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SS 2024
24.04.2024 um 10:00 Uhr in 32/110
Tim Seynnaeve (KU Leuven)
The translation-invariant Bell polytope
Bell's theorem, which states that the predictions of quantum theory cannot be accounted for by any classical theory, is a foundational result in quantum physics. In modern language, it can be formulated as a strict inclusion between two geometric objects: the Bell polytope en the convex body of quantum behaviours. Describing these objects leads to a deeper understanding of the nonlocality of quantum theory, and has been a central research theme is quantum information theory for several decades. After giving an introduction to the topic, I will focus on the so-called translation-invariant Bell polytope. Physically, this object describes Bell inequalities of a translation-invariant system; mathematically it is obtained as a certain projection of the ordinary Bell polytope. Studying the facet inequalities of this polytopes naturally leads into the realm of tensor networks, combinatorics, and tropical algebra.
This talk is based on joint work in progress with Jordi Tura, Mengyao Hu, Eloic Vallée, and Patrick Emonts.
08.05.2024 um 10:00 Uhr in 32/110
Ben Hollering (TU München)
Hyperplane Representations of Interventional Characteristic Imset Polytopes
Characteristic imsets are 0/1-vectors representing directed acyclic graphs whose edges represent direct cause-effect relations between jointly distributed random variables. A characteristic imset (CIM) polytope is the convex hull of a collection of characteristic imsets. These polytopes arise as feasible regions of an integer linear programming approach to the problem of causal disovery, which aims to infer a cause-effect structure from data. Linear optimization methods typically require a hyperplane representation of the feasible region, which has proven difficult to compute for CIM polytopes despite continued efforts. We solve this problem for CIM polytopes that are the convex hull of the imsets associated to DAGs whose underlying graph of adjacencies is a tree. Our methods use the theory of toric fiber products as well as the novel notion of interventional CIM polytopes. We obtain our results by proving a more general result for families of interventional CIM polytopes. As a demonstration of the applications of these novel polytopes, we apply our results to a real data example where we solve a linear optimization problem to learn a causal system from a combination of observational and interventional data.
15.05.2024 um 10:00 Uhr in 32/110
Mieke Fink (Universität Bonn)
A combinatorial basis for the valuative group of matroids.
The valuative group of matroids $Val(d,r)$ is the free abelian group of matroids of rank $r$ on a set of size $d$, modulo an inclusion-exclusion relation on matroid polytopes. By results of Hampe resp. Eur et al., the $Val(d,r)$ is the $2r-th$ homology group of the permutohedral resp. stellahedral variety.
Schubert matroids form basis of valuative groups and the representation of a matroid as a sum of Schubert matroids can be computed via their lattice of cyclic flats.
In this talk, I will discuss properties of matroid polytopes that relate to the lattice of cyclic flats, assuming only a superficial familiarity with matroids.
22.05.2024 um 10:00 Uhr in 32/110
M.Sc. Christian Ahring (Universität Osnabrück)
Hilbert Polynominal for Quasi- coherent sheaves on blowups
In projective algebraic geometry, the Hilbert polynominal is defined for any coherent sheaf on a projective scheme over a field k. It encodes relevant geometrical information, e.g. the rank and the degree of a vector bundle can be extracted from its Hilbert polynominal.
In this talk, we will recall some basic facts about the relations between projective geometry and graded algebras. Then we introduce a Hilbert polynominal for certain quasi- coherent sheaves on the blowup of a local ring at its maximal ideal. This Hilbert polynominal is additive in short exact sequences and the class of sheaves for which it is defined forms an abelian category. Finally, we investigate a quasi- coherent sheaf which is a natural candidate for a sheaf having a global section module of finite length.
29.05.2024 um 10:00 Uhr in 32/110
Yassine El Maazouz (RWTH Aachen)
Multivariate Gaussian distributions on local fields and sampling from p-adic algebraic manifolds
We introduce a notion of Gaussian distributions on vector spaces over non-archimedean local fields. These measures share a certain number of properties with their archimedean counterparts. In this talk we shall present one such similarity: the Gaussian entropy map. Using these measures, we give a method of sampling from p-adic algebraic manifolds. In particular, this allows us to sample from the Haar measure on an elliptic curve.
