11.04.2023 um 14:15 Uhr in 32/372
Jhon Bladimir Caicedo Portilla (Universität Osnabrück)
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
In this talk, I will present an efficient approach for counting roots of polynomial systems, where each polynomial is a general linear combination of fixed, prescribed polynomials. Our tools primarily rely on the theory of Khovanskii bases, combined with toric geometry, the BKK theorem, and fiber products.
I will demonstrate the application of this approach to the problem of counting the number of approximate stationary states for coupled Duffing oscillators. We have derived a Khovanskii basis for the corresponding polynomial system and determined the number of its complex solutions for an arbitrary degree of nonlinearity in the Duffing equation and an arbitrary number of oscillators. This is the joint work with Paul Breiding, Mateusz Michałek, Javier del Pino and Oded Zilberberg.
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.