Current Projects

• Well, I am working on it rather than writing about it here. Sorry. Talk to me in person ;-)

Projects I've done in the past

• Programming: ArticleChurner, a filter for articles
• Programming/Kids: From Bambam to Lisp (or: teach your toddler human-computer interaction)
• For Math: a tutorial why and how mathematicians should use (distributed) version control systems such as git.
• Math: My PhD project (under supervision of Matthias Wendt): Motivic Cell Structures for Projective Spaces over split Quaternions.
• Math: german diploma thesis (on Matsumoto's theorem in A¹-homotopy theory) under supervision of Matthias Wendt and Annette Huber, in Freiburg.
• Math (with Matthias Wendt) generalizing from my diploma thesis, we wrote an article: A¹-fundamental groups of isotropic groups (accepted for publication in CRAS).
• Programming (with Reimer Backhaus): Wesen: Learn Python via AI Programming
• Math workshops organized: Computer Tools in Pure Math and univalent foundations (in Freiburg); Topology and Big Data (with Anja Wittmann, in Freiburg)
• Math seminars organized: Rational Homotopy Theory (with Jan Weidner) and Homogeneous Spaces, with GK1821 in Freiburg. Grad student seminar on Motives, in Freiburg.
• Blog: I once kept a list of resources for math PhD students (no longer maintained) which might still be useful.
• Blog: I wrote some (now outdated) tutorials on how to organize various kinds of information and OCR.
• Programming (for Max Voelkel): Google Web Toolkit implementation of (parts of) the Hypertext Knowledge Workbench, precursor of Max' Denkwerkzeug.

Projects in Preparation

... in preparation (what else did you expect?)

Projects in Preparation

• Math/Fun with F_un (with Reimer Backhaus): In incidence geometry, one has some axioms to exclude bad cases. One of these cases shouldn't be excluded in our opinion. This leads to some nice observations which make the theory more coherent (for lack of a better word). Maybe this will become an expository preprint, maybe a blogpost - most of what we figured out should be well-known to the experts, and then there are some nice ideas on relating incidence geometry with scheme theory over F_un - whatever that is.
• Math: H-space structures on motivic spheres. There are two approaches, one by using geometric realizations (very, very easy) and then the classical results, another by re-doing classical proofs in algebraic K-Theory (Aravind Asok assured that this goes through without any change - which is why no one wrote it up yet). I consider this a nice exercise in K-theory operations, so it could be worth writing up (for me) nevertheless. Questions around this would be how the classical related results on division algebras and parallelizability translate. To give a stupid example, it would give a proof of the dimension bound on composition algebras over real closed fields without appealing to Tarski's elementary equivalence theorem (but that's not surprising).
• Math: A geometric proof of the Quillen-Suslin property for complements of hyperplane arrangements. Sadly, someone else was faster in publishing the theorem, and the proof is essentially the same as mine, written in algebra rather than geometry. So this will probably result in a blogpost rather than a paper.
• Fun: The Fair Trade Theorems Movement, Label and LaTeX package; raising awareness for the problems of drinking lots of coffee without spending lots of money.
• Programming/Math (with Daniel Harrer): A database of counterexamples for algebraic geometry, using James Dabbs' wonderful pi-base software. This can be thought of as expansion of my cheat sheet on algebraic geometry.
• Math/Data/Programming (with Felix Wellen): Try some topological clustering algorithms on datasets about us (by which I mean Mathematicians), like where funding money goes, citation graphs, PhD family trees, etc. Use persistent homology to get an understanding of the topology before one blindly clusters. First sub-goal: see how to improve the MSC scheme by figuring out a "natural" clustering of mathematical subjects.
• Math: Comparison of Categories of (Artin-)Tate Motives (say, over a perfect field or the integers, with rational coeff's or not ... whatever is possible): Voevodsky's DM, Nori's MM, various versions of DTM, MTM and D(MTM). This would probably help to learn more about the various categories of motives and their comparison and yield an expository article I wish I had when I learned this stuff.
• Math: Hopf monoids in categories of Motives - with a view towards getting clean proofs of motivic decompositions of reductive groups, their classifying and loop spaces, with rational coefficients.

Some Vague Project Ideas

These are ideas and drafts for projects that have been on my mind for some time. If someone wants to collaborate on these or something similar, don't hesitate to contact me. I'd be happy about very minor contributions (such as criticism), too.

• Math: What is the motivic homotopy type of things classically called algebraic loop spaces (like loop groups and Kapranov-Vasserot loop varieties)? For abelian groups, the loop group looks like it is the motivic free $\mathbb{G}_m$-loop space. For non-abelian groups, it would be interesting to figure out what the differences are, if any.
• Math: Once upon a time I thought I had found a proof of an algebraic version (and the usual version) of Ishida's theorem on Bott towers - until I found a mistake in a computation. Now I think one could rewrite that mistake into an obstruction class, but that requires some more work.
• Math: What is known about unstable exceptional K-Theory (the homotopy groups of $BG^+(R)$ for $G$ an exceptional group)? Do they give interesting invariants?
• Math: The $(\infty,1)$-category of motivic spaces is not an $(\infty,1)$-topos (as is well known by now), but there are still several univalent families. Is this a defect or actually a good thing? Can one "repair" this defect and how would a "repaired" version of motivic spaces look like? Better or worse? More accessible or completely irrelevant? I expect the "repaired" version to be badly behaved, but that would already be a result I'd like to understand precisely. A very much related question: Can there be a motivic homotopy type theory, where we capture the yoga of motives (such as the six functors) in operations of motivic homotopy types? My guess on this question is an "almost yes", and I'd like to know to which extent this fails, and why. If it doesn't fail it would be spectacular, of course.
• Programming/Math: A code golf variant of this blog post; A website where you can submit code (in Haskell?) to compute an integer sequence, associated to a sequence in the OEIS. The shortest submitted code wins, and also gives an estimate for the Kolmogorov complexity of the sequence, ultimately resulting in an approximation to the order of all integer sequences given by the computational complexity.
• Programming/Math: 4D Visualization of (algebro-)geometric objects
• Math/Fun: Turing Motives - a brand name that deserves to be filled with beautiful mathematics. I imagine to equip the category of terminating Turing machines (or an equivalent computational model) with enough extra structure that it becomes Tannakian, so that we can talk about its Galois/fundamental group, which will for example automatically contain the motivic Galois group, as periods (which are a torsor under the motivic Galois group) are certain computable numbers. The large Galois group of this category I'd like to call Turing Motives can be seen as ultimate symmetry group of (terminating) computation, thus related to physics from the point of view of theory of science. I'm not sure any results actually come out of this, but I like the philosophical idea, and hope to find time to write this up some day.
• Writing/Programming: a tutorial on how to use wikis for mathematics, along a gentle tutorial for installing Instiki (maybe publishing the changes I made to my Instiki wiki, mostly of cosmetic nature to better suit its use as a PKM - personal knowledge management - tool); mostly an update to my first attempt.
• Writing/Econ: Basic Income Experiment Path
• Writing: What if every kid would learn to code?
• Programming: Side-by-side comparison module
• Math: An audiobook on higher mathematics, conveying the philosophy of some stuff like Motives, Derived Algebraic Geometry, Homotopical Algebra, etc. Without anything written, one has to explain things differently, which can add to the available resources. It would also be a bit like a conversation at a conference meal, where you'd head off to pen and paper afterwards. I did some experiments around basic category theory, and that is already pretty hard.
• Math: Database of motivic homotopy types, motivic decompositions and related invariants (to be honest, I'm writing up examples for my thesis anyway)
• Math: Visualizing branched covers over the real numbers (with applications to visualizations in Z/2-equivariant and motivic homotopy theory).
• Math/Writing/Programming: Textbook 2.0 in Linear Algebra (more of a dream about the many uses of sotware to replace the classical notion of a book than a project, so this would be about experimenting around).

I'd be interested to learn more about: Geometric Complexity Theory, Perverse Motives, Operadic recognition principles for $\mathbb{G}_m$ or $\mathbb{P}^1$-loop spaces in motivic homotopy theory and configuration spaces, Somekawa K-groups, Rational Motivic Homotopy Theory, Derived Algebraic Geometry, Witt Motives and hermitian K-Theory, ...

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