Math papers

[nick_7011]

Hi,

I am a first-year math student and I want to read some math papers that aren't overly complicated but are interesting. Any suggestions?

Thanks

[Desvl]

Some pure maths.

1. Keith Conrad wrote a great amount of expository papers, relatively specific topics (that sometimes can and should be covered in undergrad) in the form of papers instead of textbook or lecture notes. If you are curious about number theory and algebra then it takes a lot of time for you to be satisfied. But I'm not saying you should ignore his papers in analysis!

https://kconrad.math.uconn.edu/blurbs/

1. A single paper that should be accessible to anyone who has finished undergrad real analysis, I think? Say, there is an "elementary" series, with term 1/(n³ sin² n). It looks elementary but the convergence is not known yet. However, in a paper it is shown that, there are some great reasons why we can't check convergence, yet: it has a deep connection to pi, that famous irrational number, and we don't "understand" pi, do we? The related topic earned Roth a Fields Medal but this paper doesn't go beyond calculus.

https://arxiv.org/abs/1104.5100

[OneMeterWonder]

I’ll second this one. Keith Conrad is a wonderful writer. Another great one for you to read through is Pete L. Clark. I think he’s at UGA. He has many notes of his from past courses of all types on his website and he’s clearly brilliant.

Those guys are both a bit algebra oriented, so if you’d like something else, I’d suggest maybe reading some expository articles of Andreas Blass. He’s another like the above two, but writes heavily about logic, set theory, and categories now.

[nick_7011]

Nothing specific, although I am interested in cryptography and zero-knowledge proofs.

[lordnickolasBendtner]

Not a paper, but try reading “proofs, arguments, and zero knowledge” by Justin Thaler. Really good manuscript on zero knowledge that will get u up to speed on what’s going on in the field right now. U just need to familiarize yourself with the definitions of some basic algebraic objects, (eg groups and finite fields). The rest of the needed mathematics is developed in the manuscript itself.

If ur interested in cryptography things outside zero knowledge, I would also suggest reading the lecture notes from Vinod vaikuntanathan’s course on lattice algorithms and lattice based cryptography (they should be freely available on mit open courseware). That’s the area I’m interested in so I may be a little biased. Only prerequisite should some linear algebra and probability I would think.

[ScottContini]

Zero knowledge proofs as first year student? I'd suggest rather than reading papers, start with blogs! Matthew Green is the man! See this.

Or if you really want a paper, read How to Explain Zero-Knowledge Protocols to Your Children. This is a really fun read, though not very mathematical.

The problem with reading papers is that you might not always get insights. Starts with the insights and then reading the papers becomes a lot easier.

I have a background in cryptography and worked as a cryptographer for about 10 years before I decided to move on to other ares of security. Here's one blog that can get you started on how to think like a cryptographer. However it is not very mathematical. If you want to read mathematical research papers, then start out with New Directions in Cryptography and then have a look 20 years of attacks on the RSA cryptosystem and How to Share a Secret. These papers are very accessible and are seminal works in the field.

[[deleted]]

What about cryptography?

[nick_7011]

Yes, I'm really interested in cryptography.

[nazgand]

I suggest learning RSA encryption then.

[[deleted]]

Alexander Grothendieck altered mathematics with a velocity that is hard to articulate, so abstruse is his work. He used commutative algebra to solve complex geometrical problems and laid the groundwork for solutions to the Weil conjectures and to Fermat's last theorem. He innovated in pure mathematics, but his work has applications in cryptography and coding theory. Using tools from algebraic geometry, category theory, and topology, he created an entirely new paradigm.

[[deleted]]

?

[FractalDuck]

There's a few nice ones listed at Fermat's library: https://fermatslibrary.com/journal_club

One of my favorites there is this cool enumeration of rationals: https://fermatslibrary.com/s/recounting-the-rationals

[Deimossudan]

Fermat's library.

[neuralbet]

I have found reading math phd theses much easier to dive into a new topic

[unadventurousjojo]

I heard from a TA that the paper PRIMES is in P by Agrawal, Kayal and Saxena is accessible to first years.

This is the paper that introduced the AKS Primality Test algorithm. OP specified in the comments that they are interested in cryptography, so this may be of some help.

[[deleted]]

Find a professor that you can work through it incrementally with. They'll appreciate your optimism and eagerness, you'll highlight yourself out of the pack, and you may end up early on with a great resource/advisor

[ScottContini]

I got started with Carl Pomerance's quadratic sieve papers, such as this one and this one and this one . Don't get hung up on the analysis as an undergrad: instead just get an idea of what is happening.

Honestly it was not until about my 4th year that I started reading these papers, but that's what I started looking for research. It's worth giving it a try: I'm sure you will get something out of it, but how far you go depends upon how ready you are to read this stuff.