Yassine El Maazouz

A matrix is totally positive if all of its minors are positive. This notion of positivity coincides with the type A version of Lusztig's more general total positivity in reductive real-split algebraic groups. Since skew-symmetric matrices always have nonpositive entries, they are not totally positive in the classical sense. The space of skew-symmetric matrices is an affine chart of the orthogonal Grassmannian OGr(n,2n). Thus, we define a skew-symmetric matrix to be totally positive if it lies in the totally positive orthogonal Grassmannian. We provide a positivity criterion for these matrices in terms of a fixed collection of minors, and show that their Pfaffians have a remarkable sign pattern. The totally positive orthogonal Grassmannian is a CW cell complex and is subdivided into Richardson cells. We introduce a method to determine which cell a given point belongs to in terms of its associated matroid.

We investigate determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We study the ideals of relations among functions in the matrix entries that serve as building blocks for conformal correlators.

We present a formulation of the three- and four-point amplitudes on the Coulomb branch of $N=4$ SYM as integrals over the symplectic Grassmannian. We demonstrate that their kinematic spaces are equivalent to symplectic Grassmannians $\mathrm{SpGr}(n,2n)$. For the three-point case, we express the amplitude as an integral over the symplectic Grassmannian in a specific little group frame. In the four-point case, we show that the integral yields the amplitude up to a known kinematic factor. Building on the four-dimensional analysis, we also express the six-dimensional $N = (1,1)$ SYM amplitude in terms of four-dimensional variables in a form that makes its symplectic Grassmannian structure manifest.

The spinor-helicity formalism in particle physics gives rise to natural subvarieties in the product of two Grassmannians. These include two-step flag varieties for subspaces of complementary dimension. Taking Hadamard products leads to Mandelstam varieties. We study these varieties through the lens of combinatorics and commutative algebra, and we explore their tropicalization, positive geometry, and scattering correspondence.

The Plücker positive region $\mathrm{OGr}_{+}(k,2k)$ of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian $\mathrm{OGr}_{+}(k,n)$ for general values of $k,n$. We determine the boundary structure of the quadric $\mathrm{OGr}_{+}(1,n)$ in $\mathbb{P}^{n−1}_{+}$ and show that it is a positive geometry. We show that $\mathrm{OGr}_{+}(k,2k+1)$ is isomorphic to $\mathrm{OGr}_{+}(k+1,2k+2)$ and connect its combinatorial structure to matchings on $[2k+2]$. Finally, we show that in the case $n>2k+1$, the \emph{positroid cells} of $\mathrm{Gr}_{+}(k,n)$ do not induce a CW cell decomposition of $\mathrm{OGr}_{+}(k,n)$.