Parallel spinors, Calabi-Yau manifolds, and special holonomy
Montag, 2.11.20, 16:15-17:15, vSR318 (Kasparov)
Discontinuous Galerkin methods on arbitrarily shaped elements and their application to interface problems.
Dienstag, 3.11.20, 14:15-15:15, Raum 226, virtuell Euwe
Motivated by the problem of numerical treatment of curved boundaries and interfaces in numerical PDEs, will review some recent work on the development of discontinuous Galerkin (dG) methods which are able to be applied on meshes comprising of essentially arbitrarily-shaped elements [1,2]. The use of such element shapes makes possible the fitted representation of curved geometries, by moving the variational crime challenge from the domain representation (as is the case for classical FEM/dG) to the quadrature evaluations. The second part of my talk will focus on the application of these ideas to the specific problem of proof of a posteriori error bounds for elliptic and parabolic interface problems on curved interfaces [3,4,5]. \n\n[1] A. Cangiani, Z. Dong, and E. H. Georgoulis. hp–Version discontinuous Galerkin methods on essentially arbitrarily-shaped elements. Submitted for publication. PDF\n\n[2] A. Cangiani, Z. Dong, E. H. Georgoulis and P. Houston. hp–Version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs in Mathematics (2017)\n\n[3] A. Cangiani, E. H. Georgoulis, and Y. Sabawi. Adaptive discontinuous Galerkin methods for elliptic interface problems. Mathematics of Computation 87(314) pp. 2675 – 2707 (2018) \n\n[4] Stephen A. Metcalfe. Adaptive discontinuous Galerkin methods for nonlinear parabolic problems. PhD Thesis, University of Leicester (2015).\n\n[5] Younis A. Sabawi. Adaptive discontinuous Galerkin methods for interface problems. PhD Thesis, University of Leicester (2017).
Nu-Invariants of Extra Twisted Connected Sums
Montag, 9.11.20, 16:15-17:15, Virtueller SR 318 (Kasparov)
We analyse the possible ways of gluing twisted products of circles\n with asymptotically cylindrical Calabi-Yau manifolds to produce\n manifolds with holonomy \(G_2\),\n thus generalising\n the twisted connected sum construction of Kovalev and Corti,\n Haskins, Nordström, Pacini.\n We then express the extended \(\bnu\)-invariant\n of Crowley, Goette, and Nordström in terms of fixpoint and gluing\n contributions, which include different types\n of (generalised) Dedekind sums.\n Surprisingly, the calculations\n involve some non-trivial number-theoretical arguments connected with\n special values of the Dedekind eta-function and the theory of complex \n multiplication.\n One consequence of our computations is\n that there exist compact \(G_2\)-manifolds that are not \(G_2\)-nullbordant.
Derivation of a bending plate model for nematic liquid-crystal elastomers via Gamma-convergence
Dienstag, 10.11.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract: Liquid-crystal elastomers (LCEs) are a class of materials, whose shape can be controlled via external stimulation. Here, we introduce a three-dimensional model describing the deformations. Its terms include the elastomer's hyperelastic energy (coupled to the liquid-crystal structure) and the liquid-crystal's Oseen-Frank energy. Using Gamma-convergence, we then derive and examine a dimension-reduced model, effectively describing the bending behaviour for thin LCE-plates.
Derivation of a bending plate model for nematic liquid-crystal elastomers via Gamma-convergence
Dienstag, 10.11.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Abstract: Liquid-crystal elastomers (LCEs) are a class of materials, whose shape can be controlled via external stimulation. Here, we introduce a three-dimensional model describing the deformations. Its terms include the elastomer's hyperelastic energy (coupled to the liquid-crystal structure) and the liquid-crystal's Oseen-Frank energy. Using Gamma-convergence, we then derive and examine a dimension-reduced model, effectively describing the bending behaviour for thin LCE-plates.
Verständnisorientierter Stochastikunterricht am Gymnasium: Anforderungen an die Lehrerbildung
Dienstag, 10.11.20, 19:30-20:30, Hörsaal Rundbau, Albertstr. 21a
Der gymnasiale Stochastikunterricht ist momentan von einer starken Rezeptorientierung geprägt, der auch geltende Bildungspläne Vorschub leisten. Zudem bringt ein Großteil der Lehrkräfte entweder keine Stochastik-Kenntnisse aus dem Studium mit, oder diese Kenntnisse sind in erster Linie durch eine rein mathematische Stochastik mit Elementen der Maßtheorie geprägt, wobei insbesondere Kenntnisse der Statistik -- wenn überhaupt -- nur rudimentär vorhanden sind. Stochastik gilt gemeinhin als schwierig, weil sie im Spannungsfeld zwischen Mathematik, Modellbildung und persönlichen Erfahrungen mit stochastischen Vorgängen steht. Im Vortrag stelle ich mein Konzept für eine grundständige, im Wesentlichen auf die Tafel verzichtende Stochastik-Vorlesung vor, die auf den Kenntnissen des ersten Studienjahres aufbaut und obigem Spannungsfeld Rechnung trägt.
Geometric Coding Theory
Freitag, 13.11.20, 10:30-11:30, SR 404
I present a geometric approach to error correcting (quantum) codes. \nBy layering hyperbolic surfaces and cyclic codes in the style of a Lasagne I present families of (quantum) codes with best known asymptotic behavior. The ingredients include Coxeter groups, finite groups of Lie type, fiber bundles and a degenerate spectral sequence. \nThis is a joint project with Nikolas Breuckmann (UCL).
Formoptimierung eindimensionaler Strukturen für den Haupteigenwert zweidimensionaler Gebiete
Dienstag, 17.11.20, 14:15-15:15, Hörsaal II, Virtual raum Lasker
Zusammenfassung: \nBetrachtet man die Eigenschwingungen einer dünnen Membran, so verändern sich diese durch die Anbringung einer eindimensionalen Versteifung. Aus diesem Sachverhalt lässt sich ein Formoptimierungsproblem für den Haupteigenwert der Eigenschwingungen herleiten. Während es noch recht einfach ist die Existenz geeigneter Lösungen nachzuweisen, so ist es umso schwerer Aussagen über die Strukturen optimaler eindimensionaler Mengen zu treffen. Wir werden in diesem Vortrag Eigenschaften der Löser des Formoptimierungsproblems analytisch untersuchen und durch numerische Experimente die Struktur möglicher Optimierer besser kennenlernen.\n
Tag der offenen Tür
Mittwoch, 18.11.20, 11:00-12:00, BigBlueButton. Synchrone und asynchrone Angebote, siehe Webseite (auf Titel klicken)
Bogomolov's inequality and its applications
Freitag, 20.11.20, 10:30-11:30, SR 404
Bogomolov's inequality is an inequality bounding the degree of the second Chern class of a semistable vector bundle on a smooth algebraic variety. I will talk about various applications of this type of result and its possible possible variants in the Chow ring of the variety.\n
Noncommutative differential forms
Freitag, 20.11.20, 14:15-15:15, vSR TF4 (Krush)
Starting with a ring (possibly noncommutative), how can one develop calculus in such a way that, if we start with the ring of functions on an algebraic variety, we get the usual calculus of differential forms? We will do this from the very beginning and without requiring any prior knowledge. Namely, we will start with the basic construction of noncommutative differential forms and explain what has to be added to get a nontrivial theory. We will recover Hochschild and cyclic homology of rings, both in their original version and in the version of Ginzburg and Schedler. We will also show the connection with crystalline cohomology and its generalisation to noncommutative rings. \n
A geometric model for weight variations and wall-crossing on moduli spaces of parabolic Higgs bundles over the Riemann sphere
Montag, 23.11.20, 16:15-17:15, vSR318 (Kasparov)
In this talk I will describe an ongoing project that aims to reconstruct the hyperkähler geometry of Hitchin metrics on moduli spaces of parabolic Higgs bundles over the Riemann sphere in terms of explicit geometric models. By\ndefinition, these moduli spaces depend on a polytope of real parameters called parabolic weights. This dependence induces wall-crossing phenomena, whose incarnation in the models is structurally analogous to a problem of variation of non-reductive GIT-quotients as introduced by Berczi-Jackson-Kirwan. In the smallest possible dimension, these ideas are suited to study the hyperkähler geometry of gravitational instantons of ALG type in terms of the work of Fredrickson–Mazzeo–Swoboda–Weiss.
Neue Ideen zum Einsatz von DGS-Software und Tabellenkalkulationen im Geometrieunterricht der Sekundarstufe I
Dienstag, 24.11.20, 19:30-20:30, Hörsaal Rundbau, Albertstr. 21a
Seit mehr als 30 Jahren werden Vorschläge entwickelt, wie man dynamische Geometriesysteme (DGS) und andere Computerprogramme gewinnbringend im Geometrieunterricht der Sekundarstufe I einsetzen kann. Die Fülle des Materials wird inzwischen unüberschaubar. Dieser Vortrag versucht trotzdem, einige neue Ideen zu diesem Thema vorzustellen, die im Rahmen der Lehrbuchreihe "Mathe 21" entstanden sind.
tba
Freitag, 27.11.20, 10:30-11:30, SR 404
Exponential periods and o-minimality
Freitag, 27.11.20, 10:30-11:30, online: lasker
In this talk I will present on joint work with Philipp\nHabegger and Annette Huber. Let α ∈ ℂ be an exponential period. We show\nthat the real and imaginary part of α are up to signs volumes of sets\ndefinable in the o-minimal structure generated by ℚ, the real\nexponential function and sin|_[0,1]. This is a weaker analogue of the\nprecise characterisation of ordinary periods as numbers whose real and\nimaginary part are up to signs volumes of ℚ-semialgebraic sets; and it\npoints to a relation between the theory of periods and o-minimal\nstructures.\n\nFurthermore, we compare the definition of naive exponential periods to\nthe existing definitions of cohomological exponential periods and\nperiods of exponential Nori motives and show that they all lead to the\nsame notion.
On SU(2)-bundles on 1-connected spin 7-manifolds
Montag, 30.11.20, 16:15-17:15, vSR318 (Kasparov)