Entropy + Algebra + Topology = ?

Today I'd like to share a bit of math involving ideas from information theory, algebra, and topology. It's all in a new paper I've recently uploaded to the arXiv, whose abstract you can see on the right. The paper is short — just 11 pages! Even so, I thought it'd be nice to stroll through some of the surrounding mathematics here.

To introduce those ideas, let's start by thinking about the function $d\colon[0,1]\to\mathbb{R}$ defined by $d(x)=-x\log x$ when $x>0$ and $d(x)=0$ when $x=0$. Perhaps after getting out pencil and paper, it's easy to check that this function satisfies an equation that looks a lot like the product rule from Calculus:

Functions that satisfy an equation reminiscent of the "Leibniz rule," like this one, are called derivations, which invokes the familiar idea of a derivative. The nonzero term $-x\log x$ above may also look familiar to some of you. It's an expression that appears in the Shannon entropy of a probability distribution. A probability distribution on a finite set $\{1,\ldots,n\}$ for $n\geq 1$ is a sequence $p=(p_1,\ldots,p_n)$ of nonnegative real numbers satisfying $\sum_{i=1}^np_i=1$, and the Shannon entropy of $p$ is defined to be

Now it turns out that the function $d$ is nonlinear, which means we can't pull it out in front of the summation. In other words,  $H(p)\neq d(\sum_ip_i).$ Even so, curiosity might cause us to wonder about settings in which Shannon entropy is itself a derivation. One such setting is described in the paper above, which shows a correspondence between Shannon entropy and derivations of (wait for it...) topological simplices!

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Topology Book Launch

This is the official launch week of our new book, Topology: A Categorical Approach, which is now available for purchase! We're also happy to offer a free open access version through MIT Press at topology.mitpress.mit.edu.

Inside, you'll find a presentation of basic, point-set topology from the perspective of category theory, targeted at graduate students in a first-semester course on topology. The idea is that most of these students are already somewhat familiar with the point-set ideas through a course on analysis or undergraduate topology. For this reason, many graduate-level instructors are tempted to rush through point-set topology or to skip it altogether to reach algebraic topology, which can be more fun to learn and to teach.

Our book presents an alternative to this approach. Rather than skipping over the basic ideas, we view this as an excellent opportunity to introduce students to the modern viewpoint of category theory. 

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Topology: A Categorical Approach

I've been collaborating on an exciting project for quite some time now, and today I'm happy to share it with you. There is a new topology book on the market! Topology: A Categorical Approach is a graduate-level textbook that presents basic topology from the modern perspective of category theory. Coauthored with Tyler Bryson and John Terilla, Topology is published through MIT Press and will be released on August 18, 2020. But you can pre-order on Amazon now! 

Here is the book's official description:

This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory—a contemporary branch of mathematics that provides a way to represent abstract concepts—both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics.

After presenting the basics of both category theory and topology, the book covers the universal properties of familiar constructions and three main topological properties—connectedness, Hausdorff, and compactness. It presents a fine-grained approach to convergence of sequences and filters; explores categorical limits and colimits, with examples; looks in detail at adjunctions in topology, particularly in mapping spaces; and examines additional adjunctions, presenting ideas from homotopy theory, the fundamental groupoid, and the Seifert van Kampen theorem. End-of-chapter exercises allow students to apply what they have learned. The book expertly guides students of topology through the important transition from undergraduate student with a solid background in analysis or point-set topology to graduate student preparing to work on contemporary problems in mathematics.

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Is the Square a Secure Polygon?

In this week's episode of PBS Infinite Series, I shared the following puzzle:

Consider a square in the xy-plane, and let A (an "assassin") and T (a "target") be two arbitrary-but-fixed points within the square. Suppose that the square behaves like a billiard table, so that any ray (a.k.a "shot") from the assassin will bounce off the sides of the square, with the angle of incidence equaling the angle of reflection. Puzzle: Is it possible to block any possible shot from A to T by placing a finite number of points in the square?

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Brouwer's Fixed Point Theorem (Proof)

Today I'd like to talk about Brouwer's Fixed Point Theorem. Literally! It's the subject of this week's episode on PBS Infinite Series. Brouwer's Fixed Point Theorem is a result from topology that says no matter how you stretch, twist, morph, or deform a disc (so long as you don't tear it), there's always one point that ends up in its original location.

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Topology vs. "A Topology" (cont.)

This blog post is a continuation of today's episode on PBS Infinite Series, "Topology vs. 'a' Topology." My hope is that this episode and post will be helpful to anyone who's heard of topology and thought, "Hey! This sounds cool!" then picked up a book (or asked Google) to learn more, only to find those formidable three axioms of 'a topology' that, admittedly do not sound cool. But it turns out those axioms are what's "under the hood" of the whole topological business! So without further ado, let's pick up where we left off in the video.

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"One-Line" Proof: Fundamental Group of the Circle

Once upon a time I wrote a six-part blog series on why the fundamental group of the circle is isomorphic to the integers. (You can read it here, though you may want to grab a cup of coffee first.) Last week, I shared a proof* of the same result. In one line. On Twitter. I also included a fewer-than-140-characters explanation. But the ideas are so cool that I'd like to elaborate a little more.

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The Sierpinski Space and Its Special Property

Last time we chatted about a pervasive theme in mathematics, namely that objects are determined by their relationships with other objects, or more informally, you can learn a lot about an object by studying its interactions with other things. Today I'd to give an explicit illustration of this theme in the case when "objects" = topological spaces and "relationships with other objects" = continuous functions. The goal of this post, then, is to convince you that the topology on a space  is completely determined by the set of all continuous functions to it.

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Clever Homotopy Equivalences

You know the routine. You come across a topological space $X$ and you need to find its fundamental group. Unfortunately, $X$ is an unfamiliar space and it's too difficult to look at explicit loops and relations. So what do you do? You look for another space $Y$ that is homotopy equivalent to $X$ and whose fundamental group $\pi_1(Y)$ is much easier to compute. And voila! Since $X$ and $Y$ are homotopy equivalent, you know $\pi_1(X)$ is isomorphic to $\pi_1(Y)$. Mission accomplished.

 Below is a list of some homotopy equivalences which I think are pretty clever and useful to keep in your back pocket for, say, a qualifying exam or some other pressing topological occasion.

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(Co)homology: A Poem

I was recently (avoiding doing my homology homework by) reading through some old poems by Shel Silverstein, author of The Giving TreeA Light in the Attic, and Falling Up to name a few. Feeling inspired, I continued to procrastinate by writing a little poem of my own - about homology, naturally!

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Classifying Surfaces (CliffsNotes Version)

My goal for today is to provide a step-by-step guideline for classifying closed surfaces. (By 'closed,' I mean a surface that is compact and has no boundary.) The information below may come in handy for any topology student who needs to know just the basics (for an exam, say, or even for other less practical (but still mathematically elegant) endeavors) so there won't be any proofs today. Given a polygon with certain edges identified, we can determine the surface that it represents in just three easy steps:

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Topological Magic: Infinitely Many Primes

A while ago, I wrote about the importance of open sets in topology and how the properties of a topological space $X$ are highly dependent on these special sets. In that post, we discovered that the real line $\mathbb{R}$ can either be compact or non-compact, depending on which topological glasses we choose to view $\mathbb{R}$ with. Today, I’d like to show you another such example - one which has a surprising consequence!

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The Fundamental Group of the Real Projective Plane

The goal of today's post is to prove that the fundamental group of the real projective plane, is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ And unlike our proof for the fundamental group of the circle, today's proof is fairly short, thanks to the van Kampen theorem! To make our application of the theorem a little easier, we start with a simple observation: projective plane - disk = Möbius strip. Below is an excellent animation which captures this quite clearly....

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The Fundamental Group of the Circle, Part 6

Welcome to the final post in a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. Today we prove two lemmas (the path- and homotopy-lifting properties) that were used in parts four and five. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

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A Recipe for the Universal Cover of X⋁Y

Below is a general method —a recipe, if you will —for computing the universal cover of the wedge sum $X\vee Y$ of arbitrary topological spaces $X$ and $Y$. This is simply a short-and-quick guideline that my prof mentioned in class, and I thought it'd be helpful to share on the blog. To help illustrate each step, we'll consider the case when $X=T^2$ is the torus and $Y=S^1$ is the circle.       

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The Fundamental Group of the Circle, Part 5

Welcome to part five of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our homomorphism from $\mathbb{Z}$ to $\pi_1(S^1)$ is injective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

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The Fundamental Group of the Circle, Part 4

Welcome to part four of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our homomorphism from $\mathbb{Z}$ to $\pi_1(S^1)$ is surjective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

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The Fundamental Group of the Circle, Part 3

Welcome to part three of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our map from $\mathbb{Z}$ to $\pi_1(S^1)$ is a group homomorphism. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

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The Fundamental Group of the Circle, Part 2

Welcome to part two of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we justify a shortcut that we never actually use in the remainder of this series, so the reader is welcome to skip this post. But I've included it since, in this series, we're closely following section 1.1 of Hatcher's Algebraic Topology.

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The Fundamental Group of the Circle, Part 1

Welcome to part one of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we define a map from $\mathbb{Z}$ to $\pi_1(S^1)$ and make some simple observations via pictures and an animation! The proof follows that found in Hatcher's Algebraic Topology</a>, section 1.1.

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Open Sets Are Everything

In today's post I want to emphasize a simple - but important - idea in topology which I think is helpful for anyone new to the subject, and that is: Open sets are everything! What do I mean by that? Well, for a given set $X$, all the properties of $X$ are HIGHLY dependent on how you define an "open set."

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On Connectedness, Intuitively

Today's post is a bit of a ramble, but my goal is to uncover the intuition behind one of the definitions of a connected topological space. Ideally, this is just a little tidbit I'd like to stash in The Back Pocket. But as you can tell already, the length of this post isn't so "little"! Oh well, here we go!

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Compact + Hausdorff = Normal

The notion of a topological space being Hausdorff or normal identifies the degree to which points or sets can be "separated." In a Hausdorff space, it's guaranteed that if you pick any two distinct points in the space -- say $x$ and $y$ -- then you can always find an open set containing $x$ and an open set containing $y$ such that those two sets don't overlap.

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