Bran tubs are an olden-days thing. They're part of a carnival game - someone fills a tub with bran (the most boring substance on earth) and hides a few treasures in it. Maybe they'll hide a few Apples, or a bag of lollies. You fish your arm around the bran tub and see what kind of treasures you can find.

"Your mind is like a bran tub"

The premise is that your mind is so full of crap, but every now and then you find a real treasure.  

A Japanese art where you smash a piece of pottery and then repair it with gold. Kintsukuroi is about understanding that something can be more beautiful for having been broken.

Chris Martin used the comparison to describe the heart, and opening yourself up to love. Your heart may be broken, but when repaired it will be more wonderful.

There may be a spoiler alert in this chapter. If you're reading The Fault in Our Stars, you might want to stop reading here :)

John Green writes about all lives and moments, no matter how big or small, as a little infinity.

“There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities."

When a character dies, the narrator beautifully speaks...

I want more numbers than I'm likely to get, and God, I want more numbers for [name redacted] than they got. But, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful.”

You gave me forever within the number of days. Effing Incredible.