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I read sometimes mathematical works of others outside my undergraduate studies. I think i can not follow the understanding of the proofs of theorems sometimes. What should i do? Should i read other things connected to the proofs i read to understand the proofs?

I want to make my own theorems and proofs but i can not solve the open problems i read, although they are considered difficult from others.What should i do to make theorems and proofs? At least for them to be new and accepted from the scientific community.

Could i make my own open problems and conjectures or questions and try to prove them? If i do it, what should i read and what do most of the mathematicians do to solve them?

When reading theorems without the proofs and learning them, how will i know if a problem needs those theorems to be solved? About the proofs, how should i use what i learn from them on other possible solutions of problems?

What does someone learn from proofs?

Thank you.

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    $\begingroup$ Are you an undergraduate major in mathematics? Do you plan on graduate work in mathematics? The point of a graduate education in mathematics is to learn how to investigate a field of mathematics independently, and learn how to conjecture and prove theorems on your own. $\endgroup$
    – Steven Gubkin
    Commented Jan 12, 2022 at 12:34
  • $\begingroup$ One of the greatest mathematicians of 20th century — Ramanujan — couldnt care about proofs. Does it mean proofs don't matter? Not so! Just that it's a more culture thing than you may imagine $\endgroup$
    – Rushi
    Commented Jan 12, 2022 at 13:30
  • $\begingroup$ Thank you for your answers. Yes Steven Gubkin, i am an undergraduate student in math and if i get my degree i would want to continue studies, perhaps graduate education. What should i do? Should i wait to have my degree and start research when i start graduate studies? Or should i start from now on my own on research? But i do not know which ways to go to in research and what fields to pursue probably. I have more questions regarding this topic. $\endgroup$
    – plants
    Commented Jan 12, 2022 at 14:09
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    $\begingroup$ You mention reading "mathematical works of others outside my undergraduate studies". What sort/level of works are they? Are they journal articles, textbooks, monographs, something else? Please elaborate. $\endgroup$
    – J W
    Commented Jan 12, 2022 at 14:55
  • $\begingroup$ @plants I would suggest seeking out a math REU for the summer and/or ask some professors at your university to mentor you in a research project. Does your university have a "senior thesis" requirement? $\endgroup$
    – Steven Gubkin
    Commented Jan 12, 2022 at 20:07

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It sounds to me like you need exposure to mathematical topics beyond what is covered in an undergraduate math major. Here's one recommendation:

Fuks, Dmitrij Borisovič, and Serge Tabachnikov. Mathematical Omnibus: Thirty Lectures on Classic Mathematics. American Mathematical Soc., 2007.

      Omnibus

There are similar collections, but this one is both broad and quite accessible. From a review by Harriet Pollatsek:

"The thirty lectures in Omnibus are organized into eight chapters (number of lectures in parentheses): Arithmetic and Combinatorics (3), Equations (5), Envelopes and Singularities (4), Developable Surfaces (3), Straight Lines (4), Polyhedra (6), Surprising Topological Constructions (2), and Ellipses and Ellipsoids (3)."

Added. An example from the last chapter, Lecture 28 is entitled "Billiards in Ellipses and Geodesics on Ellipsoids," drawing on material from Tabachnikov's book Geometry and Billiards. American Mathematical Society, 2005. There is considerable current research on this topic:

Reznik, Dan, Ronaldo Garcia, and Jair Koiller. "Can the elliptic billiard still surprise us?." The Mathematical Intelligencer 42, no. 1 (2020): 6-17. Springer link.

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  • $\begingroup$ Thank you Joseph O'Rourke for your answer, Are these topics accepted as research by the mathematical community, like straight lines and ellipses and ellipsoids today? If they are what could i do to contribute in some ways? I have heard in the past about AMS, one of my professors gave me a collection of papers called Bulletin and some other collections of papers like journals. How could i know what topics are accepted as research and what topics are not accepted? $\endgroup$
    – plants
    Commented Jan 13, 2022 at 6:04
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    $\begingroup$ @plants: Yes, these topics are "accepted as research by the mathematical community." I edited my answer with an example. $\endgroup$
    – Joseph O'Rourke
    Commented Jan 13, 2022 at 14:02

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