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SS 2023

26.04.2023 um 17:15 Uhr in Raum 69/125

Prof. Dr. Wolfgang Hackbusch (Max-Planck-Institut für Mathematik Leipzig)

On Nonclosed Tensor Formats

Usually, the dimension of tensor spaces is too large for storing tensors by their entries. Instead, other formats are used which on the other hand represent only a subset F of tensors. A representation if called [non]closed if F is [non]closed.
In the case of nonclosed formats F, there exist 'border tensors' in the closure of F outside of F. Approximating such border tensors causes a numerical instability corresponding to the cancellation error of numerical differentiation. We prove a uniform minimal strength of this unstable behaviour.
In a second part we discuss the case of infinite dimensional tensor spaces. Here the weak [non-]closedness of a format is of interest. We prove for the k-term format that weak closedness and standard closedness coincide and that even in infinite dimensions the instability is the same as for finite dimensional tensor spaces up to constants which are explicitly known.

Finally, we give explicit results for the instability of the 2-term format (for border tensors of border rank 2).

03.05.2023 um 16:00 Uhr in Helikoniensaal/Gebäude 64

Prof. Dr. Hanna Döring und Prof. Dr. Alexander Salle

Antrittsvorlesungen

10.05.2023 um 17:15 Uhr in Raum 69/125

Dr. Alexey Ananyevskiy (LMU München)

Non-Vanishing Sections of Algebraic Vector Bundles and Trivial Chern Classes

It is well known that one may study the problem of existence of nowhere vanishing sections of vector bundles using methods of algebraic topology, in particular, characteristic classes. The simplest incarnation of this principle is that an oriented rank d real vector bundle over a dimension d compact manifold admits a nowhere vanishing continuous section if and only if the Euler class of this bundle vanishes. In the realm of algebraic geometry the picture is more intricate: a classical result of Murthy says that a rank d vector bundle over a smooth affine variety of dimension d over an algebraically closed field (e.g. over complex numbers) admits a nowhere vanishing (algebraic) section if and only if its top Chern class vanishes, while over a general field this does not hold with the counterexample given by the tangent bundle to the 2-sphere over the real numbers. Nevertheless, for the field of real numbers Bhatwadekar, Das, Mandal and Sridharan proved that triviality of the top Chern class is equivalent to the existence of a nowhere vanishing section provided that either d is odd, or when some additional assumption on the variety holds. In the talk I will outline how one can apply the methods of motivic homotopy theory, in particular, motivic Euler classes introduced by Morel, to this problem to give a new proof of these results and generalize them to the fields of virtual cohomological dimension at most 1.

24.05.2023 um 17:15 Uhr in Raum 69/125

Prof. Dr. Katharina Jochemko (KTH Stockholm)

Weighted Ehrhart Theory

In 1962 Ehrhart proved that the number of lattice points in integer dilates of a lattice polytope is given by a polynomial — the Ehrhart polynomial of the polytope. Since then Ehrhart theory has developed into a very active area of research at the intersection of combinatorics, geometry and algebra. The Ehrhart polynomial encodes important information about the polytope such as its volume and the dimension. An important tool to study Ehrhart polynomials is the h*-polynomial, a linear transform of the Ehrhart polynomial which is given by the numerator of the generating series. By a famous theorem of Stanley the coefficients of the h*-polynomial are always nonnegative integers. In this talk, we discuss generalizations of this result to weighted lattice point enumeration in rational polytopes where the weight function is given by a polynomial. In particular, we show that Stanley’s Nonnegativity Theorem continues to hold if the weight is a sum of products of linear forms that a nonnegative over the polytope. This is joint work with Esme Bajo, Robert Davis, Jesús De Loera, Alexey Garber, Sofía Garzón Mora and Josephine Yu.

07.06.2023 um 17:15 Uhr in Raum 69/125

Prof. Dr. Michael Gnewuch (Universität Osnabrück)

High-Dimensional Integration and Approximation: Randomized Algorithms and
Their Analysis

Consider an integration or a function recovery problem where the input functions depend on a very large or even infinite number of variables and belong to some reproducing kernel Hilbert space. The following questions arise naturally:

1. What kind of structure of the functions or of the function space, respectively, avoids the curse of dimensionality and ensures that the problem is computationally tractable?

2. What kind of algorithms give us almost optimal convergence rates?

3. How does their analysis depend on the specific function space and its norm? In this talk we want to discuss these questions in the setting where we are allowed to use randomized algorithms, where the error criterion is given by the randomized worst-case error (i.e., the worst-case root mean square error over the norm unit ball of the function space), and where the cost of evaluating a function in some point x depends in a reasonable way on the number of “active variables” of x. In the course of the talk, we will see that the questions above may be answered in the following way:

Helpful concepts to lift the curse of dimensionality are tensor product spaces, weights that moderate the importance of different groups of variables or increasing smoothness of the input functions (where “increasing” is meant with respect to the ordered variables). Depending on the chosen cost model, multilevel algorithms or multivariate decomposition methods, based on good building block algorithms that take care of lower dimensional sub- problems, may achieve convergence rates arbitrarily close to the optimal order. Furthermore, we want to present an elaborate framework for the embedding of different (scales of) Hilbert spaces, which enables us to transfer tractability results from specific Hilbert spaces to larger classes of spaces.

14.06.2023 um 16:15 Uhr in Raum 69/125

Prof. Dr. Kathlén Kohn (KTH Stockholm)

3D-Rekonstruktion aus Bildern und Algebraische Geometrie

This is an Osnabrücker Maryam Mirzakhani Lecture

21.06.2023 um 17:15 Uhr in Raum 69/125

Prof. Mihyun Kang, Ph.D. (TU Graz)

Supercritical Percolation on the Hypercube

We consider a random subgraph obtained by bond percolation on the hypercube in the supercritical regime and derive expansion properties of its giant component. As a consequence we obtain upper bounds on the diameter of the giant component and the mixing time of the lazy simple random walk on the giant component. We also extend the results to random subgraphs of high-dimensional product graphs. This talk is based on joint work with Sahar Diskin, Joshua Erde, and Michael Krivelevich.