arXiv subjects
Publications and source records attributed to Bernd Sturmfels.
Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.
Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.
We present a study of cubic surfaces from the novel perspective of positive geometry. Our positive geometries have dimension two (the surface minus its 27 lines), dimension three (its complement in 3-space), and dimension four (the moduli space). In each case we explore the positive arrangement, its combinatorial rank, and the canonical forms.
Momentum twistors for scattering amplitudes in particle physics are lines in three-space. We develop Landau analysis for Feynman integrals in this setting. The resulting discriminants and resultants are identified with Hurwitz and Chow forms of incidence varieties in products of Grassmannians. We study their degrees and factorizations, and the kinematic regimes in which the fibers of the Landau map are rational or real. Identifying this map with the amplituhedron map on positroid varieties, and the associated recursions with promotion maps, yields a geometric mechanism for the emergence of positivity and cluster structures in planar N=4 super Yang-Mills theory.
Landau analysis in momentum twistor space can be formulated as the study of varieties of lines in three-dimensional projective space, together with their projections and discriminants. Within this framework, we define enumerative invariants (LS degrees) that count leading singularities. Leading Landau singularities (LS discriminants) arise as discriminants detecting the collision of leading singularities. We uncover a recursive mechanism underlying Landau singularities, governed by substitution maps between Grassmannians. Applying this framework, we prove positivity and factorization into cluster variables for the LS discriminant of a large class of Landau diagrams at arbitrary loop order. This provides a first-principles explanation for the emergence of positivity and cluster algebra structures in the singularities of planar N=4 super Yang-Mills theory.
The Hurwitz form of a projective variety characterizes linear spaces of complementary dimension which meet the variety non-transversally. We extend this notion to varieties in a product of projective spaces. This parallels the multigraded Chow forms due to Osserman and Trager. We study the degrees of multigraded Hurwitz forms. An explicit degree formula is given for complete intersections. This offers a new tool for elimination theory that has many applications, ranging from Nash equilibria to Feynman integrals.
The square of a skew-symmetric matrix is a symmetric matrix whose eigenvalues have even multiplicities. When the matrices have rank two, they represent the Grassmannian of lines, and the squaring operation takes Pl\"ucker coordinates to projection coordinates. We develop metric algebraic geometry for varieties of lines in this linear algebra setting. The Grassmann distance (GD) degree is introduced as a new invariant for subvarieties of a Grassmannian. We study the GD degree for Schubert varieties and other models.
We consider configurations of lines in 3-space with incidences prescribed by a graph. This defines a subvariety in a product of Grassmannians. Leveraging a connection with rigidity theory in the plane, for any graph, we determine the dimension of the incidence variety and characterize when it is irreducible or a complete intersection. We study its multidegree and the family of Schubert problems it encodes. Our spanning-tree coordinates enable efficient symbolic computations. We also provide numerical irreducible decompositions for incidence varieties with up to eight lines. These constructions with lines play a key role in the Landau analysis of scattering amplitudes in particle physics.
The CHY scattering equations on the moduli space $M_{0,n}$ play a prominent role at the interface of particle physics and algebraic statistics. We study the scattering correspondence when the Mandelstam invariants are restricted to a fixed graph on $n$ vertices.
Maxout polytopes are defined by feedforward neural networks with maxout activation function and non-negative weights after the first layer. We characterize the parameter spaces and extremal f-vectors of maxout polytopes for shallow networks, and we study the separating hypersurfaces which arise when a layer is added to the network. We also show that maxout polytopes are cubical for generic networks without bottlenecks.
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.
We develop novel tools for computing the likelihood correspondence of an arrangement of hypersurfaces in a projective space. This uses the module of logarithmic derivations. This object is well-studied in the linear case, when the hypersurfaces are hyperplanes. We here focus on nonlinear scenarios and their applications in statistics and physics.
Copositive matrices and copositive polynomials are objects from optimization. We connect these to the geometry of Feynman integrals in physics. The integral is guaranteed to converge if its kinematic parameters lie in the copositive cone. Pólya's method makes this manifest. We study the copositive cone for the second Symanzik polynomial of any Feynman graph. Its algebraic boundary is described by Landau discriminants.
This article serves as an introduction to the special volume on Positive Geometry in the journal Le Matematiche. We attempt to answer the question in the title by describing the origins and objects of positive geometry at this early stage of its development. We discuss the problems addressed in the volume and report on the progress. We also list some open challenges.
We study statistical models that are parametrized by squares of linear forms. All critical points of the likelihood function are real and positive. There is one critical point in each region of the projective hyperplane arrangement defined by the linear forms. We examine the ideal and singular locus of the model, and we give a determinantal presentation for its likelihood correspondence. We characterize tropical degenerations of the MLE, we describe the log-normal polytopes, and we explore connections to determinantal point processes.
We study stratifications of regions in the space of symmetric matrices. Their points are Mandelstam matrices for momentum vectors in particle physics. Kinematic strata in these regions are indexed by signs and rank two matroids. Matroid strata of Lorentzian quadratic forms arise when all signs are non-negative. We characterize the posets of strata, for massless and massive particles, with and without momentum conservation.
The CEGM formalism offers a general framework for scattering amplitudes, which rests on Grassmannians, moduli spaces and tropical geometry. The physical implications of this generalization are still to be understood. Conventional wisdom says that key features of scattering amplitudes, like factorization at their poles into lower-point amplitudes, are associated to their singularities. The factorization behavior of CEGM amplitudes at their poles is interesting but complicated. Recent developments have revealed important properties of standard particle and string scattering amplitudes from factorizations, known as splits, that happen away from poles. In this paper we introduce a kinematic subspace on which the CEGM amplitude splits into very simple rational functions. These functions, called simplex amplitudes, arise from stringy integrals for the multivariate beta function, and also from restricting the biadjoint scalar amplitude in quantum field theory to certain kinematic loci. Using split kinematics we also discover a specific class of zeros of the CEGM amplitude. Our construction rests on viewing positive moduli space as a product of simplices, and it suggests a novel approach for deriving scattering amplitudes from tropical determinantal varieties.
An arrangement of hypersurfaces in projective space is strict normal crossing (SNC) if and only if its Euler discriminant is nonzero. We study the critical loci of arbitrary Laurent monomials in the equations of the smooth hypersurfaces. The family of these loci forms an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the minimal generators of the bihomogeneous prime ideal that defines this variety.