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WS 2024/25

29.10.2024 um 12:00 Uhr in 66/101

M.Sc. Viktoriia Borovik (Universität Osnabrück)

Proudfoot-Speyer degenerations of scattering equations

We study scattering equations of hyperplane arrangements from the perspective of combinatorial commutative algebra and numerical algebraic geometry. We formulate the problem as linear equations on a reciprocal linear space and develop a degeneration-based homotopy algorithm for solving them. We also investigate the regularity of the corresponding homogeneous ideal and apply our methods to CHY scattering equations. This is joint work with B.Betti and S.Telen.

05.11.2024 um 12:00 Uhr in 66/101

Prof. Dr. Somayeh Moradi (Universität Osnabrück)

Ideals with componentwise linear powers

Let S be a polynomial ring over a field K, and let A be a finitely generated standard graded S- algebra. In terms of the initial ideal of the defining ideal of A, we give a criterion which implies that all graded components of A are componentwise linear. A typical example of such algebra is the Rees algebra of a graded ideal. Applying our criterion to the Rees algebra of cover ideals of graphs, we investigate and construct cover ideals whose powers are componentwise linear. This talk is based on joint works with Jürgen Herzog and Takayuki Hibi.

12.11.2024 um 14:00 Uhr in 66/101

Dr. Sarah Eggleston (Universität Osnabrück)

Typical ranks of random order- three tensors 

We study typical ranks of real m*n*l tensors. For (m-1)(n-1) < l < mn+1 the typical ranks are contained in {l, l+1}, and l is always a typical rank; we provide a geometric proof. We express the probabilities of these ranks in terms of the probabilities of the numbers of intersection points of a random linear space with the Segre variety. For m=n=3, the typical ranks of real 3*3*5 tensors are 5 and 6; we link the rank probabilities to the probability of a random cubic surface having real lines.

19.11.2024 um 12:00 Uhr in 66/101

Erin Connelly (Universität Osnabrück)

Algebraic Geometry in Computer Vision 

A camera is a linear projective map P³->P², represented by a full rank matrix in P(C³ˣ⁴), which sends world points q in P³ to image points p in P². A multi-view arrangement is a collection of cameras Aᵢ, world points qⱼ and image points pᵢⱼ satisfying Aᵢqⱼ=pᵢⱼ. We study the problem of reconstructing such systems from partial data. For the problem of reconstructing from the image data alone (3D Image Reconstruction) we find answers by studying the conditions under which k rank one tensors xᵢ⊗yᵢ are linearly dependent for 2≤k≤9. We also consider reconstruction from both the camera and image data (Triangulation) and from both the world point and image data (Resectioning). We utilize Carlson-Weinshall Duality and existing results for Triangulation to produce dual results for Resectioning.

26.11.2024 um 12:00 Uhr in 66/101

Laura Casabella (MPI Leipzig)

Regular subdivisions of hypersimplices

The secondary fan is a polyhedral fan which stratifies the regular subdivisions of a polytope. Using mathematical software, we compute some of these fans for specific polytopes, namely the hypersimplices Δ(2,7) and Δ(3,6). We also find new families of special rays for Δ(k,n) for general parameters. We also explain the connection between our results and the theory of finite metric spaces. This is joint work with Michael Joswig and Lars Kastner.

10.12.2024 um 12:00 Uhr in 66/101

Mahrud Sayrafi (MPI Leipzig)

Splitting of Vector Bundles on Toric Varieties

In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud- Erman- Schreyer for products of projective spaces, by Schreyer for Segre- Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.

17.12.2024 um 12:00 Uhr in 66/101

Barbara Betti (MPI Leipzig)

Khovanskii bases in computer algebra

In this talk we will recall the definition of Khovanskii bases, also known as Sagbi bases, and make a parallel with properties of Gröbner bases. Inspired by several applications of Gröbner bases in solving 0- dimensional polynominal systems, we will provide analogous applications in computer algebra using Khovanskii bases. These include the introduction of an eigenvalue algorithm based on the assumption that the equations are homogeneous with respect to a finite Khovanskii basis, homotopy continuation methods to solve systems over the Grassmannian and applications to particle physics. The results discussed are based on joint works with V. Borovik, L. Kayser, M. Panizzut and S. Telen