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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.
05.06.2024 um 10:00 Uhr in 32/110
Uwe Nagel (University of Kentucky)
Schemes Arising from Hypersurface Arrangements
A hypersurface arrangement in projective space is said to be free if it´s Jacobian ideal is Cohen- Macaulay. We consider the Cohen- Macaulayness of two related unmixed ideals: the intersection of height two primary components and the radical of the Jacobian idela. In joint work with Migliore and Schenck we showed that in the case of a hyperplane arrangement these ideals are Cohen- Macaulay under a mild hypothesis. We discuss extensions for hypersurface arrangements obtained in joint work with Juan Migliore.
12.06.2024 um 10:00 Uhr in 32/110
Dr. Sebastian Debus (TU Chemnitz)
Symmetric quartics at infinity and inequalities in power sums
The fundamental theorem of symmetric functions states that any symmetric polynomial can be uniquely expressed in terms of power sum polynomials. Using products of power sum polynomials as a linear basis, we identify the sets of nonnegative homogeneous symmetric quartic polynomials. We present a test set for being nonnegative in any number of variables and give an instance of such a polynomial, which is also never sum of squares. Moreover, we combinatorially describe all valid homogeneous binomial inequalities in products of power sums by understanding the tropicalization of the image of the Vandermonde map. This talk is based on joint work in progress with Jose Acevedo, Greg Blekherman, and Cordian Riener.
12.06.2024 um 11:00 Uhr in 32/110
Svala Sverrisdóttir (UC Berkeley)
Toric degenerations arising from quantum chemistry
The high dimensional eigenvalue problem that encodes the electronic Schrödinger equation can be approximated by a hierarchy of polynomial systems at various levels of truncation, called the coupled cluster (CC) equations. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. We determine the number of complex solutions to the CC equations over the Grassmannian. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.
26.06.2024 um 10:00 Uhr in 32/110
Leonie Mühlherr (Universität Bielefeld)
Graphic hyperplane arrangements and their freeness
Graphic hyperplane arrangements are an interesting example of arrangements, since they are the subarrangements of the well-studied braid arrangement and have a strong connection to graph theory. This makes it possible to use graph theoretical tools to study them and specifically their algebraic properties. This talk will give an introduction to concepts of both fields of study and showcase their connections with a specific focus on projective dimension of the module of logarithmic derivations. This talk is based on joint work with Takuro Abe, Lukas Kühne and Paul Mücksch.
03.07.2024 um 10:00 Uhr in 32/110
M. Sc. Fynn Pörtner (Universität Osnabrück)
Semistability of monomial filtered syzygymodules
The concept of semistability is an important tool in algebraic geometry. A few years ago Holger Brenner found a nice combinatorical criterium for the semistability of a vector bundle, which is given by thr kernel of a monomial map. We try to translate these concepts to filtered modules over a filtered local regular ring and found a similar combinatorical criterium.
07.08.2024 um 10:00 Uhr in 69/125
Benjamin Biaggi (Universität Bern)
Border subrank via a generalised Hilbert-Mumford criterion
The subrank of a bilinear map is the maximal number of independent scalar multiplications that can be linearly reduced to the bilinear map. Given a sufficiently general complex nxnxn tensor, we give an upper bound on the growth rate for the border subrank. Since this matches the growth rate for the generic (non-border) subrank recently established by Derksen-Makam-Zuiddam, we find that the generic border subrank has the same growth rate. In our proof, we use a generalisation of the Hilbert-Mumford criterion that we believe will be of independent interest. This talk is based on joint work with Chia-Yu Chang, Jan Draisma and Filip Rupniewski.
15.08.2024 um 11:15 Uhr in 69/125
Prof. Dr. Takayuki Hibi (Osaka University, Japan)
A special class of pure O-sequences
03.09.2024 um 14:15 Uhr in 69/E15
Prof. Dr. Margherita Barile (University Bari)
What are Barile-Macchia resolutions?
The aim of the talk is to present recent developments in the theory of minimal graded free resolutions of monomial ideals. In this connection, some authors introduced the expression “Barile-Macchia resolution” to denote a notion that is rapidly changing its meaning.
10.09.2024 um 11:00 Uhr in 69/E15
Graham Keiper (University of Catania)
Symbolic Powers of Toric Ideals
This talk will discuss some recent joint work conducted at the University of Catania with Giuseppe Favacchio and Elena Guardo relating to symbolic powers of toric ideals. We will go over the necessary backround on toric ideals as well as symbolic powers. We will then discuss two new results useful in the computation of symbolic powers of toric ideals including a novel method of computing the symbolic powers of toric ideals via tensors. We will conclude with some conjectures and future research directions.
