Articles of research

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 […]

“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 […]

How do you describe your mathematical research in layman's terms?

“You do research in mathematics! Can you explain your research to me?” If you’re a research mathematician, and you have any contact with people outside of the mathematics community, I’m sure you’ve been asked this question many times. For years now, I’ve struggled to find a satisfying answer. I think an ideal answer to this […]

Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”

I couldn’t find Charles Radin’s Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. Thus, while sorely disappointed for the moment, I have decided to try and solicit recommendations for other books like Radin’s Miles of Tiles, which share these particular features: Theme: “In […]

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? […]