# Making plots of modular forms¶

Inspired by the images and ideas of Elias Wegert, I thought it might be interesting to attempt to implement a version of his colorizing technique for complex functions in sage. The purpose is ultimately to revisit how one plots modular forms in the LMFDB (see lmfdb.org and click around to see various plots — some are good, others are less good).

The challenge is that plotting a function from $\mathbb{C} \longrightarrow \mathbb{C}$ is that the graph is naturally 4-dimensional, and we are very bad at visualizing 4d things. In fact, we want to use only 2d to visualize it.

A complex number $z = re^{i \theta}$ is determined by the magnitude ($r$) and the argument ($\theta$). Thus
one typical approach to represent the value taken by a function $f$ at a point $z$ is to represent the magnitude of $f(z)$ in terms of the brightness, and to represent the argument in terms of color.

For example, the typical complex space would then look like the following.

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## Twenty Mathematicians, Two Hard Problems, One Week, IdeaLab2013

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. In later posts, I’ll talk about each of the two major topics and what sorts of ideas came out of the process.