(folgt)
Wednesday, 2.11.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
(folgt)
Grenzwertaussagen in chemischen Reaktionsnetzwerken
Friday, 4.11.22, 14:15-15:15, Raum 232, Ernst-Zermelo-Str. 1
Concordances in Positive Scalar Curvature and Index Theory
Monday, 7.11.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
Scalar curvature is a local invariant of a Riemannian manifold. It measures\nasymptotically the volume growth of geodesic balls. Understanding the topological space of\nall positive scalar curvature metrics on a closed manifold has been an active field of study\nduring the last 30 years. So far, these spaces have been considered from an isotopy\nviewpoint. I will describe a new approach to study this space based on the notion of\nconcordance. To this end, I construct with the help of cubical set theory a comparison space\nthat only encodes concordance information and in which the space of positive scalar\ncurvature metrics canonically embeds. After the presentation of some of its properties, I will\nshow that the indexdifference factors over the comparison space using a new model of real\nK-theory that is based on pseudo Dirac operators.
Rendering Models for Scattering from Specular Rough Surfaces
Tuesday, 8.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Wave optics can be used to describe linearly polarized light that propagates in free\nspace in the form of waves. Thus, it enables to explain many optical phenomena such\nas interference, diffraction, dispersion and coherence. Surfaces with structures the\nsize of the wavelength of the incident light lead to such effects, which are essential to\nthe natural appearance.\n\nIn the scope of this work, the Helmholtz equation endowed with the impedance\nboundary condition is used to model sunlight incident on rough metallic surfaces.\nAfter proving unique solvability of this electromagnetic scattering problem, the\nboundary integral equation method is used to calculate such solutions for micro\nsurface patches. Numerically, this is done by means of the boundary element method,\nfor which a GPGPU implementation is introduced. This setup allows the local\ndescription of the aforementioned wave-optical phenomena, which are presented in\nthe form of BRDFs. The results are then used to assess one particular prior work.\nThere, approximate wave optics are employed for which it is not entirely clear how\nthese simplifications affect the quality. Although our results contain systematic\ndifferences, the overall agreement is good, confirming the validity of the more efficient\nprior work.\n
Simplicity of the automorphism group of fields with operators
Tuesday, 8.11.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In a recent preprint with T. Blossier, Z. Chatzidakis and C. Hardouin, we have adapted a proof of Lascar to show that certain groups of automorphisms of various theories of fields with operators are simple. It particularly applies to the theory of difference closed fields, which is simple and hence has possibly no saturated models in their uncountable cardinality. \n \n
Homotopy theory via o-minimal geometry
Friday, 11.11.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
O-minimality is a branch of model theory, with roots in real algebraic geometry, that provides a setting for "tame topology". This talk will describe the construction of a homotopy theory of spaces based on a given o-minimal structure, and give a taste of how algebraic topology can be developed in this framework.
Learning the time step size in Deep Neural Networks
Tuesday, 15.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\n\nAbstract: Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. Here, we are defining the discretization parameter (time step-size) to be an additional variable in the DNN. Hence, the time step-size can vary from layer to layer and is learned in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional-DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. To illustrate the advantages, the proposed approach is applied to an ill-posed 3D-Maxwell's equation.
Failure of GCH on a Measurable Cardinal
Tuesday, 15.11.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Let GCH hold in \(V\), and let \(\bkappa\) be a cardinal with a definable elementary embedding \(j:V\brightarrow M\) such that \({\brm crit}(j)=\bkappa\), \({}^{\bkappa}M\bsubseteq M\) and \(\bkappa^{++}=(\bkappa^{++})^{M}\) (in particular, \(\bkappa\) is measurable). H. Woodin proved that there is a cofinality preserving generic extension in which \(\bkappa\) stays measurable and GCH fails on it. This is achieved by using an Easton support iteration of Cohen forcings for having \(2^{\balpha}=\balpha^{++}\) for every inaccessible \(\balpha\bleq\bkappa\), and then adding an additional forcing to ensure the elementary embedding extends to the generic extension. Y. Ben Shalom proved in his thesis that this last forcing is unnecessary for the construction, and further extended the result to get \(2^{\bkappa}=\bkappa^{+\bgamma}\) assuming \(\bkappa^{+\bgamma}=(\bkappa^{+\bgamma})^{M}\), for any successor ordinal \(1<\bgamma<\bkappa\). We will present these results in some detail, and further extend the result of Ben Shalom for \(\bgamma=\bkappa+1\) assuming \(\bkappa^{+\bkappa+1}=(\bkappa^{+\bkappa+1})^{M}\).
Tag der Offenen Tür
Wednesday, 16.11.22, 10:30-11:30, BigBlueButton (online)
Polymorphic Uncertainty Quantification for the Additive Manufacturing of Elastic Rods.
Tuesday, 22.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We derive comprehensive models enabling an efficient uncertainty quantification of mechanical\nproperties for additively manufactured rod-shaped elastic solids Orod = (0, L) × hS(x1) in terms\nof errors introduced within the manufacturing process. Here, we consider the fused-filamentfabrication process where the main sources of uncertainties are given by variations of material\nproperties caused by fluctuation of material density and geometric deviations of the printed object from the designed object, see [CKB+18, PVB+19, KRJM+18]. The 3d-printed objects investigated in this work are made of polycarprolactone, a bioresorbable, biocompatible, polymer-based\nmaterial, which is used in the engineering of patient specific bone scaffolds, see [VDF+19].\nWe then deduce a comprehensive modelling approach in three space dimensions for determining\nthe effective mechanical properties of randomly perturbed elastic rods considering aleatoric and\nepistemic uncertainties in the representation of the random perturbations. To do so, we use the\npolymorphic uncertainty model of fuzzy structural analysis from [MGB00] which includes the\nrepresentation of random perturbations as fuzzy random fields (e.g. [PRZ93, Kwa78]) and MonteCarlo simulations (e.g. [KNP20, CGST11]) combined with finite element methods.\nFurthermore, we introduce an one-dimensional surrogate model for rod-shaped structures Orod =\n(0, L) × hS with a fixed cross-section S ⊂ R\n2\n. By this the problem can be reduced to an onedimensional optimization problem requiring only the solution of a system of ordinary differential equations. This leads to a marked reduction of computational effort compared to the threedimensional model concerning the computation of mechanical properties of randomly perturbed\nelastic rods.
Predicting with Diamond Sequences and with Ostaszewski Club Sequences
Tuesday, 22.11.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
An Ostaszewski club sequence is a weakening of Jensen's diamond.\nIn contrast to the diamond, the club does not imply the continuum hypothesis.\nNumerous questions about the club stay open, and we know only few models in which\nthere is just a club sequence but no diamond sequence. In recent joint\nwork with Shelah we found that a winning strategy for the completeness player\nin a bounding game on a forcing order does not suffice to establish the club\nin the extension.
"L-functions, Euler systems, and the Birch-Swinnerton-Dyer conjecture"
Wednesday, 23.11.22, 16:15-17:15, Hörsaal II, Albertstr. 23b
Effective toughness of brittle composite laminates
Thursday, 24.11.22, 11:30-12:30, Raum 226, Hermann-Herder-Str. 10
We consider a periodic layer of brittle elastic materials. As the layers become fine, the composite behaves elastically as a spatially homogeneous (averaged) material, whose stiffness modulus can be computed in terms of the relative volumes and the elastic modulus of the single layers. However, in the presence of a crack evolving through the layers it is still unclear if the quasi-static evolution is still represented in terms of a spatially homogeneous material with a crack. In particular not much is known on the effective (or averaged) toughness. Experimental measures, numerical simulations and theoretical estimates show surprising features of the effective toughness: it depends not only on the toughness and the size of the layers but also on their elastic moduli, and it may be even larger than the toughness of the layers (which is known as toughening effect).\nIn this framework, we provide a theoretical study and a couple of examples. We provide an abstract formula for the (possibly non-constant) effective toughness, then we prove convergence of the evolution and convergence of the energy identities, as the size of the layers vanishes. As a by-product we link the toughening effect to the micro-instabilities of the evolution, occurring at the interfaces between the layers of the composite. The two examples provide instead explicit calculations of the effective toughness, one of which presents a toughening effect.
Homological Bondal-Orlov localization conjecture
Friday, 25.11.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
An old conjecture going back to Bondal and Orlov predicts a precise relation between the derived categories of a variety with rational singularities and its resolution of singularities. I will explain the proof of the surjectivity part of this conjecture, based on an argument from Hodge theory. This is joint work with Mirko Mauri.\n\n
Machine Learning about Implementable Portfolios
Friday, 25.11.22, 12:00-13:00, online: Zoom
We develop a framework that integrates trading-cost-aware portfolio optimization with machine learning (ML). While numerous studies use ML return forecasts to generate portfolios, their agnosticism toward trading costs leads to excessive reliance on eeting small-scale characteristics, resulting in poor net returns. We propose that investment strategies should be evaluated based on their implementable ecient frontier, and show that our method produces a superior frontier. The superior net-of-cost performance is achieved by integrating ML into the portfolio problem, learning directly about portfolio weights (rather than returns). Lastly, our model gives rise to a new measure of "economic feature importance".
Variational methods for a class of mixed local-nonlocal operators
Tuesday, 29.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Problems driven by operators of mixed local and nonlocal type have\nraised a certain interest in the last few years, for example in\nconnection with the study of optimal animal foraging strategies. From a\npure mathematical point of view, the superposition of local and\nnonlocal operators, such as the Laplacian and the Fractional Laplacian,\ngenerates a lack of scale invariance that can lead to unexpected\ncomplications. Our goal is to prove the existence of solutions of\nsemilinear elliptic problems governed by these operators and dependent\non a real parameter: when the parameter is sufficiently large, our\nexistence results are known or applications of standard variational\nmethods, but when the real parameter is too small, the situation\nsuddenly becomes more delicate, especially since the operator is no\nlonger positive-definite, the naturally associated bilinear form does\nnot induce a scalar product nor a norm, the variational spectrum may\nhave negative eigenvalues, and even the maximum principle may fail. In\nthis talk, I show how to overcome these difficulties and obtain the\nexpected existence results.\n