For questions that don't admit a definitive answer. Please do not ask too many of these.
10
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151 views
definite and indefinite sums and integrals
It just occurred to me that I tend to think of integrals primarily as indefinite integrals and sums primarily as definite sums. That is, when I see a definite integral, my first approach at solving it ...
9
votes
0answers
101 views
Writing Hard Problems
Having done a fair bit of contest- type math, I was interested in trying to write similar sorts of problems, only to find myself immediately stumped. How does one write "elegant problems" of the kind ...
9
votes
0answers
355 views
How do Greek mathematicians name variables?
I've always wondered how people in Greek name variables that other people use greek letters e.g. $\theta$. They use latin?
8
votes
0answers
84 views
Undergraduate thesis advice
I am presently in my senior year and I am considering fluid mechanics for my thesis. What area of research of fluid mechanics or models of transport mechanism in Biological systems which is purely ...
8
votes
0answers
101 views
What are the “correct” modules over locally ringed spaces?
$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent ...
8
votes
0answers
118 views
Is there a collection of alternative mathematical notation? (Semi-soft Question)
I'm interested in alternative systems of notation for mathematics. I've often heard how mathematical notation is illogical, inconsistent, filled with grandfather clauses that serve no purpose, and ...
8
votes
0answers
232 views
Which is the newest and most unexplored branch in pure math?
I recently heard about a branch called tropical geometry developed by Professor Bergman. I was wondering if there´s a newest but yet unexplored math branch, and by newest I mean developed in the last ...
7
votes
0answers
185 views
Is a sustained,lifelong dedication & commitment to Mathematics worth every bit of it?
It may not be the kind of question I am supposed to ask here.I apologize if I violate any terms and conditions.But I really need an answer to it.I am in my mid 20s and just another average guy.But for ...
7
votes
0answers
198 views
Are there independent research groups?
I wanted to ask if someone knows how to find independent researchers or research groups. I mean, people interested in doing Math in a research level that are not related to universities neither ...
7
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0answers
159 views
Graham's Number : Why so big?
Can someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !
7
votes
0answers
189 views
The status of $\mathbb{R}$ in homotopy theory.
The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
6
votes
0answers
70 views
Working with subsets, as opposed to elements.
Especially in algebraic contexts, we can often work with subsets, as opposed to elements. For instance, in a ring we can define
$$A+B = \{a+b\mid a \in A, b \in B\},\quad -A = \{-a\mid a \in A\}$$
...
6
votes
0answers
66 views
Examples of falsified (or currently open) longstanding conjectures leading to large bodies of incorrect results.
In general, the way that modern mathematical research is conducted isn't the way that many would assume is the ideal method of research. That is, mathematics is not the linear progression of ...
6
votes
0answers
122 views
Articles on ideas in the history of mathematics notation?
I'm teaching a course this term on the history of scripts (writing systems) and rather than talking interminably about Semitic and Chinese and their spawn, I'd like to give students a more varied ...
6
votes
0answers
372 views
How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?
I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
6
votes
0answers
476 views
Efficient ways to read and learn a new topic
I started reading the book "Topology without tears" by Sidney A Morris and lecture notes on "Elementary Number Theory" by WWL.Chen. To get the maximum out of the book and understand the material ...
6
votes
0answers
181 views
Irrationality of e
I have a question about the irrationality of $e$:
In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
6
votes
0answers
579 views
Which is better strategy to learn and read books, traditionally one by one OR re-read carefully on perfect books
(Just focus on how to learn and master the stuff pretty well, not involve the aspect of courses or exam)
Because recently I always feel that the time and energy are pretty limited, I want to try ...
5
votes
0answers
57 views
Simplest examples of real world situations that can be elegantly represented with complex numbers
Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
5
votes
0answers
63 views
How to understand convex duality intuitively
Is there an intuitive way to understand the convex duality? If the primal problem is minimization, the dual is maximization over another set of variables - but I would love to have a geometric ...
5
votes
0answers
72 views
Proof vs Practice
I've been brushing up on a lot of basic arithmetic, algebra, and logic as I work towards a review of calculus (and beyond), and I keep noticing that in order to fully understand many principles in the ...
5
votes
0answers
246 views
formulating theories in math
One of my profs mentioned that sometimes people formulate theories about some type of object, but then later realize that those objects do not exist. Can someone given me an example of such a theory? ...
5
votes
0answers
363 views
What is magical about Cartan's magic formula?
Why is Cartan's magic formula
$$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$
called "magic"?
Should it be considered a highly surprising result? Does it "magically" prove several other ...
5
votes
0answers
96 views
Reference suggestion: eigenvalues of tridiagonal matrices
I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals).
I have seen authors use continued ...
5
votes
0answers
119 views
Convex Hulls vs Shrink Wrap
I was recently explaining to a friend what the convex hull of a set of points is using the analogy of an elastic band around a set of nails hammered into a board. I was about to say that we can ...
5
votes
0answers
113 views
Why are injective $\mathscr{O}$-modules flasque?
Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
5
votes
0answers
269 views
Transitioning from Statistics to Pure Math at the PhD level
I was not sure if I should even post this question at this site because it is not directly related to mathematics, but rather, careers in mathematics.
First of all, I have a B.S. in pure math. ...
4
votes
0answers
60 views
Soft question: why are there non-smooth manifolds?
Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...
4
votes
0answers
43 views
Hopf Algebras in Combinatorics
I know that many examples of Hopf algebras that come from combinatorics. But I'm interested in knowing how Hopf algebras are applied in solving combinatorial problem.
Are there examples of open ...
4
votes
0answers
27 views
Interpretation for the Functional Determinant
Let $S:V \rightarrow V$ be a linear operator on the function space $V$. It is possible to define a functional determinant for $S$ via the zeta function regularization process.
In specific we define ...
4
votes
0answers
78 views
The high road to learn algebraic geometry
Suppose that a student has a basic knowledge in commmutative algebra at the level of the Atiyah-MacDonald.
What do you think about the following steps to learn algebraic geometry?
step 1: read ...
4
votes
0answers
69 views
Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds
The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
4
votes
0answers
253 views
Using distance formula to find slope, any reason to use the concluding equation?
So, today I was observing a class that I will be a TA for this semester and the professor started to talk about the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Well, my mind wandered a little ...
4
votes
0answers
109 views
Recommendation textbooks on D-module
I am going to take part in a seminar on D-module and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, and A Primer of Algebraic D-Modules
...
4
votes
0answers
128 views
Forum for students - your experiences, recommendations, suggestions
Have you ever used online discussion forum or something similar for students of your course? (Or, if you are a student, have you attended a class, where something like this was used?)
If yes, I'd ...
4
votes
0answers
69 views
What is good about simple Lie algebras?
Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple.
I wonder what beautiful properties does a ...
4
votes
0answers
144 views
(Mathematical) Applications of Quantum Group
What are some mathematical applications of Quantum Groups?
I have tried researching and found that it is used to find solutions to Yang-Baxter equation.
Any other applications?
Thanks a lot.
4
votes
0answers
118 views
Status of videos from the Arf-Kervaire invariant problem conference at MSRI
This is not a real maths question (math-related though). A couple of years ago Mike Hill, Mike Hopkins and Doug Ravenel solved the Arf-Kervaire invariant problem. There was a conference at the MSRI ...
4
votes
0answers
238 views
homology groups, physical interpretation
One can roughly interpret Betti numbers as the number of n-dimensional holes of our space. Is there a reasonable interpretation for torsion coefficients? Moreover, for 'physical' applications, are ...
4
votes
0answers
181 views
Interesting applications of the cofinite topology?
Background: I'm doing some expository writing on intuitionistic logic and I have been toying with the idea of demonstrating its applicability via models where the denotations are taken from a Heyting ...
3
votes
0answers
30 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 ...
3
votes
0answers
67 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 ...
3
votes
0answers
120 views
Is it possible to learn mathematics right from the source instead of reading textbooks. By studying the masters and not their pupils
i was wondering if mathematics learning process require the use of textbooks.
When i was a high school student, i read as a preparation for university, Legendre book on Elements of geometry and ...
3
votes
0answers
70 views
$M_n=2^n-1$ Mersenne numbers in mathematics
Did the Mersenne numbers turn out to be interesting in other fields of mathematics besides the Numbers Theory?
In other words, the function $M(n)=M_n$
where
$$M_n=2^n-1$$
or the recursive realtion
...
3
votes
0answers
52 views
Why is the Euclidean metric called the prime at infinity?
I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the ...
3
votes
0answers
102 views
Why don't we introduce the concept of base for a topology in a minimal way?
Why don't we introduce the concept of base for a topology in a minimal way exactly as we did in Linear Algebra?
Edit: A topology can be obtained from a base by considering all possible unions of the ...
3
votes
0answers
55 views
References about finite group theory
In your opinion which are the best books regarding the theory of finite groups? I think that a wonderful one is "Finite Group Theory - Michael Aschbacher".
Many thanks.
3
votes
0answers
65 views
A few questions about nonabelian cohomology of finite groups.
I apologize in advance if these questions are broad or basic. I tried to read about them at the Wikipedia, but everything is written in the language of category theory, in which I have had no formal ...
3
votes
0answers
228 views
Looking for discrete math research topics for undergraduates
I am looking for a discrete math research problem that a talented undergraduate can work on and prove something new within a few years, with the help of a faculty mentor (me). Although "applied ...
3
votes
0answers
56 views
Is there a way to determine how many solution does “ The hundred Fowls problems” have looking at the coefficients?
I was looking at the different versions of the hundred fowls problems. I came across the problem posed by Chinese mathematicians posed in 5th century,and then Indians in 9th, 10th centuries. Alcuin of ...
