The wave equation with acoustic boundary conditions on non-locally reacting surfaces
Dienstag, 10.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The subject of the talk are some recent results on the wave equation posed in a suitably regular open domain of R\nN , supplied\nwith an acoustic boundary condition on a part of the boundary and an homogeneous Neumann boundary condition on the\n(possibly empty) remaining part of it, contained in the recent joint book with Delio Mugnolo.\n\nIn this talk we shall focus on non–locally reacting boundaries without any Dirichlet boundary conditions. We first give well–\nposedness results in the natural energy space and regularity results. Hence we shall give precise qualitative results for solutions\nwhen Ω is bounded and r = 2, ρ = Const., k, δ ≥ 0. In particular we shall exhibit some physically inexplicable trivial solutions\nwhich make the problem not asymptotically stable, even with an effective damping, while the problem is asymptotically stable\nwhen the initial data are restricted to a 1-codimensional subspace, which is invariant under the flow.\nThis mathematical result motivated a re-thinking of the physical derivation of the Acoustic Wave Equation, found in most\ntexbooks in Theoretical Acoustic and Classical Mechanics, and of the specific Acoustic Boundary Conditions. The main outcome\nof this detailed analysis is described as follows. In both the Eulerian and the Lagrangian frameworks, due to Hooke’s law, the\nPDEs appearing in it need to be integrated with the integral constraint found in the stability analysis in order to correctly model\nthe physical problem. This fact was never observed in the existing literature
Informationsveranstaltung zu Auslandsaufenthalten
Mittwoch, 11.1.23, 18:00-19:00, Raum 226, Hermann-Herder-Str. 10
Derived categories of singular projective varieties
Freitag, 13.1.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
Valuation and Risk Management of Guaranteed Minimum Death Benefits (GMDB) by Randomization
Freitag, 13.1.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
Randomization is a technique in Finance to replace known quantities (like the time to maturity) by random variables. This sometimes gives moments or quantiles of the payoff in closed-form, avoiding any kind of integration, Fourier inversion or simulation algorithm. We apply this idea to insurance and Guaranteed Minimum Death Benefits (GMDB) where payoff dates are per se random. The remaining lifetime is expanded in terms of a Laguerre series while the financial market follows a regime switching model with two-sided phase-type jumps. For European-type GMDBs, we obtain the density of the payoff in closed form as a Laurent series. Payoff distributions of contracts with path-dependent guarantee features can be expressed in terms of solutions of Sylvester equations (=matrix equations of the form AX + XB =C).\n\nThis is joint work with Griselda Deelstra (Université Libre de Bruxelles).\n\nA paper version is available here: Deelstra, Griselda and Hieber, Peter, Randomization and the Valuation of Guaranteed Minimum Death Benefits, https://ssrn.com/abstract=4115505.
Das Babuška-Paradoxon
Dienstag, 17.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Das Babuška-Paradoxon ist beim linearen Biegen von Platten zu beobachten. Es tritt bei einer stückweise linearen Approximation eines gekrümmten Plattenrandes. Die Lösung auf dem somit definierten Gebiet konvergiert demnach nicht gegen die „echte“ Lösung. Eine Rolle spielt es somit insbesondere beim Berechnen numerischer Lösungen z.B. mit der Finite Elemente Methode. In diesem Vortrag werden wir eine mathematische Herleitung des Paradoxon nachvollziehen und versuchen dessen Auftreten durch numerische Experimente zu verifizieren. Hierbei werden wir jedoch einige Diskrepanzen zwischen unseren Ergebnissen und etablierter Theorie feststellen.\n\n
(Non-archimedean) SYZ fibrations for Calabi-Yau hypersurfaces
Freitag, 20.1.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. A recent result by Li relates explicitly the non-archimedean approach to the classical SYZ conjecture.\n\nIn this talk, I will give an overview of this subject and I will focus on families of Calabi-Yau hypersurfaces in P^n. For this class, in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey, we solve a non-archimedean conjecture proposed by Li and deduce that classical SYZ fibrations exist on a large open region of CY hypersurfaces.
A phase field model for soma-germline interactions in Drosophila oogenesis
Freitag, 20.1.23, 12:00-13:00, online: Zoom
In [1], we study the signals mediating the mechanical interaction between somatic epithelial cells and the germline of Drosophila. We discover that, during the development of the egg chamber, the transcriptational regulator “Eyes absent” (Eya) modulates the affinity of the apical surface of epithelial cells to the nurse cells and the oocyte in the egg chamber. Using a phase field model, we develop a quantitative, mechanical description of epithelial cell behavior and demonstrate that the spatio-temporal expression of Eya controls the epithelial cells’ shape and movement during all phases of Drosophila oogenesis to establish a suitable match between epithelial cells and germline cells. Further we show that differential expression of Eya in follicle cells also controls oocyte growth via cell-cell affinity.\n\n \n\n[1] V. Weichselberger, P. Dondl, A.-K. Classen (2022): Eya-controlled affinity between cell lineages drives tissue self-organization during Drosophila oogenesis. Nat Commun 13(1):6377. DOI: 10.1038/s41467-022-33845-1
Positive scalar curvature and 2-type: an analysis via the Gromov--Lawson--Rosenberg conjecture.
Montag, 23.1.23, 16:15-17:15, Hörsaal II, Albertstr. 23b
I will talk about a conjecture that claims the PSC space of a (spin) manifold only depends on its 2-type. In particular, focusing on fundamental groups that satisfy the Gromov--Lawson--Rosenberg conjecture one can obtain positive results in certain cases. The latter conjecture claims that the non-vanishing of a certain cobordism invariant represents a total obstruction to positive scalar curvature. Index theory and surgery theory are at the base of the whole argument.
A variational approach to the regularity of optimal transportation
Dienstag, 24.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk I want to present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced by Goldman—Otto. Following De Giorgi’s strategy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, which leads to a one-step improvement lemma, and feeds into a Campanato iteration on the C^{1,\balpha}-level for the displacement.\n\nThe variational approach is flexible enough to cover general cost functions by importing the concept of almost-minimality: if the cost is quantitatively close to the Euclidean cost function |x-y|^2, a minimiser for the optimal transport problem with general cost is an almost-minimiser for the one with quadratic cost. This allows us to reprove the C^{1,\balpha}-regularity result of De Philippis—Figalli, while bypassing Caffarelli’s celebrated theory. (This is joint work with F. Otto and M. Prod’homme)
Algorithmic Solution of Elliptic Optimal Control Problems with Control Constraints by Means of the Semismooth Newton Method“
Dienstag, 24.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
I will consider the standard elliptic optimal control problem governed by the Poisson equation where the control space \(U\) is chosen to be either \(L^2(\bOmega)\) or \(H^1_0(\bOmega)\). In particular, I will introduce an abstract framework that provides q-superlinear convergence of the semismooth Newton method and which can be applied to get this convergence rate for any of the choices of \(U\). Further, semismoothness results and characterizations of the elements in the generalized differential will be done. The talk will focus on the infinite dimensional setting, but error estimates and numerical results will also be provided.
A Proof of the Halpern-Läuchli Partition Theorem without Metamathematical Argumentation
Dienstag, 24.1.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
The Halpern-Läuchli Theorem is a fundamental Ramsey type principle concerning partitions of finite products of trees. Historically the proof of the theorem was given using meta-mathematical reasoning. We will show a direct proof given by S.A Argyros, V. Felouzis and V. Kanellopoulos that uses anly standard mathematical arguments. The theorem talks about finite dimensional products of trees, but (time permitting) we will give a discussion of the infinite dimensional case.
Hyperbolic Localization and Extension Algebras
Freitag, 27.1.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
A smooth projective variety with a nice torus action, such as a Grassmannian, can be decomposed into attracting cells (Białynicki-Birula stratification). In this talk we give a cohomological description of the extension algebra of constant sheaves on the attracting cells based on Drinfeld-Gaitsgory's account of Braden's hyperbolic localization functor. This algebra describes the gluing data of the category of constructible sheaves and, in the case of flag varieties, plays an important role in the representation theory of reductive algebraic groups/Lie algebras.
Generalized Covariance Estimator
Freitag, 27.1.23, 12:00-13:00, Hörsaal II, Albertstr. 23b
We consider a class of semi-parametric dynamic models with iid errors, including the nonlinear mixed causal-noncausal Vector Autoregressive (VAR), Double-Autoregressive (DAR) and stochastic volatility models. To estimate the parameters characterizing the (nonlinear) serial dependence, we introduce a generic Generalized Covariance (GCov) estimator, which minimizes a residual-based multivariate portmanteau statistic. In comparison to the standard methods of moments, the GCov estimator has an interpretable objective function, circumvents the inversion of high-dimensional matrices, and achieves semi-parametric efficiency in one step. We derive the asymptotic properties of the GCov estimator and show its semi-parametric efficiency. We also prove that the associated residual-based portmanteau statistic is asymptotically chi-square distributed. The finite sample performance of the GCov estimator is illustrated in a simulation study. The estimator is then applied to a dynamic model of commodity futures.\nChristian Gourieroux & Joann Jasiak (2022): Generalized Covariance Estimator,Journal of Business & Economic Statistics, DOI:10.1080/07350015.2022.2120486
Localization methods and the Witten genus
Montag, 30.1.23, 16:15-17:15, Hörsaal II, Albertstr. 23b
In this talk I will give a brief introduction to equivariant cohomology and the localization formula. Applying the formula to infinite dimensional siutations one recovers interesting invariants like the A-hat genus or the Witten genus. In this representation one finds a natural explanation for the modularity of the Witten genus.
tba
Dienstag, 31.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Multiscale computer model of bone regeneration within scaffolds in T2DM
Dienstag, 31.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\nBone has the fascinating ability to self-regenerate. However, under certain conditions, such as\nlarge bone defects, this ability is impaired. The treatment of large bone defects is a major\nclinical challenge. There are a number of existing treatments such as the addition of growth\nfactors or autologous bone grafting, but with many associated side effects (Roddy et al., 2018).\n3D printed scaffolds might help to overcome these challenges by providing guiding cues for\ntissue regeneration (Pobloth et al., 2018), however their design remains challenging and mainly\nbased on an experimental trial and error approach. Moreover, the treatment of large bone defects\ngets even more challenging when comorbid with Type 2 diabetes mellitus (T2DM). T2DM is a\nchronic metabolic disease known by the presence of elevated blood glucose levels that is\nassociated with reduced bone regeneration, high fracture risk and non-union. Currently, the\ntreatment of bone defects mostly depends on clinical imaging with some references to patients’\nphysiology and medical history (Wiss, D.A. and W.B. Stetson. 1996), but lacks details about\nthe patient-specific healing potential. Omics offer quantifiable biological indication for\npatients’ intrinsic bone regenerative capacity. In this study, in silico computer modelling\napproaches are used to 1) investigate the effect of scaffold design on the bone regeneration\noutcome, 2) understand the underlying mechanisms behind impaired bone regeneration in\nT2DM and 3) investigate the potential of using an omics informed computer model to predict\npatient-specific bone regeneration.\nA multiscale in silico approach was used that combines finite element and agent based models,\nwhich allow to quantify the mechanical environment within the defect region and to simulate\nthe cellular response. Using these models, bone regeneration was investigated in healthy and\nT2DM conditions, as well as within different scaffold designs. Moreover, the potential of\nomics-driven cellular activity rates to predict bone regeneration was investigated.\nOur validated models suggest that scaffolds with strut-like architecture are beneficial over\nscaffolds with gyroid architecture promoting cell migration towards the scaffold core, both in\nhealthy and T2DM conditions. Impaired MSC proliferation, MSC migration and osteoblast\ndifferentiation were identified as key players behind impaired bone regeneration in T2DM.\nOur results show that bone regeneration is influenced by scaffold architecture agreeing with\nexperimental studies showing different healing outcomes for different scaffold designs. The\nidentification of the key cellular activities behind impaired bone regeneration in T2DM should\nallow the optimization of the scaffold design to promote bone regeneration in co-morbid\npatients.
Brüche verstehensorientiert unterrichten - adaptiv und sprachsensibel
Dienstag, 31.1.23, 19:30-20:30, Hörsaal II, Albertstr. 23b
Das Unterrichten in heterogenen Lerngruppen und die damit verbundenen Herausforderungen haben in den letzten Jahren eine erhöhte Aufmerksamkeit in der Schulpraxis und in der Forschung erfahren. Neben der fachlichen Leistung rücken zunehmend auch andere lernrelevante Bereiche der Heterogenität in den Blick, wie etwa die sprachlichen Voraussetzungen der Lernenden. Dass Sprache für den Lernerfolg in Mathematik eine zentrale Rolle spielt, ist vielfach belegt. Doch wo genau liegen die Unterschiede zwischen sprachlich schwachen und sprachlich starken Lernenden und wie können beide Lernendengruppen im Unterricht angemessen berücksichtigt werden? Dieser grundlegenden Frage soll exemplarisch für den Lerngegenstand Brüche nachgegangen werden. Im Vortrag wird die Entwicklung und Evaluation eines adaptiven sprachsensiblen Lernangebotes zur Förderung des konzeptuellen Wissens zu Brüchen beschrieben, das binnendifferenzierend im Unterricht eingesetzt wurde. Aus der Erprobung des Materials werden qualitative und quantitative Ergebnisse vorgestellt, die zeigen, wie sich die sprachliche Unterstützung auf sprachlich stärkere und sprachlich schwächere Lernende auswirkt.