A fractal spectrum of tales
- Riemann rearrangement theorem in the browser
One of the first exercises when studying convergence of infinite series asks to show that $\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent while $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ is convergent. In more technical terms, $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ is a conditionally convergent series.
This is in contrast to absolutely convergent series, that is, infinite sums $\sum_{n=1}^{\infty} a_n$ such that $\sum_{n=1}^{\infty} |a_n|$ is also convergent.
What is peculiar about conditionally convergent series, is that it is possible to rearrange the terms in the sequence so that the sum of the rearrangement converges to any arbitrary number. This is called Riemann’s rearrangement theorem.
2 min read - Dec 14, 2024 - Running LLMs Locally With Ollama
In my previous post, I explored various ways to run Large Language Models locally. Since that post, I have been pointed to try another powerful tool for this purpose: Ollama. This open-source project makes it incredibly easy to run LLMs on your local machine, offering a great balance between simplicity and flexibility.
While it does not seem as flexible as the
llmpython library I presented in the other post and it can scare some users with its command-line interface, I was impressed by its ease of use and the wide range of models it supports. I am not overselling this, its simplicity is staggering: you can get started with just a few commands.
4 min read - Jul 26, 2024 - Running LLMs locally
Large Language Models (LLMs) are powerful tools for generating human-like text responses. You might be familiar with them through services like ChatGPT, Anthropic Claude, Google Gemini, and Perplexity AI, Nowadays people are using them for editing purposes, writing, brainstorming, and even for generating code snippets. When used responsibly and critically as a tool to assist human creativity, they can be very helpful.
Recently, I spent some time playing with these models and I found them fascinating. However, due to privacy concerns and their high environmental costs, I don’t feel comfortable using cloud-based services. This post is an account of my experience with running LLMs locally on my machines. This can be quite straightforward, and if you have 8-16GB of RAM and a decent GPU, you can run these models on your own computer without significant issues.
7 min read - Jul 24, 2024 - A brief introduction to git, with an eye towards mathematicians
This is a followup to my previous post on git from 11 years ago. I’ve been using git for a while now and I’ve learned a few things since then, that I think are worth sharing.
I am not going to explain in detail what git is, how to install it and how it works; there are plenty of resources for that. It will be enough for our purposes to know that
gitis a version control system that allows you to keep track of changes in your files and collaborate with others.
12 min read - Feb 29, 2024 - Giving Good Maths Talks
The upcoming block I have to coordinate a course on giving good talks. This also meant looking for good extra material for the students to reflect on what it means to give a good talk, what to be aware of and what tips can help to do a good job. Here is a collection of the resources I liked so far:
- MAA has a curated list of resources on giving mathematics talks which includes plenty of examples of good mathematics presentations
- There is an essay by John McCarthy on how to give a good colloquium talk with a response blog post on how to give a good 20min talk
- On the blog of the AMS there is a nice post on preparing a conference presentation, from what it means to many resources on how it can be done
- What characterizes a good presentation? There is a nice summary from an analogous course at MIT
- Terence Tao has a famous blog post on giving talks
- Jordan Ellenberg has a post with tips on giving talks that links an interesting pdf on the same topic
- A nice essay on giving talks by Bryna Kra
Let me know on twitter or mastodon (links on the left bar) if you have other suggestions!
1 min read - Apr 7, 2023 - Panel discussion on the Future of Higher Education
Recently I was invited to participate to a panel discussion as part of a symposium on the future of higher education hosted by our faculty for the launch of the Centre for Learning and Teaching.
I joined other four incredible panelists and we have been given in advance five quite broad (and though) questions on some important aspects of (high) education. I want report here my contribution, with some extra commentary at the end.
6 min read - Mar 27, 2023 - So You Are Moving to Mastodon
After Elon Musk announced its interest in acquiring twitter and even more now that he did it, there has been a growing influx of people to Mastodon, a decentralized social media platform inspired by twitter but in some sense further influenced by forums and emails.
There are a few tools and suggestions that have been shared around recently, and I thought of blogging them here to save them. The culture on Mastodon is quite different and more thoughtful than the one in Twitter. It takes a bit of getting used to it and what to do or avoid. Overall it makes it a much more pleasant experience (in my opinion).
2 min read - Nov 4, 2022 - The Man Who Saved Geometry
The other day I found an old file with links to interesting videos. Some of them were no longer available, some I still have to check, but one caught my attention and the other evening I decided to watch it.
It is a little documentary on the last conference of Donald Coexeter, and I enjoed watching it very much. You can find it embedded below, and linked above. And while collecting links for this post I have also discovered a nice writeup.
1 min read - Jan 12, 2022 - Research Minutes
Since September I am a member of the Young Academy Groningen. One of the nice perks of being part of such incredible group of people (yes, I still wonder how I got there :P) was the opportunity to take a short course on dealing with the media and to record my “research minutes”, a short, 2 minutes long, video to popularize my work.
The video was published on Twitter some time ago, and is now available on youtube. I hope you will enjoy it as much as I did enjoy recording it :)
1 min read - Nov 11, 2021 - A Micro Course on Sub-Riemannian Geometry
Last year I had the opportunity to give the first talk at the The Dutch Differential Topology & Geometry seminar (DDT&G). This was a 3 hour long event in which I had the chance to give a high-level introduction to sub-Riemannian geometry and present some interesting results and open questions partly related to my research.
In the first part of the seminar I introduced what are sub-Riemannian structures, where do they come from and what are some of the major properties and open questions, using some famous examples to drive the discussion. For this part of the talk I assumed basic knowledge of Riemannian geometry and calculus of variation and took time to provide a reading list for the interested reader.
2 min read - Oct 21, 2021