Explorations in math and programming
David Lowry-Duda

July has been an exciting and busy month for me. I taught number theory 3 hours a day, 5 days a week, for 3 weeks to (mostly) devoted and motivated high school students in the Summer@Brown program. In the middle, I moved to Massachusetts. Immediately after the Summer@Brown program ended, I was given the opportunity to return to ICERM to participate in an experimental program called an IdeaLab.

IdeaLab invited 20 early career mathematicians to come together for a week and to generate ideas on two very different problems: Tipping Points in Climate Systems and Efficient Fully Homomorphic Encryption. Although I plan on writing a bit more about each of these problems and the IdeaLab process in action (at least from my point of view), I should say something about what these are.

Models of Earth's climate are used all the time, to give daily weather reports, to predict and warn about hurricanes, to attempt to understand the effects of anthropogenic sources of carbon on long-term climate. As we know from uncertainty about weather reports, these models aren't perfect. In particular, they don't currently predict sudden, abrupt changes called 'Tippling points.' But are tipping points possible? There have been warm periods following ice-ages in the past, so it seems that there might be tipping points that aren't modelled in the system. Understanding these form the basis for the idea behind the Tipping Points in Climate Systems project. This project also forms another link in Mathematics of Planet Earth.

On the other hand, homomorphic encryption is a topic in modern cryptography. To encrypt a message is to make it hard or impossible for others to read it unless they have a 'key.' You might think that you wouldn't want someone holding onto an encrypted data to be able to do anything with the data, and in most modern encryption algorithms this is the case. But what if we were able to give Google an encrypted dataset and ask them to perform a search on it? Is it possible to have a secure encryption that would allow Google to do some sort of search algorithm and give us the results, but without Google ever understanding the data itself? It may seem far-fetched, but this is exactly the idea behind the Efficient Fully Homomorphic Encryption group. Surprisingly enough, it is possible. But known methods are obnoxiously slow and infeasible. This is why the group was after 'efficient' encryption.

So 20 early career mathematicians from all sorts of areas of mathematics gathered to think about these two questions. For the rest of this post, I'd like to talk about the structure and my thoughts on the IdeaLab process.

20 mathematicians, 5 days, 2 intractable problems. What is to be done? The first day was spent almost entirely on introductions. Each of the 20 participants prepared a 6 minute, 3 slide introduction about his or her own work and background. There was an interesting design plan here: all 20 participants spoke to each other, even though they were going to split into 2+ groups within the next 24 hours. This might be because the 20 people hadn't officially precommitted to one or the other of the two problems. You could still decide. Conceivably, there might be cross-pollination and members from each half could inform each other. But while this sounds like a good idea, it also seems not very likely. The two problems were very disparate: climate models seem to rely very fundamentally on dynamical systems (which I know nothing about, really). Cryptography seems to rely on one-way or trapdoor functions, thus far with a number theory bent.

Then again, the goal of the week was for cross-pollination and spreading ideas. Both tipping points and homomorphic encryption need an influx of new ideas. So perhaps it wasn't a bad idea? What I do know is that the opening introductions did not help people bridge the gap. The introductions were full of jargon, acronyms, and a general assumption of a base level of knowledge that was perhaps true of those who would work on the same problem, but which was beyond the level of those on the other problem. For example, some did work on dynamical systems and gave explicit formulas in terms of massive systems of differential equations with dozens of parameters. An unintended consequence of having a 6-minute, 3-slide intro is that there's not much space to say anything. So to fit an accurate representation of your thesis topic (which many did), you have to be fast and dense - not conducive to general understanding.

I don't mean to say the number theorists were any better. On our part, there were Dirichlet series and L-functions being acted on with Hecke operators, all without definitions given due to time/space/planning consequences. As a result, I think that the introductions largely served to grow the divide between the two problem groups. I didn't anticipate this beforehand, but I think the next IdeaLab should really think about this. Either split up the groups earlier so that there is more conversation sooner, or somehow advertise that the introductions need to be aimed at a lower audience.

After the introductions, the organizers (relative experts on the two problems) gave an overview of the problems and the current level of progress towards their solution. The group hadn't split yet, and I must say I enjoyed learning about problems facing modelers of climate systems and the potential for tipping points. This presentation was given by Christopher Jones, and he instilled within me sufficient interest for me to have at least kept tabs on the climate group's progress over the week (although I did not contribute). I also enjoyed Henry Cohn's explanation of homomorphic encryption, but I was already familiar with the material: the dramatic power help less sway over me. Perhaps this is how the applied mathematicians felt about the climate talk?

This was the end of the first day, but by the following morning the participants needed to decide which group they were going to attend since almost every subsequent meeting was split. I came to work on homomorphic encryption as a student of Jeff Hoffstein, one of the organizers of the cryptography portion (and the reason why me, as a grad student, felt comfortable enough to attend a conference/workshop aimed at early-career mathematicians), so I didn't need to make a decision. But I don't think anyone was actually on the fence. The disciplines were very separate.

The second day started with more talks given by the experts on the subject. The material was more pointed, more technical. Unlike the first day, I learned a lot very quickly. Or perhaps I should say that many things I didn't know were quickly taught to me, and I picked up many bits and pieces. The goal,'' they continually reminded us,is not to completely solve fully homomorphic encryption. That's not a reasonable goal for a week. The goal is to come up with new ideas, to reinvigorate, inform, or create new connections to the field.'' Then afternoon came, and the experts were about to set us loose.

This was a very exciting time. Ten of us are sitting together, none of us cryptographers, suddenly to be set loose. I don't know what we would have done, but someone had the foresight to have us all sit and just brainstorm a large list of potential ideas/areas/hard problems to explore or investigate. I think this stage was essential: it gave us some sort of direction. This is my second big piece of advice to future IdeaLabs: keep up this sort of brainstorming session. After we had a good sized list and nothing else struck us, we split into two smaller groups of roughly five each to pursue subsets of the ideas. The climate group did roughly the same, except that three subgroups formed (I think - I wasn't there, so I won't really say much more about their process throughout the week).

What this meant for the cryptography group is that the five of us with a number theory background formed a subgroup, and those with a graph theory/logic/probability background formed another. From this moment on, we were more or less left to our own devices by the experts. This was intentional - the idea is that if they were around, the temptation to look to them for guidance would likely lead group progress towards already-established ideas in the field. And why would you want that?

We worked a lot harder and on a lot more different directions than I had anticipated. We toyed with tropical geometry, Hecke algebras, lattices, rings, etc. We bounced ideas off each other, and often split into subsubgroups that varied in composition and goal. I liked the fluidity, but I don't know if we were too unfocused. Perhaps we were too 'free' in accepting that we weren't going to solve the problem? We certainly generated a lot of ideas, albeit not wholly fleshed out ideas.

One thing I really enjoyed was seeing how others worked and were informed by their backgrounds. For the second time, I attended an ICERM program as the most junior mathematician around. It's very inspiring to be around more experienced mathematicians. Some tried to modify existing protocols. Some sought to find new 'hard problems' to create new trapdoor functions. Some were excited to jump into areas of math far from their own discipline because it seemed fun or interesting. The graph theorists had some really interesting ideas that are completely different from modern cryptography (as far as I can tell). It wouldn't surprise me if there were some really nice deep material in what they presented. All in all, it was exciting and rewarding.

All too soon, Friday came, and we presented our ideas to all 20 participants and a panel of visitors from institutions like the NSF. I didn't expect the time to go by so soon. The visitors and experts had many pointed and well-thought questions. I was particularly impressed with one of the climate presentations and the question and answer session between them and the panel. In a later post, I'll talk more about that.

Finally, the panel told us about grants, grant sources, institutes of collaborative mathematics (like ICERM), and so on. It was a good experience, and I would absolutely recommend it to others. Perhaps more importantly, I'm still thinking about some of the things I worked on at IdeaLab. I have two encryption schemes in particular that I really like and that seem interesting but different to current schemes. I'm also very happy to know the people I met. We run in tiny circles like hamsters sometimes, and I'm sure that we'll cross paths again. If nothing else, I think this program allowed me to expand my collaborative sphere. I'm really tempted to delve deeper into cryptography too. There seems to be room for more mathematicians, as opposed to computer scientists and programmers, in the field.

I should also mention that another member from my subgroup, Adriana Salerno, has written a post to her blog PhD+epsilon about her IdeaLab experience too. You should read it to get another perspective!

There's a cute story here too. During the first working day (Tuesday), Adriana was the first person daring enough to go up to the white board and just start going down idea paths. (There's really nothing to be afraid of, but she was the first to overcome the hesitation). She filled up the whole board, and I got an idea and went up to the board. One minute later, while I'm explaining what I was thinking about, the ICERM staff takes a picture and posts it to their site.

An action shot of math in progress!

As a final note, I'd like to mention one last thing I would add to the IdeaLab experience. There were many talks and many slides, and the final presentations were done beamer-style in slides. But they aren't available (or at least, not yet). But why not? Shouldn't the ideas from the IdeaLab be made available? Nonetheless, I had a great time, I learned a lot, and I'd recommend IdeaLabs to everyone.

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