**Project Heads**

Michael Joswig

**Project Members**

Jorge Alberto Olarte

**Project Duration**

01.01.2021 − 31.12.2021

**Located at**

TU Berlin

The Grassmannian Gr(k,n) comprises the k-dimensional sub- spaces of an n-dimensional vector space over a given field K. As a projective algebraic variety, the Grassmannian can be considered through its Plücker embedding; it is the vanishing locus of quadratic equations on variables indexed by subsets of [n] of size k. When K is an ordered field, the positive (non-negative) Grassmannian consists of all subspaces whose Plücker coordinates are positive (non-negative, respectively). The positive Grassmannian enjoys very rich combinatorial structures, from cluster algebras to matroids. These combinatorial gadgets offer diverse ways of stratifying and parametrizing the Grassmannian.

A while ago, physicists realized and mathematicians realized that the Grassmannians, and especially their positive parts, provide relevant models for studying scattering amplitudes in quantum field theory. This is about modelling the collision of any k out of n massless particles in a model called Supersymmetric Yang Mills (SYM). Such model is known not to represent actual physical reality, but has been proven useful as test-grounds for theories before trying them on more complicated models. Through this connection, physicists realized that the combinatorics of the positive Grassmannian could be applied to simplify Feynman integrals for the SYM model. Such computation is reduced to calculating the volume of a geometrical object called the amplituhedron. Therefore finding triangulations of the amplituhedron has become highly desirable and to that end matroids are key.

The tropicalization of the Grassmannian is a polyhedral shadow of the Grassmannian which also acts as the moduli space of tropical linear spaces, which are equivalent to valuated matroids. Even more so than in the classical case, matroids play a protagonist role in the study of the tropical Grassmannian, specially through the lens that view matroids as polytopes. Recently it has been observed that the positroid parametrization of the positive Grassmannian translate well into the tropical context. Moreover, one can obtain triangulations of the amplituhedron from subdivisions of the hypersimplex into positroids.

The goal of this project is to deepen our understanding of the connection between the tropical Grassmannian with quantum field theory. By studying the combinatorics of tropical linear spaces and positroids we can provide a solid ground to:

- Obtain geometrical information about the amplituhedron.
- Develop efficient algorithms that produce triangulations of the amplituhedron and compute scattering amplitudes for the SYM model.
- Find new combinatorial representations of the non-negative Grassmannian, its strata and their tropical counterparts.

**External Website**

**Related Publications
**

- M. Joswig. Essentials of tropical combinatorics, volume 219 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2021.
- George Balla and Jorge Alberto Olarte:
*The tropical symplectic Grassmannian. International Mathematics Research Notices, 10 2021. rnab267. arxiv:2103.16512*

**Related Pictures
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