About me
I am a metrology engineer at ASML. I develop algorithms to improve the accuracy of the interferometry systems that ASML produces. I mainly study process-induced damage to the alignment gratings on the wafer, which distort the interferometric readout, and I look for algorithms that attempt to correct for this distortion.
Prior to my work at ASML I was a PhD student in mathematics at the GeoTop centre of the University of Copenhagen, researching applications of homotopy theory to other fields of mathematics. For more on my research in mathematics, see below.
Outside of work I like cooking and toying around with electronics. I enjoy reading about history and religion when I find the time. I’m also an amateur checkers player.
Contact
- E-mail: meer@vivaldi.net
- LinkedIn: jmeer
- PGP: vandermeer.asc
- Telegram: @jw_meer
Please note that if you expect an e-mail from me, it’s not unlikely that my e-mail got caught in your spam folder.
Blog
I’ve made a few writeups on random topics, primarily within mathematics. I hope that they will be of some use to people, in which case I would be happy to hear about it. Feedback and questions are always welcome.
Mathematical research interests
Back in my PhD days, I studied the application of homotopy theory to other branches of mathematics. Historically, homotopy theory was a branch of algebraic topology in which spaces were studied ‘up to continuous deformation’; in modern days, however, homotopy theory has grown out of its native environment into a field of its own, and has proved itself to be spectacularly useful in many other areas of mathematics, such as algebraic geometry, number theory, algebraic K-theory, symplectic geometry, and type theory.
My research mainly concerned itself with the application of homotopy theory to group theory. Specifically, I proved that certain invariants arising from the modular representation theory of finite groups can be reinterpreted in higher-categorical language, after which I was able to invoke powerful homotopical machinery to explicitly compute these invariants.
Below are some papers I wrote during my years as a mathematician. Thanks to the Internet, I deemed publishing papers in so-called ‘professional’ journals to be redundant, and I’m happy to say that I could get away with never publishing a single work.
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Higher-algebraic Picard invariants in modular representation theory.
This PhD thesis consists of three main parts, prefaced by a general introduction.
The first part is based on a paper joint with Richard Wong. We exhibit an $\infty$-categorical decomposition of the stable module category of a general finite group, and we show that, in the case of certain particularly simple finite $p$-groups, this decomposition can be interpreted as an instance of Galois descent. We then use this perspective to produce proof-of-concept calculations of the group of endotrivial modules for these $p$-groups.
In the second part, we move on to computations for more complicated groups. Of particular interest will be the case of the extraspecial groups, which have traditionally played a fundamental role in the theory of endotrivial modules. We analyse the Picard spectral sequence for the extraspecial groups and show that the $E_2$-page inherits a great deal of structure from a certain Tits building of isotropic subspaces with respect to a quadratic form.
In the third and final part, we move on to study the Dade group of endopermutation modules. We investigate how it can be realised as the Picard group of a certain $\infty$-category of genuine equivariant spectra. On our way, we produce a general framework for studying modules whose endomorphisms are trivial up to a specified subcategory of the representation category. This produces invariants that interpolate between the group of endotrivial modules and the Dade group, as well as other more exotic invariants that are of independent interest.
PDF here
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Endotrivial modules for cyclic $p$-groups and generalized quaternion groups via Galois descent.
We use tools from homotopy theory to investigate a certain class of modular representations called endotrivial modules for cyclic $p$-groups and generalized quaternion groups. We show how, for these groups, the classical phenomenon of Quillen stratification can be interpreted as an instance of Galois descent. We then work out the associated homotopy fixed points spectral sequence on Picard spectra, yielding a proof-of-concept homotopical computation of the group of endotrivial modules.
Joint with Richard Wong. arXiv:2107.06308
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A stack-theoretic perspective on $\mathrm{KU}$-local stable homotopy theory.
We study the structure of the $\mathrm{KU}$-local stable homotopy category using the language of algebraic stacks. Specifically, we show that the $\mathrm{KU}$-local stable homotopy category can be described in terms of coherent sheaves on the moduli stack of formal groups.
PDF here