I love mathematics, but physics is far away from my interest. I see that recent mathematics research is strongly connected to mathematical physics which is something doesn’t interest me! I love mathematical entities and structures, stuff like relativity and quantum mechanics are not even close to my area of interest, so I wonder if I […]

The Bockstein homomorphism can be generalized for $\mathbb{Z}_n$ values, $$\beta_n: H^m(M^d,\mathbb Z_n) \to H^{m+1}(M^d,\mathbb Z_n),$$ and $$\beta_n x =\frac1n d x \text{ mod } n,$$ $$x \in H^m(M^d,\mathbb Z_n),$$ $$\beta_n x \in H^{m+1}(M^d,\mathbb Z_n).$$ When $n=2$ for $\mathbb{Z}_2$, we can study the relation between Bockstein homomorphism and Steenrod square. For general $\mathbb{Z}_n$, can we find […]

question: What is the relation between Bockstein homomorphism and Steenrod square? For example, can one explain why the following relation works in the case of cohomology group with $\mathbb{Z}_2$ coefficient? For $x \in H^m(M^d,\mathbb{Z}_2)$, $$ \beta_2 x=\frac{1}{2} d x \text{ mod } 2 \in H^{m+1}(M^d,\mathbb{Z}_2) $$ which is the Bockstein homomorphism. It turns out that […]

If a non-mathematician wanted to conjecture something and had strong numerical evidence to support the conjecture, how would he/she go about doing so? Would the mathematical community (a) take it seriously? (b) even look at it?

What books or broad survey articles survey the mathematics of the last 50-100 years? The ones I’ve read do a good job conveying mathematics from the ground up but typically assume a complete beginner or high school student audience and therefore reach only as far as the advanced undergraduate curriculum (middle of the 19th century). […]

This question already has an answer here: In the history of mathematics, has there ever been a mistake? 20 answers

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ Background: The well known 1983 IMO Problem 6 is the following: Let $ a$, $ b$ and $ c$ be the lengths of the […]

I am an 3rd year undergrad interested in mathematics and theoretical physics. I have been reading some classical differential geometry books and I want to pursue this subject further. I have three questions: 1) What are the current research topics in differential geometry? How is scope in those areas? 2) How should I go about […]

For a high school student interested in majoring in math and learning more about math, what kinds of mathematical research can a student in high school get involved in? How can a high school student get involved? If a high school student does conduct research, what mathematical journals would be willing to publish their work? […]

I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the Career Advice section of Terence Tao’s blog, and I am amazed by the intelligence of the pieces of advice given […]

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