The band that wouldn’t stop playing

I had the pleasure Friday night of attending the dramatic presentation by Colin Adams and the Moubiusbandaid Players (hence the title, which I appropriated from a chapter in The Heart of Mathematics, by Ed Burger and Mike Starbird). As usual, we were treated to a few short plays or sketches which were featured plenty of math and humor. The room was packed, as expected, and the mood was light and fun. I think it’s great that this is held on the last night of the meetings (Saturday is usually a half-day, and the Convention Center has a ghost town feel). Instead of a long description which will do no justice to Adams’ genius writing, I will share some pictures I took of the different pieces.

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Lattices and secrets

Jill Pipher

Jill Pipher

Jill Pipher gave a great lecture Friday on lattice-based cryptography. As is common for crypto talks, Pipher described a bit about the theory and history behind Public Key Cryptography. In particular, PKC relies on being able to find good one-way functions, that is, something that is easy to compute but hard to reverse, unless one has some additional information, known as the “trapdoor”. Most of these have come from ideas in  number theory, like factorization and the discrete log problem.

Lattice-based cryptography is based on two computational problems related to lattices: the closest vector problem and the shortest vector problem. The shortest vector problem (or SVP) is, given a lattice, to find the shortest vector in that lattice. This problem is solvable by using the LLL algorithm, but only for low dimensions.

After some attempts to use these problems by various talented mathematicians, Pipher together with Jeff Hoffstein and Joseph Silverman developed NTRU, in which the hard problem (one-way function) can be reduced to an SVP in a class of lattices with a lot of symmetry. Pipher then told us how this was first presented in the Crypto 96 rump session, subsequently submitted to Crypto 97 (and rejected!) and was finally published in ANTS (Algorithmic Number Theory Symposium) 98. It was fully accepted as an IEEE standard in 2008. According to the NIST, lattice-based constructions are most resistant to quantum computer attacks (2009). Finally in 2011 it was accepted for use in financial services.

Pipher then proceeded to explain the details behind NTRU. I will not go through them here but there are several places you can go if interested, such as Pipher’s book with Hoffstein and Silverman “An Introduction to Mathematical Cryptography”. It all boils down to creating a lattice for which we know a good basis and a bad one, and the former becomes our private key and the latter our public key (this is not quite accurate because they do everything in terms of polynomials, which give rise to a lattice).

At the end, Pipher talked again about the hottest topic in crypto at the moment, that of fully homomorphic encryption, which I wrote about after I attended Alice Silverberg’s talk.

Pipher joked at the beginning that we were going to learn a lot of acronyms during the talk, and boy she wasn’t kidding!

My Mathematics – Poetry by Karen Morgan Ivy

I had the opportunity to attend a very small part of a poetry reading. There were several people who either read poetry related to mathematics from other sources or read their own poetry for us. The result was a beautiful montage of mathematical self-expression. One of my favorites was Karen Morgan Ivy’s “My Mathematics.” She is a professor of mathematics at New Jersey City University.

Although I didn’t record her recitation of this wonderful poem, you can see the lyrics below [1] and view this YouTube video of her profile. Hopefully we can get a full video of her recitation of the poem.

What are your thoughts on the poem?

My Mathematics
Karen Morgan Ivy

When I want to reach you
I always know that I can speak to you
through my mathematics
because you see
my mathematics is the closest
anyone can ever get to really knowing me

I can’t bare my soul specically
so I paint its reection whimsically
onto your walls
brush-stroking
heart-provoking codes
fifty-five digits at a time
composing cadences of inordinate rhyme

I said I can’t bare my soul specically
so I tattoo its scent onto your olfactory nerves
welcoming the cohesion of realism and abstraction
enclosed in elliptic curves
because you see
my mathematics is the closest
anyone can ever get to really knowing me

no, my mathematics isn’t always the greatest translation
but think of it like emancipation
like a civil right
a “we the people” fight
understanding that the enlightenment
only manifests and beckons the call
when there is liberty and justice for all

my mathematics isn’t always my feelings
but it is the way that I bleed
so I seed
it and hope it sprouts in your mind
because sometimes the war crimes
that we have committed
against our spirits
refuse to be fitted
casually into conversation

I gather them and bury them
beneath my most artistic-mathematical creations
in hopes that your fascination
with deciphering things leads you there
and everywhere
that you encounter a mathematical sentence
accept that as my personal defense
that with hating mathematics comes repentance

my mathematics is one of my favorite things
one of my best things
the way through which I hotwire
the world and if you inquire
too much beyond that
you might miss the fact
that I am not really insane
I simply feign
for mathematics

you see…I just remain that poetic feminine energy
digging mathematical synergy
hello my name is Karen
and I am a mathoholic
I dwell among the suspended
symbolic images of
iota mu epsilon
revealing the real I mu E a.k.a. me
who resurrected from the remains of pantheons

the real I mu E a.k.a. me
who sometimes hesitates to speak
but never keeps silent
because my mathematics is brave
and shouts about revolution
all while my fingers slave
as a matter of fact
I’m writing this piece in between keystrokes
while my thoughts are making minimum wage

thoughts linked to my mathematics
inked onto pages and pages
sonnets, theorems, haikus, and equations
solidifying conscious thought-invasions
I keep plenty of ammunition in my arsenal
free verse words as meaningful and powerful as they are artful
peer through the stained glass of my temple
and you will see
that I am nothing and yet everything
the manifestation of all that my mother dreamed

in keeping with my mathematical state of mind
I offer my mathematics in hopes you will discover
that speaking mathematics
doesn’t require living and loving undercover
and this becomes the way that i reach you
when I want to speak to you
I always know that I can speak to you
through my mathematics
because you see
my mathematics is the closest
anyone can ever get to really knowing me

[1] Morgan Ivy, Karen (2013) “my mathematics,” Journal of Humanistic Mathematics: Vol. 3: Iss. 2, pages 144-146.
Available at: http://scholarship.claremont.edu/jhm/vol3/iss2/12

The poster session at the end of the earth

Ursula Whitcher and myself, presenting a poster.

Ursula and myself, presenting a poster.

On Day 2, I presented a poster at the YMN/Project NExT poster session with my collaborator Ursula Whitcher. The Young Mathematicians Network is a great online community of young mathematicians (from undergraduates to professors in early stages of their careers), and Project NExT is a professional development program focused on helping early career mathematicians. So I was very surprised when: 1) there were not a lot of poster presenters, and 2) not a lot of people went to see the posters. I guess #2 is not so surprising given that the exhibit hall for poster sessions was in a very remote location, and this is probably my main gripe. It is not a poster session you would accidentally stumble upon, and I feel in previous years they have had posters on hallways or exhibit halls and it’s much easier to wander across them. This is the same hall in which the huge and well-attended undergrad poster session was held, too, but this is one that is well-advertised and has tons of submissions. Anyway, the poster session was well-organized, but perhaps not well-advertised, and since it involves young mathematicians, I suspect part of the lack of submission had to do with the fact that the meetings were late enough this year that most people have started teaching. One wouldn’t go to the meetings only to present a poster, so unless you’re going for other reasons (like myself), you wouldn’t even think of signing up.

I am happy I did this because it gave me a couple of hours to chat with my collaborator in front of our poster, and in that sense restarted our thinking about this problem. We did have a couple of visitors and chatted with them, although I did get yelled at by someone from the NSA for asking if he knew someone at the NSA. Apparently I’m not supposed to do that.

Still, I think this could be a successful poster session if it was better advertised, and better located.

The Public Face of Mathematics

Epic and wonderful. That is how I would describe yesterday’s AMS panel discussion “The Public Face of Math,” moderated by Arthur Benjamin. They put together a top-flight panel of today’s most relevant math communicators.

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Starting from the far left, we have Keith Devlin, a professor at Stanford and NPR’s resident “Math Guy” — you can hear his math stories on Weekend Edition. Next up, Steven Strogatz, a professor at Cornell, and author of many great book about math and lots of great articles about math for the New York Times (if you haven’t read it yet, you should). Next, Cathy O’Neil, of mathbabe.org, a truly original blogger and important voice in the conversation about our ethical responsibilities in the emerging world of data science. Next we have Tom Siegfried, a freelance science journalist. And a surprise guest to the panel, Congressman Jerry McNerney, who I believe is the only person in the house of representatives with a PhD in math. Then on the far right is our moderator, Arthur Benjamin, a professor at Harvey-Mudd, and America’s favorite mathemagician.

The panel addressed lots of important issues, like the role of social media in bringing math to the public. Steven Strogatz was strongly in favor, and made particular note of the connections he’s been able to make with K-12 teaching using Twitter. Tom Siegfried was a little more hesitant, suggesting that social media could have some benefits, but was perhaps not the best use of time. Cathy O’Neil encouraged anyone with something to say to write a blog — it’s free, it’s easy, and did you know that you can embed LaTeX in WordPress? (I did not!)

When asked about the role of civic engagement in the promotion of mathematics, Congressman McNerney urged academics to get to know their representatives, pay them a visit, and be nice to them. He also suggested finding a billionaire, and starting a super PAC with the sole mission of stacking congress with math and science types.

Artists at JMM 2014

I always love going through the exhibit hall to see what kinds of goodies there are. This year, there have been some really great exhibits. My favorite though has been the art exhibit. If you have not had a chance to see it, I highly recommend checking it out. A list of pieces and descriptions of them can be found at http://gallery.bridgesmathart.org/exhibitions/2014-joint-mathematics-meetings. Below are some photos I took of some of the pieces.

I had the pleasure of meeting with Shanthi Chandrasekar and was able to discuss some of her artwork. Her work called “Kolam” consists of a 93×93 grid of “dots” with a continuous loop around each dot. The art form comes from Indian tradition and “symbolize the scientific and philosophical patterns innate to and infinite throughout the cosmos.”

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Friday Morning Math Fun

Today has been quite a busy day. If only I had the ability to duplicate myself, I might have the chance to attend everything that interests me. My morning started by representing the AMS Graduate Student Blog at the Graduate Fair.

Afterwards, Ron Graham gave a talk about “Paul Erdös and Egyptian Fractions.” It was my first time hearing Graham speak and it was truly a delight. He informed the audience that every good talk should have one proof, one joke, and the two should not be the same. The main topic of his talk involved Erdös’ contributions to research regarding Egyptian fractions. Egyptian Fractions involve the idea of writing rational numbers as the sum of unit fractions.

I also had the pleasure of attending two talks by Gil Strang. The first talk discussed a neat property of matrices. An example he gave was   A*A=\begin{pmatrix} 1 & 2\\ 3 & 4\end{pmatrix} \begin{pmatrix} 1 & 2\\ 3 & 4\end{pmatrix}=\begin{pmatrix} 7 & 10\\ 15 & 22\end{pmatrix}=B. Notice that the ratio  \dfrac{a_{12}}{a_{21}}=\dfrac{2}{3} .  Furthermore, notice that \dfrac{b_{12}}{b_{21}}=\dfrac{10}{15}=\dfrac{2}{3}. This nice pattern continues for A^3,A^4, \ldots. It is natural to wonder if this works for matrices of higher dimensions. It does not always.

Reverse Mathematics on Day 3

One of my favorite things about the JMM is that is feels like a family reunion! Running into former professors from my undergrad days, old grad school office-mates, collaborators, and all of the conference buddies that I’ve made over the years — it starts to feel like the people you know at the JMM might actually be a set of non-trivial density!

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I found two of my favorite conference buddies Aly Deines and Lola Thompson at the Sage booth.

megan and bonita

…and two of old grad school accomplices, Bonita Graham and Megan Heenehan enjoying art in the exhibit hall.

It’s fun to see who is on the job market (definitely wearing a tie), who is giving a talk (likely wearing some kind of shirt with buttons on it), and who is just kicking it (probably wearing a hoodie).

But sometimes you go to a packed talk so far out of your area that you look around the room and realize you don’t recognize a soul, and you are reminded of what a gigantic conference this is.

For example, this morning I went the Association for Symbolic Logic invited Address “New Directions in Reverse Mathematics,” by Damir Dzhafarov. Reverse mathematics (which I have to admit I’d never heard of!) asks the following question: which axioms do we really need to prove a given theorem? The talk was great, and Dzhafarov explained all of the new ideas that currently evolving about Ramsey’s Theorem using the techniques of reverse mathematics.

After this talk I got back into my comfort zone, and hit up the MAA session about the influence of Paul Erdős on number theory. Erdősian (is that a word?) number theory has been a big focus of this JMM since this year marks Erdős’ centennial. Florian Luca gave a great talk about counting sets of special integers, like odds, or primes, or ones that arise as image sets of a function. He gave a great talk and furthered solidified in my mind the tremendous influence of Erdős on our current understanding of number theory.

Effective thinking and mathematics

Mike Starbird.

Mike Starbird

“Most people like to talk about how in college we need to develop critical thinking skills”, said Mike Starbird near the beginning of this talk yesterday, “but really, who wants to hear “Oh, yeah, Soandso, he’s really critical”?”. This, Starbird says, is what led him and coauthor Ed Burger to coin the phrase “effective thinking”. Because that is something one would like to be called.

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Mind The Gap

Today was a big day for analytic number theorists, and by extension, everyone who loves math. During today’s AMS Special Session on Analytic Number Theory, II, James Maynard and Yitang Zang delivered a solid one-two punch.

Both Maynard and Zhang have made significant progress towards a resolution of the twin prime conjecture, which states that there are infinitely many pairs of primes separated by 1 number (like 5,7 and 11,13; not like 2,3).

For those that haven’t been glued to the Arxiv for the last 6 months, Yitang Zhang appeared from out of nowhere last spring and proved there are infinitely many primes separated by 70,000,000. Then more recently, in November, completely independently, James Maynard brought the gap down to 600. And as of last week, the Polymath Project brought it down to 270.

This is extremely rapid for a field whose problems usually sit on the shelf for 200 years! But back to todays at the JMM.

First up, James Maynard wowed us with an explanation of his recent work bounding small gaps between primes using the GPY method.

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James Maynard

After the first round of extremely long (And well deserved!) applause died down, the room immediately started buzzing. People piled into the room, shuffling turned to whispers turned into talking, whispers to talking, until eventually Andrew Granville shouted “QUIET!!” That worked.

The room was packed, there was a buzz of excitement in the air, and then Yitang Zhang took the stage. His opening line:

I was going to talk about twin primes, but a few months ago I decided to talk about something else.

Looks like Zhang, hot on the heels of his great discovery, has set his sights on something new.

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Yitang Zhang

All in all, an exciting day in Number Theory!