# **Project proposals for OSCAR Coding Sprint 2026**

Dates: 23/02/2026 - 27/02/2026
Place: TU Berlin

# MONDAY, 14:00-15:30

## A: Tropical Geometry

**Participants**
* Oliver Clarke
* Kevin Kühn
* Dante Luber
* Yue Ren
* Victoria Schleis
* Roan Talbut

**Tasks**
* Merge Puiseux Polynomials into AbstractAlgebra or Oscar (Ollie)
* Implement a first demo for nonarchimedean optimization (Ollie)
* Work on tropicalizations of polynomial ideals (Dante & Victoria)
* Work on tropical homotopy continuation (Roan)

## B: Cohomology rings

**Participants**
* Thomas Breuer
* Claus Fieker
* Michael Joswig
* Fabian Lenzen
* Gabriel Ong
* Matthias Zach

**Tasks**
* Compute (simplicial) cohomology rings; requires to develop a type for mildly noncommutative rings
* I'd broaden slightly and call it "Cohomology (rings)": I'd also like to include some of the group cohomology here, building on partial resolutions of Z[G] for various not too small groups G

**Steps/ Design**
AbstractCohomologyRings{T}
grading
ring

SimplicicalCohRing <= AbstractCohRing
SimplicialComplex
PolyRing

CohRingElem{T}
PolyRingElem...

free resolution -> CohomologyComplex -> CohRing
but also
ToricVariety -> CohRing without resolution in the background
maybe later functions to compute a resolution? useful for induced maps

## C: A better way to get randomness

**Participants**
- John Abbott
- Max Horn

**Tasks**
There is a [proposal by Max](https://github.com/oscar-system/Oscar.jl/issues/2896) and [a prototype by John](https://github.com/oscar-system/Oscar.jl/pull/4787) which however for now deviates from the proposal.

Claus:
Task 1: define the goal. Should this allow a user/ programmer to
- given a structure S
- figure out how to call rand(S)?

The problem being that the proposals and solutions allow me to do this in a readable, verbose, clear way iff I now the details about S. Suppose S is a matrix parent over polynomials over Q. Then I need to figure out the parameters allowed/ useful for base_ring(S), which involves finding out parameter spaces for polynomials and then the parameter spaces for the coefficients as well.
So we need a proposal to query the parameter spaces recursively - and to be able to set some.
Or we limit to easy structures: Q, Qx, Qxy, Z, finite structures.

Maybe define a global homework to identify actual structures where the proposal should apply?

After discussions, to make this a useful actionable point:
- implement Max's proposal for ZZ, QQ, ZZ[X], ZZ[X,Y] and similar SIMPLE rings
- wait for s.o. to complain and require more rings
- do NOT try to solve all the other stuff:
- no random_process with growing parameters
- no querying automagically for everything
- no immediate attempt to do or support matrix groups over polynomial rings over...

so: keep it simple and manageable and wait to usage/ complaints

# MONDAY 15:50

OSCAR The Movie (6min raw cut)

# MONDAY 16:00-17:30

## tba

# TUESDAY 10:00-13:00

## more movie

Claus & Max + B-Roll

## tba

# TUESDAY 14:00-15:30

## Serialization / MaRDI file format

**Participants**
- Antony Della Vecchia
- Claus Fieker
- Jeroen Hanselmann
- Max Horn
- Michael Joswig

* Int's vs strings in MaRDI file format / taking MongoDB into account
* slowly fixing the format for interoperability with gap/sage/magma/cocoa/???
* looking at the speed of reading?
* maybe slowly defining miniscule parts of the format? (ie. integer matrices, polynomials with rational coeffs?) in examples?

See also [this list of OSCAR issues on serialization](https://github.com/oscar-system/Oscar.jl/issues?q=is%3Aissue%20state%3Aopen%20label%3Aserialization)

## Visualizing objects (number fields)

**Participants**
- Claus Fieker
- Michael Joswig
- Marcel Wack
- ???

**Tasks**

As the name says: visualize objects, e.g. number fields to get pretty pictures for PR and of course deep insights.
For number fields there are suggestions:
- https://davidlowryduda.com/pcmi-vis-nf/
- https://mathematica.stackexchange.com/questions/115758/visualisation-of-the-field-of-algebraic-numbers-in-the-complex-plane
- https://math.colorado.edu/~kstange/papers/Stange-short-exp.pdf
- https://www.reddit.com/r/math/comments/2xs4t7/visualising_complex_quadratic_number_fields/

But maybe we have other objects/ ideas as well?

Other potential Objects:

- Matroids: Visualizing matroids via a point-line configurations (if possible). This already exists in SAGE
- Graphs: Connect the graph visualization with RecipesBase for more useful images. (part of Plots.jl)

## Matrix-Tensor arithmetic

OSCAR provides extensive functionality for arithmetic on matrices, but offers very limited support for higher-order tensors.
We would like to develop efficient implementations for matrix–tensor multiplication (including 8*48mode multiplication), tensor–tensor multiplication, elementary Gaussian operations on higher-order tensors, congruence transformations, and many other fundamental operations.
We test and apply this functionality in recent developments on path learning from signature tensors in rough analysis.

**Participants**
- Gabriel Riffo
- Leonard Schmitz
- ???

## Multigraded Groebner bases/resolutions

When working over a polynomial ring $\mathbb{k}[x_1,\ldots,x_N]$ with a $\mathbb{Z}^N$-grading, the algebra becomes much simpler. Also, computations become easier. OSCAR does not exploit that, at the moment. For example, if $L$ is a massive subquotient of free modules (around thousands of generators), then computing a (minimal) free resolution of $L$ is infeasible in OSCAR. Exploiting multigrading and sparsity could help this. I (Fabian) can bring interesting modules of that size to play around with.

**Participants**
- Fabian Lenzen
- Lars Kastner

## Tutte Groups

**Participants**
- Antony Della Veccia
- Marcel Wack
- Dante Luber?
- ???

Implement Tutte Groups in Oscar see [here](https://doi.org/10.1016/0001-8708(89)90013-3)

## Other potential topics
* transfer more polytope constructions from polymake to Oscar
* improving the documentation...
* defining the goal for linear algebra: identify missing/ slow areas and define interfaces

# Wednesday

## A: Tropical Geometry

### Topical Homotopy Continuation

Tasks:
* (Roan) Complete THC package for Bergman fans with the cyclic structure
* (Roan) Create an abstract class for THC package and begin implementing tropical drifts
* (Kevin) Add THC package description for TGr(2,n) to overleaf

### Puiseux polynomials & tropical varieties:

Tasks:
* (Dante) Complete reduction algorithm
* (Ollie + Victoria) Test whether Puiseux polynomial bug still persists. If yes, work on fixing it with whoever you may need.
* (Victoria) Complete test suite for tropical links
* (Ollie) Work on Puiseux polynomial merge (discuss points on GitHub)