1 -> 2 -> 4 -> 8 -> 16 -> ...

an infinite sequence! (9/n)

0 -> 0 -> 0 -> 0 -> ...

a finite sequence. The points which give a finite sequence are called "torsion points". (10/n)

So here's a question: (11/n)

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OK everyone, buckle up. I have 12 minutes until I go to seminar, and am going to try to tell you in that time about my header photo. (1/n)

The first important thing to know about is elliptic curves. These are objects that mathematicians are interested in because they can carry three really important and different types of structure: (2/n)

(1) geometric structure: every elliptic curve can be written as the set of pairs of complex numbers (x,y) satisfying an equation of the form y^2 = x^3 + ax+b for some fixed complex numbers a and b. (3/n)

This means - among other things - that when you zoom in to an elliptic curve, it just looks like a little patch of the complex plane. (4/n)

(2) algebraic structure: there is a way to define the sum of two points on an elliptic curve which behaves just like you want addition to: there is a unique zero element, every point P has an inverse -P, and other nice features like P + Q = Q + P. (5/n)

(3) arithmetic structure: everything that I said is not just true for curves y^2 = x^3+ax+b with a,b complex numbers, but generally for any type of numbers a and b (appropriately modified in case (1)) - numbers in modular arithmetic, p-adic numbers, functions, and so on. (6/n)

So for example, if a and b are real numbers, then we can study the elliptic curve from the point of view of real numbers *and* the point of view of complex numbers. If integers, we can study the curve from the p-adic point of view and the complex. This is very powerful! (7/n)

Now, since we can add points on any elliptic curve (call it E), we can define a function from E to itself which takes a point P and maps it to P + P. This is just like the function f(x) = 2x in real numbers, except that our new notion of addition gives a new "times 2". (8/n)

If we repeatedly apply this function to a point, sometimes we'll get an infinite sequence of points, and sometimes we won't. Thinking of f(x) = 2x as a function of the real numbers, applying repeatedly to 1, we get

1 -> 2 -> 4 -> 8 -> 16 -> ...

an infinite sequence! (9/n)

1 -> 2 -> 4 -> 8 -> 16 -> ...

an infinite sequence! (9/n)

On the other hand, applying repeatedly to 0 gives

0 -> 0 -> 0 -> 0 -> ...

a finite sequence. The points which give a finite sequence are called "torsion points". (10/n)

0 -> 0 -> 0 -> 0 -> ...

a finite sequence. The points which give a finite sequence are called "torsion points". (10/n)

Understanding how these points - which are algebraically special - interact with the geometry and arithmetic of the curve gives insight into the structure of the curve itself.

So here's a question: (11/n)

So here's a question: (11/n)

if I take two different elliptic curves, say y^2 = x^3 + ax+b and y^2 = x^3 + cx + d, how many points (x,y) are torsion points for both curves? Well, this is actually easy - there's not many, because the curves share very few points at all, without the torsion restriction. (12/n)

But up to choice of sign, y is determined by x on an elliptic curve. So let's ask instead: how many values x have the property that x is the x-coordinate of a torsion point for both elliptic curves? (13/n)

THIS is what my header photo shows! It's taking two different elliptic curves, and putting blue dots at the x-coordinates of torsion points for either. Where two blue dots overlap, we have an x-value which is a torsion x-coordinate for both curves. (14/n)

In the paper just posted: arxiv.org/abs/1901.09945 my coauthors and I prove that there is an upper bound on how much overlap there can be, no matter which two elliptic curves you choose- you can't get an arbitrarily large amount of common torsion for two different curves. (15/15)

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Until we can figure out what's going on??? I can tell you what's going on. The kids have had enough of your Liberal BS. They wear MAGA hats because they are proud of our country. Unlike your sorry, liberal, commie ass.

I have great respect for Native Americans, but this Native American activist has done a world of hurt to Native Americans across the country. What a disgrace he is to himself, his People and this country.

[ yoonmin au ]

Yoongi assumed the little sweet shop on the corner was filled with treats and delights, tasty and addictive enough to bring customers in everyday, but when he finally decides to enter the shop, he finds something a lot sweeter than the candy in the window.

Yoongi assumed the little sweet shop on the corner was filled with treats and delights, tasty and addictive enough to bring customers in everyday, but when he finally decides to enter the shop, he finds something a lot sweeter than the candy in the window.

Hi guys! It’s been a while since I posted an au but I hope you enjoy uwu

Few things:

- this will mainly be fluffy (with PG13 occasionally)

- it will include texts and tweets

- Please don’t reply to the thread since it breaks very easily!

💛💛💛

Few things:

- this will mainly be fluffy (with PG13 occasionally)

- it will include texts and tweets

- Please don’t reply to the thread since it breaks very easily!

💛💛💛

Yoongi was in the middle of a bad week. Or a bad month, if he was honest. Everything that could go wrong in his final year as a Music Production student went wrong within a matter of days.

charlottesville city council tonight is an all-you-can-comment event! unlimited public comment -- just in time for them to do absolutely nothing about anything anyone says!

ok comrades, buckle up. we’re 30 minutes away from everyone’s favorite, most thrilling content: a six hour moment by moment livetweet of a small town city council meeting!

we’re gonna have:

- lots of public comment

- a vote to confirm the new interim city manager

- some spicy discussion about that selection process!

- and a resolution allowing the new interim city manager to declare a state of emergency this weekend in the event of “civil unrest”

- lots of public comment

- a vote to confirm the new interim city manager

- some spicy discussion about that selection process!

- and a resolution allowing the new interim city manager to declare a state of emergency this weekend in the event of “civil unrest”

Ok, everyone, heads up. This is how subtle racism and not-so-subtle colonialism works. 1/

One of our students is a Brit who lived in India for 30 years and never learned Hindi. Also, shortly after she arrived here, in the middle of a community post-lecture coffee, she walked into my office - which I share with a white officemate, 2/

*looked straight at me*, addressed me by name & said, 'Some milk has been spilled, it needs cleaning up.' My office is on the way to the kitchen, which has towels, soap, and water. It didn't occur to her to do it herself, nor did it occur to her to include my white officemate 3/

If I have to, I'll explain. Promise. But the first important thing to know is that this is not a count tonight; it's unofficial and does not include absentee or provisional ballots. The formal count begins Friday, & overseas/military absentees have until next Tues to come in.

Ok, thread time.

1. Almost all the voting in these counties is done on touchscreen machines. Westmoreland County has a mix of touchscreen and optical scan. So a recount won't change things much -- only the Westmoreland paper ballots & absentees/provisionals could.

1. Almost all the voting in these counties is done on touchscreen machines. Westmoreland County has a mix of touchscreen and optical scan. So a recount won't change things much -- only the Westmoreland paper ballots & absentees/provisionals could.

2. Again, to stress: this isn't the official count. This is an informal count. Absentees/provisionals have not been counted yet. Wait until Friday on the former; on the latter voters have a week to verify their eligibility still.

Are you one of those people who's looked back on 2017 and convinced yourself that you haven't achieved anything? Worried about catching up with your mates who seem to have done much better?

Let me tell you a story about #time.

A thread 👇

Let me tell you a story about #time.

A thread 👇

A long #time ago, back when I was about twelve or thirteen years old, my parents travelled to Uyo.

Before leaving, my mother asked me to prepare Edikang ikong soup so she and my father would eat when they returned.

Before leaving, my mother asked me to prepare Edikang ikong soup so she and my father would eat when they returned.

When I was finished with the soup, it was supposed to look something like this:

*photo credit: waiter foodies.

*photo credit: waiter foodies.