Pieter Belmans

Semisimple.jl 1.0.0 is now available!

Mon, 11 May 2026 00:00:00 +0000

This is a cross-post from the HomogeneousTools website, where package-specific updates about Semisimple.jl will also appear.

I will occasionally post about Semisimple.jl here, but if you really want to keep up with development, you should head over to homogeneous.tools.

Note: this package was previously known as Lie.jl and has been renamed to Semisimple.jl to adhere to Julia registry naming conventions.

This post concerns Semisimple.jl, a Julia package for computations with semisimple Lie algebras. There is documentation.

This first stable release focuses on efficient computations with semisimple Lie algebras, of the type that are most useful to an algebraic geometer who wants to do computations for completely reducible vector bundles on partial flag varieties. A large part of that efficiency comes from leaning into Julia's type system and specialization, so fairly high-level code can still run very fast.

Here is a short session:

It is not yet at feature parity with LiE, LieART, or the corresponding functionality in SageMath, but the foundations are now in place. Let us know which features you would like to see implemented next!

HomogeneousTools website

Fri, 08 May 2026 00:00:00 +0000

Together with Javier Fernández Píriz, a PhD student at the University of Luxembourg, we are working on HomogeneousTools, a collection of packages meant to efficiently deal with homogeneous varieties and bundles on them.

For now, our main joint project isn’t live yet (more on that in a few weeks), but

the project’s website is now live
two preliminary packages I created for our main work are also live

More on those already live packages soon!

Moving from Jekyll to Hugo

Tue, 05 May 2026 00:00:00 +0000

Almost ten years ago, I moved from WordPress to Jekyll. It felt like time for another change, so I have now moved the site to Hugo. There was no particular need for it, build times weren't too bad, but it is nice to see how much faster Hugo is in building my website.

The migration is complete, and I have tried to make sure everything still works as intended. Let me know if something did break!

Intersection theory in OSCAR

Tue, 17 Mar 2026 00:00:00 +0000

5 years ago, Jieao Song wrote a really nice intersection theory package in Julia, my new favourite programming language. It is similar to the famous Schubert2 package in Macaulay2, and other, lesser-known packages, like Chow for SageMath. It is extremely efficient (being significantly faster than Schubert2).

One plan that had been around for almost 5 years was to integrate it into OSCAR, the Julia analogue of what SageMath is in Python, so that it becomes part of a stable ecosystem.

I'm happy to say that this integration is now done! This was a joint effort of Wolfram Decker and myself. I want to thank Wolfram for the great help (and patience). Aside from simply integrating the infrastructure, we have added lots of documentation, examples, tests, and some minor features. That said, it cannot be overstated how great the foundation laid by Jieao was. I feel like standing on the shoulder of a giant here, and I hope that this will be a great tool for many people in the future.

It is not yet in the latest version (which is 1.7.0 at the time of writing) of OSCAR, but you can use the development version per these instructions.

If you want to see what it is capable of, see the documentation (which for now still refers to the development branch; in the latest branch it is still the older version), e.g., check out:

the different constructions and properties, or
one of the many examples it can work with.

It is still in the experimental phase, and we are eager to hear your feedback and comments!

Hyperkaehler.info: an update

Mon, 09 Mar 2026 00:00:00 +0000

Last week I updated Hyperkaehler.info a bit, and now it also contains information on:

Especially the latter is really cool, it is a refinement of the Hodge decomposition using the representation theory of a finite-dimensional Lie algebra attached to a hyperkähler variety. Jieao Song created an interactive visualization of this decomposition, and this is now integrated into the website. You can explore it for type K3^[4]. Notice how each type now also has its own canonical URL!

Full disclosure: I have used LLMs (more precisely, Claude Opus and Sonnet, via GitHub Copilot) to implement these additions. What would probably have taken me more than a day of work (including various unrelated fixes) ended up taking only one or two hours, split between some implementation work of my own and a larger amount of reviewing LLM-generated code.

It seems that LLMs have reached sufficient maturity to be useful for these things (but one has to carefully review things! they are basically very good autocomplete machines behaving like slot machines, or vice versa) and I'll be describing some more successful use cases later.

Welcome to the new members of the research group

Fri, 06 Feb 2026 00:00:00 +0000

It is all good to post job announcements, but I've come to realize that it is also good to actually publicly welcome the people who have joined the research group!

I am really glad to say that I'm being joined by

dr. Yuki Mizuno, who is joining me as a postdoc for 3 years, working on noncommutative algebraic geometry
Matthijs Holstege, who is joining me as a PhD student, working on quiver moduli and noncommutative algebraic geometry
Javier Fernández Piriz, who is joining Karin Melnick and me as a PhD student (so joint with the University of Luxembourg), working on partial flag varieties

I'm looking forward to all the interesting mathematics that we will do together!

Postdoc position in algebraic geometry

Thu, 05 Feb 2026 00:00:00 +0000

I am excited to announce a postdoc position in algebraic geometry .

The summary:

starting date is October 1 2026 (or as soon as possible thereafter)
it is for 3 years (with a 1-year probation)
the focus is (derived categories of) Hilbert schemes of points on surfaces, so some experience with these or topics closely related to it is required
it is part of a joint grant between Andreas Krug and myself, and regular visits to Hannover are anticipated
we will start reviewing and interviewing candidates after the application deadline of February 22

There is a really active research group on algebraic geometry in Utrecht, and the Netherlands is a great place in general for algebraic geometry, with many interesting activities.

The position is funded by the NWO, through the NWO+DFG Weave project HilbertMoVieS.

All information on how to apply can be found on the Utrecht University job portal. If you have any questions, please get in touch!

An assistant professorship at Utrecht University

Tue, 03 Feb 2026 00:00:00 +0000

The previous such post is only a few months old, but here is another position!

Utrecht University is hiring an assistant professor in pure mathematics, not tied to specific area. I can attest it is a great place to work, so if you fit the (broad) profile, and would like to work at Utrecht University, please consider applying!

If you have questions, you can ask me, or in case of more official questions, the head of the committee, which is Gijs Heuts.

Events related to Markus Reineke as Springer Chair

Fri, 16 Jan 2026 00:00:00 +0000

This academic year, the Springer Chair (a visiting professorship at Utrecht University) has been given to Markus Reineke. He will be visiting us in February, and in this context we are organizing several events:

a colloquium talk on Tuesday, February 3, from 16 to 17, in Minaert 201
a lecture series on February 4, 9, and 10 (see the webpage for more details)
a problem session on Wednesday, February 11, from 13h15 to 15h00, in HFG 611
a 2-day workshop on February 23 and 24 (see the webpage for more details, note that capacity is very limited)

Everyone is cordially invited to the colloquium, lecture series, and problem session!

Whilst the workshop isn't invitation-only, we could only get a small room, so if you plan to attend it, please get in touch with me at p.belmans@uu.nl.

Change of host for Hyperkaehler.info

Sun, 28 Dec 2025 00:00:00 +0000

I've been hosting all interactive websites on PythonAnywhere, but recently I felt the urge to try and host them on my Synology DS920+. Four years ago, I bought it with the possibility in mind of starting to self-host various things aside from using it as a NAS. After a failed attempt a few weeks ago (despite port forwarding being set up in a seemingly correct fashion, my ISP was blocking communication with the Let's Encrypt servers) I figured out that I could use CloudFlare Tunnels instead.

I have now done this for Hyperkaehler.info, as a little experiment. You shouldn't notice anything different, except that response times might even be a little quicker, and the SSL certificate comes from somewhere else. But in case it does go down for a prolonged period, now you know why.

If this approach works well for the next few months, I'll probably move my other interactive websites over as well (the static ones can simply continue running via GitHub Pages for the time being)