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Is Aluffis book a good second text for Algebra? However, there IS something unique and cutting edge in the book that I don't think Aluffi gets a lot of credit for-there's a whole section on the elements of homotopic algebra, which is a very significant topic of current research interest and I don't think it's covered in any of the other standard graduate algebra books!
general topology - Examples of the difference between Topological . . . 33 There is apparently cutting-edge research by Dustin Clausen Peter Scholze (and probably others) under the name Condensed Mathematics, which is meant to show that the notion of Topological Space is not so well-chosen, and that Condensed Sets lead to better behaved structures What is a simple low-tech example to see the difference?
What background is required to understand Random Matrix Theory Topics in Random Matrix Theory by T Tao; An introduction to random matrices; Spectral Analysis of Large Dimensional Random Matrices In fact, these books probably contain many of the results you were reading about in the papers you mentioned (unless what you were reading was really cutting edge) presented in a much more accessible way
number theory - Undergraduate roadmap for Langlands program and its . . . This is like a person who is just learning to count asking for a roadmap to integral calculus Focus on completing the core graduate material in analysis, geometry, and algebra -- the cutting-edge stuff will come in due time
What languages to learn for maths? - Mathematics Stack Exchange The first few for obvious historical reasons In fact, except for Chinese, the impetus is predominantly historical nowadays; internationally, most cutting-edge mathematics is being published in English
What algebraic topology book to read after Hatchers? J Peter May and Kate Ponto's More Concise Course In Algebraic Topology is a follow-up to May's introductory book containing cutting edge topics such as spectral sequences, model categories,basic homotopy theory,localization and completion of spaces and more
Slicing edges out of a high dimensional polytope. The original cone was $123$ and after cutting it it became $12p3$ because $12$ is a valid edge, so are $2p, p3, 31$ In 4D our plane generates 3 new edges, we know that it intersects the planes $2,3,4$
Is Foundational Research a Dead Field? - Mathematics Stack Exchange Of course the cutting-edge results are usually technical, but the same can be said for every other well-developed area of mathematics Nobody would read a paper by Galois and think that it is reflective of cutting edge work in algebra, or read work by Cauchy and think that is it reflective of current research in analysis
What is the difference between elementary and advanced math? Pell's equation is quite easy to solve, Pythagorean triples were known to the Babylonians, Fermat's Last Marginalium lasted hundreds of years and requires cutting-edge work on general elliptic curves Bisecting an angle is easy with straightedge and compass, trisecting an angle is impossible without neusis The area of a cirle is $\pi r^2$
Erdos conjecture about Fat sequences? - Mathematics Stack Exchange 2 Some while ago i sat with my professor Adrian Stern of the Ben-Gurion University for a little more than half an hour, we talked about a lot of open question and cutting edge methods in math to tackle these problems So he told me about "Erdos Fat sequences conjecture" to be found in Wikipedia Erdős conjecture on arithmetic progressions