a}$

I've been looking at basic complex analysis recently, and have noticed (am imagining?) something which I've never really paid attention to before: The "helper function" $$g(z) = \frac{f(z) - f(a)}{z ...
2
votes
1answer
152 views

Topology and commutative algebra.

I don't know both of these subjects, but I was wondering if there was any topology in commutative algebra. I don't need any detailed answer (since I don't know any of them yet)...So would it be ...
1
vote
1answer
100 views

What sections should I study to prove that fifth (and up) degree polynomial equations are not solvable with Fraleigh?

I'm Korean high school student who wants to study how to prove that degree ≥5 polynomial equations are not solvable. I know some of Set Theory and will study abstract algebra with 'A First Course in ...
0
votes
3answers
210 views

Does difficulty in math tend to shift over to the actual concepts? [closed]

So far (calculus 1), the math concepts that I learned have been pretty easy, and the greater difficulty is usually in solving certain problems. However, as you go higher and higher do the concepts ...
2
votes
4answers
1k views

How do you make less mistakes in math? [duplicate]

How do you make less mistakes in math? Do you try to be more alert, do you take your time more, or what? Usually I don't make that many mistakes, but sometimes (like now) I do math as I imagine I ...
4
votes
2answers
240 views

Textbook Questions to Do while Self-learning

I am working through Dummit and Foote's Abstract Algebra this summer in preparation for a class next year. However, this is my first time really trying to learn a subject through only a text. It seems ...
8
votes
7answers
608 views

Advice on self study of category theory

I'm very intrested in start to study category theory with aim to use this in algebraic geometry. I already took courses (besides the basics) on commutative algebra and general topology with a soft ...
15
votes
5answers
4k views

What is the best way to develop Mathematical intuition?

I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the ...
14
votes
5answers
899 views

Favourite applications of the Nakayama Lemma

Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result ...
11
votes
3answers
787 views

How to go About Undergraduate Research

I apologize in advance if this question is out of the scope or focus here. I was just wondering about the whole prospect of researching as an undergraduate. How to do it? Who to talk to in my ...
3
votes
1answer
203 views

What's correspondence between the model theoric and the set theoric kernel of homomorphism?

A kernel of a mapping $h$ from $\mathfrak{A}$ to $\mathfrak{B}$, generally, is an equivalence relation $\{(a,a') \in \mathfrak{A} \times \mathfrak{A} \mid h(a)=h(a')\}$. However, in model theory, ...
2
votes
0answers
53 views

Is this slightly different proof to Hibert's Theorem “different enough”?

I generally try and think up slightly different proofs to the material that I read in order to grasp some deeper possible insight, and then painstakingly record them in a latex document. I've been ...
2
votes
0answers
132 views

Symbol for functions that vanish on boundary?

If I have a domain $ M \subset \mathbb{R}^n $, is there a standard symbol for the set of functions $ f \in C^\infty(M) $ that vanish on $ \partial M$ ? I feel like I have seen this before, but I'm ...
7
votes
2answers
331 views

self studying advice on analysis

I am trying to learn analysis on my own but there are times when I can't solve the problem or I get the solution wrong after looking it up, but I will only look up the problems online after I am ...
4
votes
1answer
193 views

Learning Complex Analysis: Integrals vs. Power Series - ordering the development of results.

Over the last few months, I have been visiting elementary complex analysis. My exposure to complex analysis is pretty much limited to the material in three books: Ahlfors, Bak/Newman, and ...
28
votes
4answers
1k views

Understanding mathematics imprecisely

For a long time, it has been a complete mystery to me how any of my peers understood any math at all with anything short of filling in every detail, being careful about every set theoretic detail down ...
1
vote
2answers
112 views

Approximation of differential equations

Can someone provide me a good reference about approximation techniques in continous domain (not piecewise nor numerical methods) for differential equations?
2
votes
1answer
175 views

Convergence w.p. 1 vs convergence in probability: a “physical” example

I understand (proved) that convergence with probability one implies convergence in probability, and that the latter notion is indeed weaker; I've completed an exercise showing that a sequence of ...
9
votes
1answer
772 views

Special cases of the Stark-Heegner theorem with simple proofs

The Stark-Heegner theorem states that the ring of integers of the quadratic number field $\mathbb Q(\sqrt{m})$, where $m$ is a squarefree negative integer, is a principal ideal domain, iff ...
9
votes
8answers
287 views

Examples of “transfer via bijection”

On some occasions I have seen the following situation: We want find out whether a set of a given cardinality $\varkappa$ has some property P. If this property is invariant under bijective maps, then ...
2
votes
2answers
309 views

Is the intersection of every non-empty family of inductive sets equal to the intersection of every inductive set?

In Halmos Naive set theory, there is the following passage (excuse my french) in his section introducing natural numbers : In this language the axiom of infinity simply says that there exists a ...
7
votes
4answers
262 views

Mathematical news sources.

I'm studying my high school right now but I really like math and it would be great for me if I could find a place where I can find about what is going on in the math world nowadays. About a year ago I ...
4
votes
2answers
124 views

Why do we only consider quadratic domains as Euclidean domains with squarefree integers?

I have been reading "Introductory algebraic number theory" by Alaca and Williams, and in the opening chapters they use the quadratic domains $\mathbb{Z}+\mathbb{z}(\sqrt{m})$ for non-square $m$ and ...
14
votes
2answers
7k views

Relearning from the basics to Calculus and beyond.

Assume someone has very limited knowledge of math. (low level high school, 5-6 years ago) How would they learn from the basics of algebra, geometry and trigonometry to a solid foundation for calculus ...
3
votes
0answers
97 views

How to get interest in the mathematics of tax

In a similar vein to my previous thread, I will also be teaching about the mathematics behind taxation - to a lot of people, this is very mundane - but that is not true of everyone. The practicality ...
6
votes
2answers
208 views

The notations change as we grow up

In school life we were taught that $<$ and $>$ are strict inequalities while $\ge$ and $\le$ aren't. We were also taught that $\subset$ was strict containment but. $\subseteq$ wasn't. My ...
7
votes
2answers
144 views

Usefulness of induced representations.

I am learning representation theory from Serre's book by myself. Currently I am reading about induced representations, but I don't understand the importance. The concept looks strange and the ...
9
votes
1answer
395 views

Motivation for Jordan Canonical Form

I took linear algebra and understood the proof that linear operators on a vector space over an algebraically closed field have a Jordan Canonical Form. Why should I care about this theorem? I ...
3
votes
2answers
138 views

What's the correct name in english for “Analysis in $\Bbb R^n$”?

Well, this question may seem silly and I fear it's even out of topic here. My motivation to ask that is to know the correct terminology when talking about that here in Math.SE. The point is, here in ...
6
votes
1answer
258 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
49
votes
2answers
2k views

Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
9
votes
2answers
218 views

Ideas for a present to my topology teacher

Tomorrow is the my final lecture in my favorite course, algebraic topology. I want to give a present to my prof. as a keepsake, something along the lines of this only something I can make due ...
3
votes
1answer
107 views

What's next? $\text{Hom}(U,V)$, tensor product and so on

I don't think this question is as soft as soft-question tag, but if I not how to tag it, I would not be asking this question: I have taken second year linear algebra (Axler's Linear Algebra Done ...
3
votes
4answers
1k views

Sports that use Mathematics

What kind of sports and games use mathematics beyond simple arithmetic? How is math applied to build strategies for these games? Sailing could use mathematics in terms of astronavigation, tying ...
28
votes
4answers
1k views

Why do we believe the Church-Turing Thesis?

The Church-Turing Thesis, which says that the Turing Machine model is at least as powerful as any computer that can be built in practice, seems to be pretty unquestioningly accepted in my exposure to ...
5
votes
1answer
157 views

Algebra and skills needed for Hatcher

A couple of months ago I asked a professor by e-mail to mentor me on topology during the summer. He advised me to study general topology (Hausdorff spaces, connectedness, compactness) and algebraic ...
2
votes
0answers
49 views

Facing mostly-faced decks

In my day job, I am often called upon to take a large stack of—let's call them cards—and make sure that a large majority of them are facing in a particular direction. In most cases, there ...
3
votes
1answer
107 views

About having everything well under control [closed]

When I started doing maths I "rediscovered" some elemental things and listed them on a notebook. All things I learned were in there, like Pythagorean theorem, sine and cosine definitions, some ...
27
votes
8answers
2k views

“It looks straightforward, but actually it isn't”

In a previous topic, I asked about proof of statements which are simple but incorrect. Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, ...
19
votes
5answers
880 views

Is there a deep reason why $(3, 4, 5)$ is pythagorean? [closed]

The triple $(3, 4, 5)$ is a pythagorean triple - it satisfies $a^2 + b^2 = c^2$ and, equivalently, its components are the lengths of the sides of a right triangle in the Euclidean plane. But of ...
3
votes
1answer
101 views

Would this be an effective way to study and comprehend text's?

This is probably a grey area question but I am going to test the waters anyway. What I am thinking of doing would be to basically record myself doing examples from textbooks and making lessons for ...
4
votes
0answers
105 views

Some long and good prerequisite textbooks to the graduate probability textbook by shiryaev and boas?

Some long and good prerequisite textbooks to the graduate probability textbook by shiryaev and boas? It seems that it have a big gap between this graduate textbooks and the easier ones.