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5
votes
2answers
211 views

Are algebraic numbers analogous to group elements with finite order?

Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory? I thought this is the case since requiring an algebraic number $\alpha$ to be ...
7
votes
6answers
1k views

What is this physicist saying?

I do not want to poison this forum with politics. But I want to understand, precisely, what is meant by the bolded statement. It is made by a physicist who used to work at Harvard regarding the ...
5
votes
2answers
413 views

On the Möbius $\mu$ function

A search on wikipedia shows: $$\mu(n) = \sum_{k=1,gcd(k,n)=1}^{n} e^{2\pi i \frac{k}{n}}$$ But that uses complex numbers... and requires finding out the gcd... How useful will be a method, if that ...
6
votes
2answers
691 views

Advice: Modern vs. Classics

First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if ...
4
votes
2answers
189 views

Foundation on Diophantine Analysis and Number Theory

I want to read particularly about diophantine Analysis and Elementary Number Theory from a novice level. The books which I found on net: A Guide to Elementary Number Theory by Underwood Dudley ...
30
votes
5answers
4k views

How to pick a thesis advisor?

This sort of question is probably in bad taste for math.stackexchange, but is probably in high demand. (I tried to start a site on Area 51 to house questions like this, but my request was closed due ...
8
votes
3answers
3k views

Why is 1 raised to infinity Not defined and not “1” [duplicate]

$1$ square is $1$, so is raised $1$ to $123434234$. My maths teacher claims that $1$ raised to infinity is not $1$, but not defined. Is there any reason for this? I know that any number raised to ...
7
votes
4answers
239 views

Why are homomorphisms usually assumed to be functions?

A group homomorphism is usually defined to be a function $\phi$ such that if $x * y = z$, then $\phi x \times \phi y = \phi z$. However, this can be generalized. We could define that a group ...
6
votes
1answer
103 views

Why are germs of functions important?

Why is it necessary to define germs of functions (in my case, for foliations, but my question is in general)? does any inconsistency arises if instead of using a germ in some context, I use ...
4
votes
2answers
473 views

Pascal's triangle and combinatorial proofs

This recent question got me thinking, if a textbook (or an exam) tells a student to give a combinatorial proof of something involving (sums of) binomial coefficients, would it be enough to show that ...
6
votes
3answers
219 views

Analogy between Integration and Summation

There are many analogies between definite integral and Summation: $$\int_a^b \leftrightarrow \sum_a^b$$, This makes me wonder if there is analogous concept of indefinite integral, derivative and ...
3
votes
2answers
255 views

Isn't seven bridges problem trivial? [closed]

What was the actual actual problem that led Euler to graph theory? By looking even at non-simplified map like this It is obvious that, if a landmass is connected by odd number of bridges, it ...
46
votes
3answers
9k views

What is the importance of Calculus in today's Mathematics?

For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a ...
14
votes
2answers
778 views

Working habits in mathematics

Short version of my question: What are good and motivating working habits for a mathematician? Note that, there are similar questions to mine: see this (reading books) or this or this. But none of ...
3
votes
1answer
68 views

Cohomology theories that arise in different fields of mathematics

During my studies in university I have encountered several cohomology theories. Part of them I've met in topology\differential geometry\analysis on manifolds courses (simplicial, singular, cell, ...
5
votes
2answers
447 views

In what order should mathematical fields be learned? [closed]

This could be considered a broader version of this question, with all fields. I know that when high-level maths are reached, the fields being to split quickly (i.e. specializing in this type of ...
2
votes
1answer
188 views

How important are the following undergrad courses when trying to pursue studies in chaos theory/dynamical systems? [closed]

I'm currently a physics major with a year left, and deciding whether to switch into mathematical physics, mathematics or applied mathematics. I'm definitely switching into one of them, as I can meet ...
5
votes
2answers
170 views

Analysis without algebra

I once heard someone say that analysis is $99 \%$ algebra. He was, of course, referring to the amount of algebraic manipulations in the exercises from any calculus course. I know that in topology, ...
20
votes
3answers
1k views

Why should “graph theory be part of the education of every student of mathematics”?

Until recently, I thought that graph theory is a topic which is well-suited for math olympiads, but which is a very small field of current mathematical research with not so many connections to ...
2
votes
2answers
263 views

Texts that define the derivative as the “Anti-integral”…?

Every text that I read starts by defining differentiation then integration... but does anyone know if there is one that goes the other way? Also is there any harm in taking this approach.... to me, ...
47
votes
14answers
2k views

How to entertain a crowd with mathematics? [closed]

I am a high school student who follows a university level curriculum, and recently my teacher asked me to hold a short lecture to a crowd of about 100 people (mostly parents of my classmates and such, ...
2
votes
1answer
35 views

Which expressions in English should I use for a morphism having a certain source and target?

Say that $f: A \rightarrow B$ is an arrow in a category $\mathcal C$. Which verbs or expressions do we use to express in an alternative way that $A$ is the source of $f$ and $B$ its target? E.g., ...
56
votes
4answers
7k views

Why do we care about dual spaces?

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
10
votes
4answers
509 views

How to introduce advanced set-theoretical objects to philosophy students?

First, I apologize if MSE is a bad fit for this question. I'm going to give a course as the last course of "elementary set theory" (the previous courses were not given by me). I planed to introduce ...
7
votes
2answers
276 views

Does $\mathsf{ZFC} + \neg\mathrm{Con}(\mathsf{ZFC})$ suffice as a foundations of mathematics?

I've heard people make the argument that: $\mathsf{ZFC}$ suffices as a foundations of mathematics because almost all theorems in the mathematics literature can be proven using $\mathsf{ZFC}$, so ...
5
votes
2answers
145 views

Why generalize the Euclidean metric?

It is well known that the Euclidean metric can be generalized to $\Bbb R^n$ by $\sqrt{(x_1-x'_1)^2+\cdots + (x_n-x'_n)^2}$, and that under this generalization it is still a metric and satisfies ...
8
votes
1answer
321 views

Can we think of an adjunction as a homotopy equivalence of categories?

There is a way in which we can think about a natural transformation $\eta: F \rightarrow G$ as a homotopy between functors $F,G:\mathcal{C}\rightarrow \mathcal{D}$. Now, an adjunction $F \dashv G$ ...
4
votes
0answers
56 views

Soft Question: Scientific applications of ordinal arithmetic?

Are there any known scientific applications of ordinal arithmetic -- either direct applications or application of results in other areas that depend even indirectly on results from the study of ...
11
votes
2answers
193 views

What's the idea of an action of a group?

I know the formal definition of an action over a set. I'm not asking this. What I'm asking is: what's the intuition of it? It is a way to define an algebra over a set? Since an action can exist in ...
2
votes
2answers
976 views

How far does BEDMAS (order of operations) go?

I was just curious to know how far does BEDMAS go? Can I always find the opposite of something and do it on both sides of an equation to remove it? I know this works for elementary math and high ...
4
votes
2answers
364 views

Motivation for Schur multipliers

What are Schur multipliers good for? I should probably clarify what I want. Here is an instructive story of how I came to appreciate complex representations and characters of groups. Basically, I ...
0
votes
1answer
461 views

What are the prerequisite for understanding complex analysis?

Which should I complete first before complex analysis? I am following Visual Complex Analysis by Tristan Needham. Is there any easier book?
24
votes
7answers
2k views

Popular math books with depth

The most wonderful book I have ever read in my life was Fearless Symmetry by Avner Ash and Robert Gross, which is a good book that gives an intuition , and reasons behind the introducing fields, need ...
12
votes
3answers
395 views

What have been some of the most revolutionary philosophical shifts in perspective in mathematics?

Often times, great revolutions in mathematics come from shifts in philosophical perspective. The shift from extrinsic to intrinsic geometry yields manifolds (and much else). The shift in focus from ...
6
votes
3answers
1k views

How to fill gaps in my math knowledge?

Just finishing highschool, even though I am doing "well" (in the context of the math course itself), I have significant holes in my actual math knowledge. As I think many people who explore math ...
6
votes
1answer
319 views

Age of Stochasticity?

Today I came across D. Mumford's 1999 article The Dawning of the Age of Stochasticity, which is quite remarkable even after more than a decade. The title already indicates the theme, but I copy the ...
12
votes
2answers
977 views

How to deal with Homeomorphisms?

I have one doubt that may be too general, I don't know, so sorry if this is not a good place to ask it. I've also seem many other people with the same problem that I have, so I think that if this ...
1
vote
1answer
52 views

Field of mathematics which deals with similarity of a set of objects each with property variables

(Contextual word of warning: Question written by mathematical novice.) I have a large set of objects. Each object has three variables. Each variable is a number between 0 and 1. For each object in ...
6
votes
6answers
2k views

Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real ...
3
votes
0answers
169 views

What analysis is needed for AG?

My question: On what level do I need to know (complex/real) analysis or diff. geometry to study algebraic geometry from Hartshorne? And AG in general? Context: I will be taken a course in ...
6
votes
2answers
679 views

Local vs. global in the definition of a sheaf

Apologies in advance that this question is inescapably soft. What I am stuck on is squishy; I have the feeling that if I could even make it precise, I'd already be satisfied. To what extent is a ...
12
votes
4answers
1k views

Why vector calculus seems inconsistent and vague

I am a senior student of engineering and I have been studying calculus for a while when I reached the part of vector calculus I felt that this part is inconsistent and there is a multiple questions ...
4
votes
1answer
152 views

Mathematics for Pleasure of a Beginner

I've just read "The Music of the Primes" by Marcus du Sautoy, it is worth a read. I'm not from a maths background, but I'd like to develop a deeper understanding of the concepts. The poetry of math is ...
9
votes
4answers
262 views

Hyperreal field extension

In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension? Is it ...
0
votes
1answer
475 views

What branch of mathematics improves logical thinking? [closed]

So, that's the question. I dare to generalize it even wider: what branch of mathematics improves the general thinking ability, intilligence, the way the person thinks, and makes it more logical? I'm ...
3
votes
3answers
173 views

Euclidean geometry in higher math?

Often, we study a lot of euclidean geometry during high school. For example,Pascal's Theorem,Cross-ratio,Ceva's Theorem` and others. I am looking for instances of theorems encountered in Euclidean ...
1
vote
0answers
45 views

How to build the largest sambusa

I was making sambusa last night. Typically when mama cooks them she has small circles of dough, but mama is not here so I went to the store and bought the dough, and it came in the shape of a ...
1
vote
3answers
88 views

A question about root of a polynomial

If we plug in $x$ into a polynomial and we get the value of $0$ as a result, can we be certain that $x$ is the root of the polynomial? If that is the case, why in this Wikipedia article, it says ...
9
votes
3answers
4k views

Choosing a PhD topic [closed]

My supervisor and I recently had a long chat about PhD related stuff. He said something to the extent that your chances of employment after finishing your PhD among other factors depends on the topic ...
11
votes
5answers
393 views

Constructing a degree 4 rational polynomial satisfying $f(\sqrt{2}+\sqrt{3}) = 0$

Goal: Find $f \in \mathbb{Q}[x]$ such that $f(\sqrt{2}+\sqrt{3}) = 0$. A direct approach is to look at the following $$ \begin{align} (\sqrt{2}+\sqrt{3})^2 &= 5+2\sqrt{6} \\ ...