Institutsöffentliche Vorstellungsvorträge Didaktik
Dienstag, 10.10.17, 10:00-11:00, Raum 404, Eckerstr. 1
Die Vorträge finden zu folgenden Zeiten statt:\n\n10:00-10:30 Uhr \n11:15-11:45 Uhr \n12:30-13:00 Uhr
Algebraic trees versus metric trees as states of stochastic processes
Donnerstag, 19.10.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
In this talk we are interested in limit objects of graph-theoretic trees\nas the number of vertices goes to infinity. Depending on which notion of\nconvergence we choose different objects are obtained.\n\nOne notion of convergence with several applications in different areas is\nbased on encoding trees as metric measure spaces and then using the\nGromov-weak topology. Apparently this notion is problematic in the\nconstruction of scaling limits of tree-valued Markov chains whenever the\nmetric and the measure have a different scaling regime. We therefore\nintroduce the notion of algebraic measure trees which capture only the tree\nstructure but not the metric distances.\nConvergence of algebraic measure trees will then rely on weak convergence\nof the random shape of a subtree spanned a sample of finite size.\nWe will be particularly interested in binary algebraic measure trees which\ncan be encoded by triangulations of the circle. We will show that in the\nsubspace of binary algebraic measure trees sample shape convergence is\nequivalent to Gromov-weak convergence when we equip the algebraic measure\ntree with an intrinsic metric coming from the branch point distribution.\nWe will illustrate this with the example of a Markov chain arising in\nphylogeny whose mixing behavior was studied in detail by Aldous (2000) and\nSchweinsberg (2001).\n\n (based on joint work with Wolfgang Löhr and Leonid Mytnik)\n
Classification of principal bundles via motivic homotopy theory
Freitag, 20.10.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will talk about recent joint work with Aravind Asok and Marc Hoyois on classification results for principal bundles over smooth affine varieties. The main point will be to explain how the general homotopical results boil down to very concrete examples of interesting octonion algebras.
Flat bundles, \(\bmathbb{R/Z}\)-K-theory and rho invariants
Montag, 23.10.17, 16:15-17:15, Raum 404, Eckerstr. 1
Atiyah, Patodi and Singer constructed the relative K-theory class \([\balpha]\) associated with a flat unitary vector bundle over a closed manifold. \nThis class is related to the spectral invariant rho of a Dirac operator by the so called index theorem for flat bundles, which computes the pairing between \([\balpha]\) and the K-homology class \([D]\) of the Dirac operator.\n\nIn this talk, after introducing the context and the needed tools, we show that \([\balpha]\) admits a canonical construction, using von Neumann algebras and that, as a secondary class, it results from Atiyah's \(L^2\)-index theorem for covering.\n\nTaking an operator algebraic point of view, we show that Atiyah's property can be encoded using KK-theory with real coefficients (which will be introduced). This permits to generalise the constructions of secondary classes of rho-type in the noncommutative setting of a discrete group \(\bGamma\) suitably acting on a \(C^*\)-algebra \(A\).\nBased on joint work with Paolo Antonini and Georges Skandalis.
Transport of a two-phase flow with sharp interface in three dimensions
Dienstag, 24.10.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Eigenschaften der J-Hierarchie und L[E]
Mittwoch, 25.10.17, 16:15-17:15, Raum 404, Eckerstr. 1
Die J-Hierarchie ermöglicht eine alternative Konstruktion des\nkonstruktiblen Universums L und erlaubt es, für jede Klasse E ein Universum\nL[E] zu konstruieren, das - wie L - ein Modell von ZFC ist. In diesem\nVortrag werden wir L[E] und die J-Hierarchie untersuchen. Dabei werden wir\nauch sehen, dass L[E] mit der Forcingerweiterung von L bezüglich E\nübereinstimmt, falls die Menge E ein generischer Filter ist.\n\n
Donnerstag, 26.10.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
K-theory of locally compact modules
Freitag, 27.10.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite different: Instead of a homotopy coherent cone construction in infinity categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact which might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just K-theory.
Trajectorial Models based on Operational Assumptions
Montag, 30.10.17, 14:15-15:15, Raum 125, Eckerstr. 1
We illustrate by example the construction of\none-dimensional models for\noption pricing based on operational and observable features of a\nsingle class of investors and a\nrisky asset. Market models are defined based on a class of investors\ncharacterized by how they operate on financial data leading to\npotential portfolio re-balances.\nOnce observable variables are selected for modeling, necessary conditions\nconstraining these variables and resulting from the operational setup are\nderived. Future uncertainty is then reflected in the construction of\ncombinatorial trajectory spaces satisfying such constraints. In the absence\nof probability assumptions, a minmax methodology is available to price option\ncontracts; numerical results are presented based on worst case estimation of\nparameters.