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SS 2023

11.04.2023 um 14:15 Uhr in 32/372

Jhon Bladimir Caicedo Portilla (Universität Osnabrück)

Abstract

18.04.2023 um 14:15 Uhr in 32/372

Janina Letz (Uni Bielefeld)

Local to global principles for generation time over commutative rings

In the derived category of modules over a commutative ring a complex G is said to generate a complex X if the latter can be obtained from the former by taking finitely many summands ans cones. The number of cones needed in this process is the generation time of X. I will present some local to global type results for computing this invariant and also discuss some applications.

25.04.2023 um 14:15 Uhr in 32/372

Yairon Cid- Ruiz (KU Leuven)

Duality and blow-up algebras

We provide a generalization of Jouanolou duality that is applicable to a plethora of situations. The environment where this generalized duality takes place is a new class of rings, that we introduce and call weakly Gorenstein. As a main consequence, we obtain a new general framework to investigate blowup algebras. We use our results to study and determine the defining equations of the Rees algebra of certain families of ideals. This is joint work with Claudia Polini and Bernd Ulrich.

02.05.2023 um 14:15 Uhr in 32/372

Sarah Eggleson (Universität Osnabrück)

The amoeba dimension of a linear space

Given a complex vector subspace V, the dimension of the amoeba depends only on the matroid of V. Here we prove that this dimension is given by the minimum of a certain function over all partitions of the ground set into nonempty parts, as previously conjectured by Rau. We also prove that this formula can be evaluated in polynomial time.

09.05.2023 um 14:15 Uhr in 32/372

Viktoriia Borovik (Universität Osnabrück)

Khovanskii bases for semimixed systems of polynomial equations

16.05.2023 um 14:15 Uhr in 32/372

Pierpaola Santarsiero (Universität Osnabrück)

The symmetric geometric rank of symmetric tensors

Inspired by recent work of Kopparty-Moshkovitz-Zuiddam and motivated by problems in combinatorics and hypergraphs, we introduce the notion of symmetric geometric rank of a symmetric tensor. This quantity is equal to the codimension of the singular locus of the hypersurface associated to the tensor. In this talk, we will first learn fundamental properties of the symmetric geometric rank. Then, we will study the space of symmetric tensors of prescribed symmetric geometric rank, which is the space of homogeneous polynomials whose corresponding hypersurfaces have a singular locus of bounded codimension. This is joint work with J. Lindberg.

23.05.2023 um 14:15 Uhr in 32/372

Georg Loho (Universität Twente)

Lower bounds on neural network depth via lattice polytopes

We study the set of functions representable by ReLU neural networks, a standard model in the machine learning community. It is an open question whether this set strictly increases with the number of layers used. We prove that this is indeed the case if one considers neural networks with only integer weights.
We show that at least log(n) many layers are required to compute the maximum of n numbers, matching known upper bounds. To obtain our result, we first use previously discovered connections between neural networks and tropical geometry to translate the problem into the language of Newton polytopes. These Newton polytopes are lattice polytopes arising from alternatingly taking convex hulls and Minkowski sums. Our depth lower bounds then follow from a parity argument for the volume of faces of such polytopes, which might be of independent interest. This is joint work with Christian Haase and Christoph Hertrich.

30.05.2023 um 14:15 Uhr in 32/372

Sudarshan Gurjar (Indian Institute of Technology, Mumbay)

Topology of Varieties

In this talk I will comment on some joint papers written in collaboration with R. V Gurjar, Buddhadev Hajra and Poonam Pokhale. I will begin by discussing some results on the homotopy and homology groups of smooth complex surfaces (both affine and projective) of non- general type. This will result in the classification of Eilenberg MacLane surfaces of non- general type.

06.06.2023 um 14:15 Uhr in 32/372

Tarek Emmrich (Uni Osnabrück)

Chebotarëv's nonvanishing minors for eigenvectors of random matrices and graphs

For a matrix M we establish a condition on the Galois group of the characteristic polynomial that induces nonvanishing of the minors of the eigenvector matrix of M. For integer matrices recent results by Eberhard show that, conditionally on the extended Riemann hypothesis, this condition is satisfied with high probability. And hence with high probability the minors of eigenvectors matrices of random integer matrices are nonzero. For random graphs this will yield a novel uncertainty principle, related to Chebotarëv's theorem on the roots of unity and results from Tao and Meshulam.

13.06.2023 um 14:15 Uhr in 32/372

Holger Brenner (Universität Osnabrück)

Some remarks on a Theorem of Steinberg in invariant theory

A theorem of Steinberg states that the invariant ring for a finite group acting linearly on a polynominal ring is itself a polynominal ring if and only if the colenght of the Hilbert ideal equals the cardinality of the group. Here, we consider more generally the corresponding ratio between this colenght and the cardinality and show that this ratio depends only on the invariant ring. In particular, this provides a new proof of Steinberg´s theorem. Our approach uses reduction to positive characteristic and Hilbert- Kunz multiplicity in positive characteristic. This is joint work with Mandira Mondal. 

20.06.2023 um 14:15 Uhr in 32/372

Sophie Rehberg (FU Berlin)

Rational Ehrhart Theory

The Ehrhart quasipolynominal of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer points in positive integral dilates of P. Ehrhart quasipolynominal were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni- Berline- Koeppe- Vergne (2013), and Stapledon (2017). We introduce a generating- function ansatz for rational Ehrhart quasipolynominals, which unifies several known results in classical and rational Ehrhart theory. In particular, we define y- rational Gorenstein polytopes, which extend the classical notion to the rational setting.
This is joint work with Matthias Beck and Sophia Elia.

27.06.2023 um 14:15 Uhr in 32/372

Paul Breiding (Universität Osnabrück)

Euclidean Distance Degree and Mixed Volume

The Euclidean Distance Degree (EDD) of an algebraic variety V counts the number of complex critical points of the distance function from V to a generic fixed point outside of V. The BKK-Theorem (Bernstein, Kushnirenko, and Khovanskii) says that the number of complex zeros of a generic sparse polynomial system is equal to the mixed volume of the Newton polytopes of the polynomials. In this talk I will explain why for a generic sparse polynomial f the EDD of the hypersurface f=0 equals the mixed volume of the Lagrange multiplier equations for the EDD. This has impact on using polynomial homotopy continuation for computing the ED-critical points. And it provides new formulas for the EDD.

04.07.2023 um 14:15 Uhr in 32/372

Svala Sverisdottir (UC Berkeley)

The variety of four dimensional Lie algebras

I will present recent joint work with Laurent Manivel and Bernd Sturmfels on the variety of Lie algebra structures on a 4-dimensional vector space. This is an 11 dimensional projective variety with four irreducible components all of dimension 11. I will show how we obtain the degree of its irreducible components using both numerical computations like HomotopyContinuation.jl and symbolic computations.