Untersuchung des Bruchverhaltens von undehnbaren, dünnen Stäben an Beispielen
Tuesday, 7.1.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In unserer Umgebung kommt eine Vielzahl von langen, dünnen, elastischen Stäben vor, wie zum Beispiel das menschliche Haar oder eine trockene Spaghetti. In diesem Vortrag wird das Bruchverhalten eines Haares bei einer Einfach-, Doppel- oder Dreifachklingenrasur und das Bruchverhalten einer trockenen Spaghetti unter Betrachtung von Krümmung und Torsion untersucht. Hierbei ist die zentrale Leitfrage: Unter welchen Bedingungen bricht die Spaghetti in genau zwei Teile?
A new proof of the Global Torelli Theorem for holomorphic symplectic varieties
Friday, 10.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In a joint work with Benjamin Bakker, we develop a theoretical framework to approach the global moduli theory of certain singular symplectic varieties. Our work is based on new results about the deformation theory of these varieties together with the notion of ergodic complex structures which has been introduced by Verbitsky and used to study for example hyperbolicity questions. I will explain how to use these techniques to prove a Global Torelli theorem for the varieties in question. Our result in particular gives a new proof of Verbitsky's Global Torelli Theorem for irreducible symplectic manifolds as soon as the second Betti number is at least 5.
A new proof of the Global Torelli Theorem for holomorphic symplectic varieties
Friday, 10.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In a joint work with Benjamin Bakker, we develop a theoretical framework to approach the global moduli theory of certain singular symplectic varieties. Our work is based on new results about the deformation theory of these varieties together with the notion of ergodic complex structures which has been introduced by Verbitsky and used to study for example hyperbolicity questions. I will explain how to use these techniques to prove a Global Torelli theorem for the varieties in question. Our result in particular gives a new proof of Verbitsky's Global Torelli Theorem for irreducible symplectic manifolds as soon as the second Betti number is at least 5.
Explicit Kähler packings of projective complex manifolds
Monday, 13.1.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In this talk I will start by introducing the concept of multipoint Seshadri constants and discuss their relationship with Nagata's conjecture on plane curves. I will then introduce the notion of a Kähler packing and show that there is a direct connection between the multipoint Seshardi constant and the existence of Kähler packings. To end I will provide an example of a Kähler packing of a toric variety which highlights a connection between Kähler packings of certain polytopes and their corresponding varieties.
Vortrag abgesagt
Tuesday, 14.1.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Deep Ritz revisited
Tuesday, 14.1.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations. In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In this notes we use notion of Gamma convergence to show that ReLU networks of growing architecture that are trained with respect to suitably regularised Dirichlet energies converge to the true solution of the Poisson problem.
Was ist guter Mathematikunterricht? Unterschiedliche Perspektiven aus Deutschland und Taiwan
Tuesday, 14.1.20, 19:30-20:30, Hörsaal II, Albertstr. 23b
Was guten Mathematikunterricht ausmacht, ist eine der zentralen Fragen, mit der sich die Mathematikdidaktik beschäftigt. Zur Beantwortung dieser Frage kann ein Vergleich von Perspektiven aus unterschiedlichen Kulturen beitragen, denn häufig führt erst ein solcher Vergleich zu einem expliziten Verständnis der eigenen impliziten Theorien und Annahmen. Anhand von konkreten Unterrichtssituationen aus dem Bereich des gymnasialen Mathematikunterrichts werden im Vortrag Beispiele für unterschiedliche Perspektiven reflektiert und diskutiert.
More on trees and Cohen reals, part 2
Wednesday, 15.1.20, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
The talk is a continuation of the topic developed by Brendan\nStuber-Rousselle during the previous Oberseminar, and it is based on our\njoint work. I will go into more details showing how the presence of a Cohen\nreal affects the nature of the ideals of P-nowhere dense and P-meager sets,\nand I will sketch out a proof of the general theorem stating that when a\ntree-forcing P adds Cohen reals under certain reasonable assumption and F is\na well-sorted family of subsets of reals, then P-measurability for all sets\nin F implies the Baire property for all sets in F. If there will be any time\nleft, I will also provide more details about some basic properties of the\nvariant of Mathias forcing introduced in our paper.
More on trees and Cohen reals, part 2
Wednesday, 15.1.20, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
The talk is a continuation of the topic developed by Brendan\nStuber-Rousselle during the previous Oberseminar, and it is based on our\njoint work. I will go into more details showing how the presence of a Cohen\nreal affects the nature of the ideals of P-nowhere dense and P-meager sets,\nand I will sketch out a proof of the general theorem stating that when a\ntree-forcing P adds Cohen reals under certain reasonable assumption and F is\na well-sorted family of subsets of reals, then P-measurability for all sets\nin F implies the Baire property for all sets in F. If there will be any time\nleft, I will also provide more details about some basic properties of the\nvariant of Mathias forcing introduced in our paper.
Deformations of path algebras of quivers with relations
Friday, 17.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk I will present ongoing joint work with Zhengfang Wang on deformations of path algebras of quivers with relations. Such path algebras naturally appear in many different guises in algebraic geometry and representation theory and I would like to explain how one can obtain concrete descriptions of their deformations. For example, deformations of path algebras of quivers with relations can be used to describe deformations of the Abelian category of coherent sheaves on any quasi-projective variety X, deformation quantizations of Poisson structures on affine n-space, or PBW deformations of graded algebras.
The Euler characteristic - an invariant with many incarnations
Monday, 20.1.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The Euler characteristic plays a major role as a topological invariant, for example in Euler's polyhedron theorem. It can also be understood as an analytic invariant. The Poincaré-Hopf-Index theorem builds a bridge between these two realms. Furthermore, it is a geometric invariant as in the theorem of Gauß-Bonnet.\n\nIn this talk, I will focus on homologies and explain the relation between the Euler characteristic and the so-called Betti numbers, which are the rank of the homology groups of an underlying complex. The Euler characteristic is therefore also an invariant of algebraic objects.
Topology of Surfaces with Finite Willmore Energy
Tuesday, 21.1.20, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we care about the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type \(C^{\balpha}\) regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in \(\bmathbb{R}^n\) with finite Willmore energy. Especially, we prove a removability of singularity of multiplicity one surface with finite Willmore energy and a uniqueness theorem of the catenoid under no a priori topological finiteness assumption.\n
Thursday, 23.1.20, 17:00-18:00, Hörsaal II, Albertstr. 23b
19th GAMM-Seminar on Microstructures 24-25.01, 2020
Friday, 24.1.20, 09:00-10:00, Hörsaal II, Albertstr. 23b
Dual complexes of log Calabi-Yau pairs and Mori fibre spaces
Friday, 24.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. They have been finding useful applications for instance in the study of degenerations of projective varieties, mirror symmetry and nonabelian Hodge theory. In particular, Kollár and Xu conjecture that the dual complex of a log Calabi-Yau pair should be a sphere or a finite quotient of a sphere. It is natural to ask whether the conjecture holds on the end products of minimal model programs. In this talk, we will validate the conjecture for Mori fibre spaces of Picard rank two.
TBA
Friday, 24.1.20, 13:00-14:00, Raum 404, Ernst-Zermelo-Str. 1
Friday, 24.1.20, 13:00-14:00, Raum 404, Ernst-Zermelo-Str. 1
Steenrod squares in differential cohomology
Monday, 27.1.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Kuno Fladt und der Mathematikunterricht im Nationalsozialismus
Tuesday, 28.1.20, 19:30-20:30, Hörsaal II, Albertstr. 23b
Über fünfzig Jahre und in drei verschiedenen politischen Systemen wirkte Kuno Fladt (1889-1977) als einflussreicher Mathematikdidaktiker, zuletzt als Honorarprofessor an der Universität Freiburg. Im Vortrag werden sein Leben und sein umfangreiches Werk näher dargestellt. Dabei richtet sich der Fokus auf die Zeit des Nationalsozialismus, in der Fladt als Zeitschriftenherausgeber und Reichssachbearbeiter für Mathematik und Naturwissenschaften im Nationalsozialistischen Lehrerbund (NSLB) eine besondere Rolle spielte. Der Vortrag bietet damit einen Anknüpfungspunkt für die eigene, kritische Auseinandersetzung mit der Geschichte des Mathematikunterrichts.\n
On automorphism groups of fields with operators
Friday, 31.1.20, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In 1993 Lacar showed with model-theoretical techniques that the group of field automorphisms of the complex numbers which fix pointwise the algebraic closure of the rationals is simple, assuming the continuum hypothesis. He later on provided a different proof without assuming CH. There are two main ingredients in Lascar's proof: First, isolating those automorphisms such that the image of a point is algebraic over the point, and secondly, amalgamating field extensions with prescribed automorphisms.\n\nIn this talk, we will present a sketch of Lascar's proof and explain how the techniques can be used in order to determine the simplicity of the automorphism group of algebraically closed fields (in all possible characteristics) with additional structure (such as a derivation or a transformal map, often arising in algebraic dynamical systems). No prior knowledge of model theory or mathematical logic is required for this talk.\n\n\n
Design risk of Constant Proportion Portfolio Insurance
Friday, 31.1.20, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
This paper introduces the notion of design risk into the portfolio insurance literature. It focus on the evaluation of path–dependency/independency of the most widespread portfolio insurance strategies. In particular, we look at constant proportion portfolio insurance (CPPI) structures and compare them to both the classical option based portfolio insurance (OBPI) and naïve strategies such as stop-loss portfolio insurance (SLPI), or a CPPI with a multiplier of one. The paper is based upon conditional Monte Carlo simulations to control for the terminal value of the underlying. We show that even in scenarios where the terminal value of the underlying is several times higher its initial value, CPPIs can get cash-locked. The likelihood of ending up cash-locked increases with the size of the multiplier and the maturity, more than on the properties of the risky underlying’s dynamics. This cash-lock problem is specific of CPPIs, it goes against the European-style nature of traded CPPIs, and it adds to the strategy a risk that is unrelated to the underlying risky asset – a design risk. Design risk does not occur for path-independent portfolio insurance strategies, like in OBPI strategies, nor in naïve strategies. This study contributes to reinforce the idea that bad designing of structure products or investments strategies, may expose investors to undesired risks.\n\nJoint work with João Carvalho and João Beleza Sousa.\n