Articles of research

Can I research in complex analysis, PDE and differential geometry without exposure to mathematical physics?

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 […]

Bockstein homomorphism for $\mathbb{Z}_n$ and “Steenrod” $n$th power

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 […]

Bockstein homomorphism and Steenrod square

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 […]

making mathematical conjectures

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?

Surveys of Current (last 50 years) Mathematics at Graduate / Research level?

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). […]

Errors in math research papers

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

A generalization of IMO 1983 problem 6

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 […]

Research in differential geometry

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 […]

How can a high schooler get more involved in mathematics?

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? […]

“Advice to young mathematicians”

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 […]