Posted by Jason Polak on 21. September 2018 · 1 comment · Categories: number-theory · Tags: , ,

Well this is strange indeed: according to this New Scientist article published today, the famous Sir Michael Atiyah is supposed to talk this Monday at the Heidelberg Laureate Forum. The topic: a proof of the Riemann hypothesis. The Riemann hypothesis states that the Riemann Zeta function defined by the analytic continuation of $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ has nontrivial zeros only on the critical line whose numbers have real part $1/2$. Check out this MathWorld article for more details.

The Riemann hypothesis is considered by many to be the outstanding problem in mathematics. Many people have tried to prove it and failed.

Is this for real?

Posted by Jason Polak on 07. September 2018 · Write a comment · Categories: books

When I was a student at McGill I loved looking at the latest Springer texts in the now-nonexistant Rosenthall library. So, I thought that I'd list some of the cool looking titles that have come out in 2018:

  1. Walter Dittrick, Reassessing Riemann's Paper: This book is an analysis of Riemann's paper "On the Number of Primes Less Than a Given Magnitude", and could be a great historical starting point into the subject
  2. Andreas Hinz, Sandi Klavžar, and Ciril Petr, The Tower of Hanoi – Myths and Maths: Looks like a fun recreational math book about the Tower of Hanoi game
  3. Wojciech Chachólski, Tobias Dyckerhoff, John Greenlees, Greg Stevenson, Building Bridges Between Algebra and Topology: It contains notes from four different mini-lectures on Hall algebras, triangulated categories, homotopy invariant commutative algebra, and idempotent symmetries!
  4. Karin Erdmann, Thorsten Holm, Algebras and Representation Theory: Nice-looking text on representation theory. Has classical things like the Artin-Wedderburn theorem and more modern topics like quivers
  5. V. Lakshmibai, Justin Brown, Flag Varieties: An introduction to flag varieties, assuming some background in commutative algebra and algebraic geometry
  6. Nicolas Privault, Understanding Markov Chains
  7. Masao Jinzenji, Classical Mirror Symmetry
  8. Berthé, Michel Rigo (editors), Sequences, Groups, and Number Theory: Now this looks interesting! It is a curious volume of lectures on the interactions between words (as in formal languages and presented groups), number theory, and dynamical systems
  9. Ibrahim Assem, Sonia Trepode (editors), Homological Methods, Representation Theory, and Cluster Algebras: A series of lectures from CRM minicourses

*N.B. I do not work for Springer and I do not get compensated for posting these links. I just think they look cool.

Posted by Jason Polak on 07. September 2018 · Write a comment · Categories: commutative-algebra · Tags: , ,

Let $R$ be a commutative ring and $M_n(R)$ denote the ring of $n\times n$ matrices with coefficients in $R$. For $X,Y\in M_n(R)$, their commutator $[X,Y]$ is defined by
$$[X,Y] := XY – YX.$$ The trace of any matrix is defined as the sum of its diagonal entries.

If $X$ and $Y$ are any matrices, what is the trace of $[X,Y]$? It's zero! That's because the trace of $XY$ is the same as the trace of $YX$. Therefore:

Any commutator has trace zero.

What about the converse? Is any trace zero matrix also a commutator? In other words, given a trace zero matrix $Z\in M_n(R)$, can we find matrices $X$ and $Y$ such that $Z = [X,Y]$? Albert and Muckenhoupt proved that you can, assuming that $R$ is a field.

What happens if you also want $X$ and $Y$ to have trace zero?

Good question. In general, this is not possible. For example, let's consider the simplest field of all, the field with two elements denoted by $\F_2$. Okay, it's actually debatable whether $\F_2$ really is the simplest field, because so many problems happen in characteristic two. For example, this problem we've been considering: in $\F_2$, if $X$ and $Y$ are $2\times 2$ matrices of trace zero, then $[X,Y]$ will have zero off-diagonal entries. So for example, the matrix
$$\begin{pmatrix}1 & 1\\1 & 1\end{pmatrix}\in M_2(\F_2)$$ cannot be written as a commutator of two matrices with trace zero.

It seems that characteristic two is the only obstruction, in the case of $2\times 2$ matrices. In fact, Alexander Stasinski proved in his paper [1] the following:

Theorem. Let $R$ be a principal ideal domain. If $n\geq 3$ then any matrix in $M_n(R)$ of trace zero can be written as the commutator of two matrices in $M_n(R)$, each having trace zero. The same holds for $n=2$ if two is invertible in $R$.

Notice how the characteristic two problem only happens in the $2\times 2$ case.

[1] Stasinski, A. Isr. J. Math. (2018).

Posted by Jason Polak on 06. September 2018 · 1 comment · Categories: technology · Tags: , ,

Some believe that if you're main profession is pure math research, you don't need a scientific calculator. That's simply not true. Although I don't use one nearly as much as when I was an undergrad, I still need a calculator and the only one I'm willing to use is the Casio FX-991MS.

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Posted by Jason Polak on 04. September 2018 · Write a comment · Categories: books · Tags:

Here they are:

Keith Devlin, The Joy of Sets

If you're not a set theorist but want to understand set theory, this book is awesome and one of a kind. I have not read it all, but what I have read I can actually understand!

Frank de Meyer and Edward Igraham, Separable Algebras over Commutative Rings

This classic book explains Galois theory but for commutative rings. Even though there are many more technicalities in the general commutative ring case compared to fields, I actually found the approach in this book more natural than the Galois theory for fields that I learned in undergrad algebra. There are some exercises and this book is easy to read.
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Posted by Jason Polak on 02. September 2018 · Write a comment · Categories: opinion · Tags: ,

I happened to come across a 1993 opinion piece, Theorems for a price: Tomorrow's semi-rigorous mathematical culture by Doron Zeilberger. I think it's a rather fascinating document as it questions the future of mathematical proof. Its basic thesis is that some time in the future of mathematics, the expectation of proof will move to a "semi-rigorous" state where mathematical statements will be given probabilities of being true.

It helps to clarify this with an example even more simple than in Zeilberger's paper. Take the arithmetic-geometric mean inequality for two variables $a,b\geq 0$. It says that
$$\frac{a + b}{2} \geq \sqrt{ab}.$$ This simple identity is just a rearrangement of the inequality $(a – b)^2 \geq 0$. For simplicity, let's say that $a,b\in [0,1]$. Instead of actually proving this inequality, we could generate uniform random numbers in $[0,1]$ and see if this inequality actually works for them. So if I test this inequality 1000 times, of course I will get that it works 1000 times.
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I've been talking a little about abelian categories these days. That's because I've been going over Weibel's An Introduction to Homological Algebra. It's a book I read before, and I still feel pretty confident about the material. This time, though, I think I'm going to explore a few different paths that I haven't really given much thought to before, such as diagram proofs in abelian categories, group cohomology (more in-depth), and Hochschild homology.

Back to abelian categories. An abelian category is a category with the following properties:
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Posted by Jason Polak on 23. August 2018 · 2 comments · Categories: book · Tags: , ,

When it comes to the philosophy of science, not many publications are relevant to modern practice. Let's take math. The current literature still talks about platonism. Look harder and you might find the rise of non-Euclidean geometry or other breakthroughs like cardinality. In short, the bulk of mathematical philosophy still consists of math that's hundreds of years old. While these topics are still important, I find it much more interesting to look at the new philosophical issues present in modern mathematics and science.

That's why I was delighted to find Lost in Math by Sabine Hossenfelder, who is also the author of a popular physics blog called Backreaction.
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Posted by Jason Polak on 17. August 2018 · 2 comments · Categories: website · Tags: ,

Update: Voting is now closed and the results of this contest will be announced on September 5, 2018.

Update #2: Due to a typo in the email I got about the contest rules, the previous update was a mistake. It seems like you can still vote until the end of August!

Update #3: Sadly, I did not win the Goosies. Perhaps next year! Thank you to everyone who voted for me.

Do you enjoy this blog and all the math on it? If so, consider voting for my blog in my host's Canadian Goosy Awards. That's right! Aleph Zero Categorical was nominated as a finalist in this contest. I am in the category for Best Blog/Personal Website. You have until August 31st to vote.

Part of the prize for this contest is free hosting for a year. If you do think I'm worthy of a vote, then your action will support this blog, which has been written my spare time for fun for over seven years. In fact, Aleph Zero Categorical is one of the longest running still-active math blogs on the Internet!

Thanks for considering :)

Let $R$ be a ring and $f:B\to C$ be a morphism of $R$-modules. The image of $f$ is of course
$${\rm im}(f) = \{ f(x) : x\in B \}.$$The image of $f$ is a submodule of $C$. It is pretty much self-evident that $f$ factors as
$$B\xrightarrow{e} {\rm im}(f)\xrightarrow{m} C$$where $e$ is a surjective homomorphism and $m$ is an injective homomorphism. In fact, there is nothing special about working in the category of $R$-modules at all. The same thing holds in the category of sets and a proof for the category of sets works perfectly well for the category of $R$-modules. This set-theoretic reasoning is very natural.

However, we can't always work with categories whose objects are sets with additional structure and whose morphisms are set functions that respect the additional structure (concrete categories). Sometimes we have to work with abelian categories. What's an abelian category? Briefly, it is a category $\Acl$ such that:
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