Multiplying Non-Numbers

In last last week's episode of PBS Infinite Series, we talked about different flavors of multiplication (like associativity and commutativity) to think about when multiplying things that aren't numbers. My examples of multiplying non-numbers were vectors and matrices, which come from the land of algebra. Today I'd like to highlight another example:

 
shapes.jpg
 

It's true! To illustrate, here's the multiplication table for a point, a line, a circle, and a square.

See the pattern? Given any two shapes $X$ and $Y$, their product $X\times Y$ is the shape whose vertical cross sections look like $Y$ and whose horizontal cross sections look like $X$! Also notice that the point acts like a multiplicative identity: a point times a shape is just the shape.

I'm really stretching my art skills today, but the idea is hopefully clear: A 'circle $\times$ line segment' is a vertical cylinder: its horizontal cross sections are circles and its vertical cross sections are lines.

 
crossection2.jpg
 
The idea behind the multiplication is that there's one circle for every point on the line segment (so, "a line segment's worth of circles") and one line segment for every point on the circle (so, "a circle's worth of line segments"). When you put it all together, the result is a cylinder!
cylinder.jpg
Similarly, a 'circle $\times$ square' is a roundish-but-square torus thing. (For some reason it reminds me of a cronut, though they're not quite the same.) Its horizontal cross sections are circles, and its vertical cross sections are squares. (So what does a 'circle $\times$ torus' look like? Or a 'torus $\times$ torus'??)
crosssections.jpg
Notice that this multiplication is not commutative! A circular torus with square cross sections is not the same as a square torus with circular cross sections! In other words, 'circle $\times$ square $\neq$ square $\times$ circle.' Likewise, a box lying horizontally is not the same as a box standing upright.

But what about associativity? Let's compare '(point $\times$ line segment) $\times$ circle' with 'point $\times$ (line segment $\times$ circle).' According to the multiplication table:
assoc.jpg
So, yes! It's associative! It doesn't matter which two shapes we multiply first---at least in this example. And it turns out that this multiplication-of-shapes is in general associative. By the way, this multiplication has a name. The product $X\times Y$ of two shapes is called the Cartesian product of $X$ and $Y$. So if you watched last week's video, you've now heard of at least four examples of multiplication* of non-numbers:
 
four.jpg
 
Now here's what's cool: shapes will make an appearance in our next epsisode on Infinite Series! As we saw last week, loop concatenation is not associative: the two ways of multiplying three loops are not equal. And in the next episode, we'll discover that there are infinitely many ways of multiplying three loops. AND - it gets better - all of those ways are encoded in a particular shape!

What shape, exactly?

You'll have to wait and see!

 

 
*While watching the video, you might've wondered why I referred to loop concatenation as a "multiplication" (or product) instead of "addition" (or sum). Technically, loop concatenation is a binary operation, which is a way to combine two things (the inputs) to get a third thing (the output). Addition and multiplication of numbers are examples of binary operations. In practice, mathematicians often use the word “addition” to describe a binary operation that is commutative. And although I don’t mention it in the video, loop concatenation is not commutative! So we call it a “multiplication” instead.

Thanks, John, for suggesting the idea for today's post!

Math3ma + PBS Infinite Series!

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Hi everyone! Here's a bit of exciting news: As of today, I'll be extending my mathematical voice from the blogosphere to the videosphere! In addition to Math3ma, you can now find me over at PBS Infinite Series, a YouTube channel dedicated to the wonderful world of mathematics.

Infinite Series (a part of PBS Digital Studios) was formally hosted by Kelsey Houston-Edwards, a PhD candidate at Cornell University. Over the past year Kelsey has written and hosted 45 marvelous videos on a wide variety of topics. Here are a few!

As Kelsey announced in her latest video, The Heat Equation, she'll be using the upcoming year to finish her dissertation. (Best of luck, Kelsey!) Physicist Gabe Perez-Giz -- former host of PBS SpaceTime -- and I will step in as hosts for Season 2.

The first two episodes, including today's "The Multiplication Multiverse," were inspired by recent posts here on the blog, and I had a lot of fun finding ways to pare down research-level mathematics. There is a vast world of beautiful, enticing math that's often hidden behind lofty language. But it's far too exciting to remain hidden! With that said, I sincerely hope that Season 2 of Infinite Series will continue to serve as a warm invitation into the wonders of mathematics.

So without further ado, here's the first episode. It's the first of a two-part series. I'll be sure to post the second video here once it's live. And after this mini-series, Gabe's got some exciting things to share too!

Be sure to subscribe to the channel, new videos come out each Thursday!

Enjoy!

And here's part 2!


PS. Big shoutout to producer Rusty Ward and animators Ray Lux and Louis Costa for making my drawings come to life!

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What is an Operad? Part 2

Last week we introduced the definition of an operad: it's a sequence $\mathcal{O}(1),\mathcal{O}(2), \mathcal{O}(3),\ldots$ of sets or vector spaces or topological spaces or most anything you like (whose elements we think of as abstract operations), together with composition maps $\circ_i\colon \mathcal{O}(n)\times\mathcal{O}(m)\to\mathcal{O}(n+m-1)$ and a way to permute the inputs using symmetric groups. We also defined an algebra over an operad, which a way to realize each abstract operation as an actual operation. Now it's time for some examples!

The Associative and Commutative Operads


Suppose $V$ is a vector space over a field $\mathbb{k}$. For each $n\geq 1$, define Assoc$(n)$ to be the 1-dimensional vector space generated by the tree with $n$ leaves. And let's not worry about permuting the leaves---there's no action of the symemtric group $S_n$ here. (Such operads are called non-symmetric operads.)
assoc.jpg
The $\circ_i$ composition is tree grafting, as introduced last time For example, to describe the compositions $$\circ_i\colon \text{Assoc}(2)\times \text{Assoc}(2)\to \text{Assoc}(3), \qquad i=1,2$$ it's enough to say where 𝖸$\circ_1$𝖸 and 𝖸$\circ_2$𝖸 land. Here, I'm using 𝖸 to depict the 2-to-1 operation that generates $\text{Assoc}(2)$. But there's only one option! Up to a scalar multiple, there's only one 3-to-1 tree in $\text{Assoc}(3)$! In other words, the following trees must be equal
 
what.jpg
 
Now, what's an algebra over this operad? As we saw last time, it's a collection of maps $\varphi\colon \text{Assoc}(n)\to\text{End}_V(n)$ for each $n=1,2,\ldots$ that's compatible with the $\circ_i$. Let's define $m:=\varphi(𝖸)\colon V\times V\to V$ to be the image of the 2-to-1 operation 𝖸. Compatibility tells us that the first and third equalities hold: $$\varphi(𝖸)\circ_1\varphi(𝖸)=\varphi(𝖸\circ_1 𝖸)=\varphi(𝖸\circ_2 𝖸)=\varphi(𝖸)\circ_2\varphi(𝖸)$$ while the second equality holds from the picture above. This amounts to the statement that $$m(m(v_1,v_2),v_3)=m(v_1,m(v_2,v_3))$$ for all $(v_1,v_2,v_3)\in V^3$, or writing $v_1\cdot v_2$ instead of $m(v_1,v_2)$, $$(v_1\cdot v_2)\cdot v_3= v_1\cdot(v_2\cdot v_3).$$ This shows that $m$ is an associative product on $V$! In other words, an algebra over the operad Assoc is an associative algebra.
comm1.jpg
We didn't consider a symmetric group action, but if we do, and if we define it to be trivial (i.e. $\sigma f=f$ for all $\sigma\in S_n$ and all $n$-ary operations $f$) then $v_1\cdot v_2= v_2\cdot v_1$ for all $v_1,v_2\in V$ since the two trees on the left must be equal. This operad is called the Comm operad, and an algebra over it is a commutative algebra.

The Associahedra Operad


The associahedra are a sequence of polytopes that encode operations that are associative up to homotopy. Let's look at an example. Suppose $X$ is a topological space and let $a,b\colon I\to X$ be loops based at point in $X$. (That is, $a$ and $b$ are continuous functions, both of which send $0,1\in I$ to the same point in $X$.) The product $a\cdot b$ gives us a new loop by "going around $a$ and $b$ each at twice the original speed." We can think of traversing $a$ in the first half-second, then traversing $b$ in the second half. This gives us a 2-to-1 operation $\Omega X\times \Omega X\to \Omega X$ where $\Omega X$ denotes the space of all based loops of $X$.
ab.jpg
Is this operation associative? Well, if we have three loops $a,b$ and $c$, there are two options:
 
two loops.jpg
 
Here, $(a\cdot b)\cdot c$ means "do $a$ on the first quarter of the interval, and do $c$ on the second half," while $a\cdot(b\cdot c)$ means "do $a$ on the first half of the interval and do $c$ on the last quarter." These two loops are not equal, so this "multiplication" is not associative. But we can get from one loop to the other simply by adjusting the speed at which we traverse $a$ (and $c$)! In other words, we can go from $(a\cdot b)\cdot c$ to $a\cdot(b\cdot c)$ continuously by traveling around $a$ a little slower and traveling around $c$ a little faster. This defines a homotopy between the two loops, which we can represent as a line segment, called $K_3$, joining two points.
 
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The vertices represent the two loops $(a\cdot b)\cdot c$ and $a \cdot(b\cdot c)$, and every point in between represents an intermediate loop. For example, the midpoint represents the loop $a\cdot b\cdot c$ in which $a, b$ and $c$ are all traversed in equal time.

Now what happens if you want to take the product of four loops $a,b,c,d$?! There are five ways to parenthesize four letters, so we have five different vertices. Some of these can be connected by edges using a homotopy, which gives us the boundary of a pentagon. Now it turns out that you can get from $((ab)c)d$ to $a(b(cd))$ via one of two homotopies, depicted as the red and blue paths below. What's more, you can get from any point on the blue path to a point on the red path in a continuum of ways. In short, we get a continuum of paths between the red and blue paths, which sweeps out the face of the pentagon! So the gray region is really a homotopy between homotopies. All the ways you can multiply four loops is captured by this 2-dimensional polytope, which we call $K_4$.

 
 
Now the next polytope $K_5$ has one vertex for each of the 14 ways way you can parenthesize five letters. There are 21 edges (corresponding to homotopies) and 9 faces (homotopies between homotopies) and 1 solid interior (a homotopy between the homotopies between the homotopies)!
K5.jpg
And the list goes on. The polytopes $K_2,K_3,K_4,\ldots$ form an non-symmetric operad (where $K_1=\varnothing$) with composition being the inclusion of faces. Each $K_n$ is $n-2$-dimensional and the vertices represent the ways of putting parentheses around $n$ letters.
 
polytopes.jpg
 
An algebra over this operad is called an $A_\infty$ space, first introduced by Jim Stasheff in the early sixties. (Take note of the word "space!" unlike our previous examples, the $n$-ary operations form a topological space* rather than a vector space!) The "A" stands for "associative" and the infinity reminds us of the infinite string of homotopies between homotopies between homotopies between homotopies between.... And the associahedra are of algebraic, geometric, and combinatorial interest, too! For instance, take a look at this survey by J. L. Loday.


The Little k-Cubes Operad

Closely related to the associahedra is the little $k$-cubes operad, where $k>0$ is a fixed integer. In this example, the set $\mathcal{O}(n)$ of $n$-ary operations forms a topological space---it's the space of all labeled configurations of $n$ $k$-dimensional rectangles within the unit $k$-cube. For example, when $k$=2, here's a picture of a point in $\mathcal{O}(5)$.
 
2cube.jpg
 
So $\mathcal{O}(5)$ is the topological space of all such configurations. That is, if we move the rectangle #4 just a little bit, the new picture we get is a new point in the space $\mathcal{O}(5)$. The $\circ_i$ composition is given by insertion of one picture into the $i$th rectangle of the other, then relabeling. For example
comp.jpg
This operad appears** in a seminal paper by topologist Peter May called The Geometry of Iterated Loop Spaces. In short, May answered the question, "Does a topological space have a particular structure if and only if it is (weakly homotopy equivalent to) a $k$-fold loop space?" The answer is
  • Yes! When $k=1$, the structure is that of an algebra over the associahedra oeprad.***
  • Yes! When $k>1$, the structure is that of an algebra over the little $k$-cubes operad.
In other words, an algebra over the little $k$-cubes operad and a $k$-fold loop space are the same in the eyes of a homotopy theorist. So if you're interested in homotopy theory,**** you'll want to get acquainted with the little cubes operad!


The Simplex Operad

Did you know that topological simplices form an operad?
 
simplices.jpg
 
The standard $n-1$-simplex is defined as $$\Delta^{n-1}=\{(p_1,\ldots,p_n)\in \mathbb{R}^{n}:\sum_{i=1}^np_i=1 \text{ and } 0\leq p_i\leq 1\}.$$ And we can think of each point $p=(p_1\ldots,p_n)$ in $\Delta_n:=\Delta^{n-1}$ as a probability distribution on a discrete set $X=\{1,2,\ldots,n\}$. For example, the point $p=(\frac{1}{2},\frac{1}{2})$ in $\Delta_2$ can represent the distribution of a fair coin toss, while $q=(\frac{1}{6},\ldots,\frac{1}{6})$ in $\Delta_6$ might represent the distribution of rolling a six-sided die.
 
coin.jpg
 
What's the composition $\circ_i\colon \Delta_n\times\Delta_m\to\Delta_{n+m-1}$? As an example, suppose $m=6$ and $n=2$ with $p$ and $q$ given as above. To compute $p\circ_2 q$, first multiply each of the entries of $q$ by $\frac{1}{2}$, then stick the result in the second entry of $p$.
 
comp2.jpg
 
Notice, the sum of the entries on the right-hand side add up to 1! So we get a bona fide point in $\Delta_{7}$. More generally, I like to think of $p$ as a $n$-ary tree whose leaves are labeled by the $p_i$. Then $p\circ_i q$ is obtained by "painting" the leaves of $q$ with "$p_i$" and then grafting the result onto the $i$th leaf of $p$. For example, the above composition can be pictured as
 
compcolor.jpg
 
Convex subsets of $\mathbb{R}^n$ are one example of an algebra over this operad, and this plays a very cool role in information theory. In a wonderful 2011 paper, John Baez, Tobias Fritz, and Tom Leinster used the simplex operad to provide a categorical/topological characterization of Shannon entropy. Baez has a nice summary of their work in this blog post, and Leinster outlined their use of the simplex operad in a recent talk at CIRM.

Other Examples


We've only looked a few examples of operads, but there are tons more! There are cyclic operads (think: Frobenius algebras), modular operads (think: moduli spaces), cacti operads (think: string topology), a phylogenetic operad (think: biology), and even a swiss cheese operad. And hey, why stop at operations with only one output? If we consider $n$-to-$m$ operations, we get something called a properad. For example, Riemann surfaces of genus $g$ with $n$ holes for inputs and $m$ holes for outputs form a properad. And an algebra over this properad is a conformal field theory. And we might even consider the disjoint union of such $n$-to-$m$ operations--called a PROP. And algebra over that gadget is a topological quantum field theory. The list goes on!

riemann.jpg
Interested in reading more? Here are a few places to start:

Enjoy!
 

 
*But there is an algebraic analogue! We can view each $K_n$ as a CW complex and consider the cellular chain complex of each. These chain complexes assemble into a new operad which is algebraic in nature---each collection of $n$-ary operations forms a differential graded algebra. This operad is called the $A_\infty$ operad and an algebra over it is an $A_\infty$-algebra. For more on $A_\infty$-algebras, check out Homotopy + Algebra = Operad by Bruno Vallette and Introduction to $A$-infinity Algebras and Modules by Bernhard Keller.

**(Added 12/10/17) Here's some history: May coined the word "operad" in his 1972 Iterated Loops Spaces paper, but the concept originated with Joachim Lambek in his 1969 paper Deductive Systems and Categories II. Lambek used the term 'multicategory' which is a generalization of an operad. Also, the little $n$-cubes first appeared in John Michael Boardman's and Rainer Vogt's 1968 Homotopy-everything $H$-spaces, which is cited by May. They also prove the recognition principle (which is the formal name for what I call "May's question" above), although their proof is a bit different than May's.

Sincerest thanks to Prof. Donald Yau for pointing out these historical remarks!

***There's a sense in which the associahedra operad and the little $1$-cubes operad (a.k.a the little intervals operad) are the same.

****Already doing homotopy-things? Be sure to say hi to the folks over at MathOverflow's homotopy chat room!

IN THIS SERIES
 
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What is an Operad? Part 1

What is an Operad? Part 1

If you browse through the research of your local algebraist, homotopy theorist, algebraic topologist or―well, anyone whose research involves an operation of some type, you might come across the word "operad." But what are operads? And what are they good for? Loosely speaking, operads―which come in a wide variety of types―keep track of various "flavors" of operations.

Read More

The Yoneda Embedding

The Yoneda Embedding

Last week we began a discussion about the Yoneda lemma. Though rather than stating the lemma (sans motivation)we took a leisurely stroll through an implication of its corollaries - the Yoneda perspective, as we called it: An object is completely determined by its relationships to other objects, i.e. by what the object "looks like" from the vantage point of each object
in the category.

But this left us wondering, What are the mathematics behind this idea? And what are the actual corollaries? In this post, we'll work to discover the answers.

Read More

The Yoneda Perspective

The Yoneda Perspective

In the words of Dan Piponi, it "is the hardest trivial thing in mathematics." The nLab catalogues it as "elementary but deep and central," while Emily Riehl nominates it as "arguably the most important result in category theory." Yet as Tom Leinster has pointed out, "many people find it quite bewildering."And what are they referring to?

The Yoneda lemma.

"But," you ask, "what is the Yoneda lemma? And if it's just a lemma then - my gosh - what's the theorem?"

Read More

Dear Autocorrect... (Sincerely, Mathematician)

Dear Autocorrect... (Sincerely, Mathematician)

Dear Autocorrect,

No.

"Topos theory" is not the theory of tops. Or coats or shoes or hats or socks or gloves or slacks or scarves or shorts or skorts or--um, actually, what is topos theory?

Zorn’s lemma” is not a result attributed to corn. Neither boiled corn, grilled corn, frozen corn, fresh corn, canned corn, popped corn, nor unicorns. Though I'm sure one of these is equivalent to the Axiom of Choice.

Read More

Commutative Diagrams Explained

Commutative Diagrams Explained

Have you ever come across the words "commutative diagram" before? Perhaps you've read or heard someone utter a sentence that went something like

"For every [bla bla] there exists
a [yadda yadda] such that
the following diagram commutes."

and perhaps it left you wondering what it all meant.

Read More

Some Notes on Taking Notes

Some Notes on Taking Notes

A quick browse through my Instagram account and you might guess that I take notes. Lots of notes. And you'd be spot on! For this reason, I suppose, I am often asked the question, "How do you do it?!" Now while I don't think my note-taking strategy is particularly special, I am happy to share! I'll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to "stick." So while some people think aloud (or quietly), 

I think on paper.

Read More