Das Mathematische Kolloquium ist eine gemeinsame wissenschaftliche Veranstaltung des gesamten Mathematischen Instituts. Es steht allen Interessierten offen und richtet sich neben den Mitgliedern und Mitarbeitern des Instituts auch an die Studierenden. Das Kolloquium findet dreimal im Semester am Donnerstag um 17:00 s.t. im Hörsaal II, Albertstr. 23b statt. Danach (gegen 16:15) gibt es Kaffee und Kekse, zu dem der vortragende Gast und alle Besucher eingeladen sind.
Replicating Portfolio Approach to Capital Calculation
Thursday, 13.10.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
The replicating portfolio (RP) approach to the calculation of capital for life insurance portfolios is an industry standard. The RP is obtained from projecting the terminal loss of discounted asset liability cash flows on a set of factors generated by a family of financial instruments that can be efficiently simulated. We provide the mathematical foundations and a novel dynamic and path-dependent RP approach for real-world and risk-neutral sampling. We show that the RP approach yields asymptotically consistent capital estimators. We illustrate the tractability of the RP approach by two numerical examples.
Thursday, 20.10.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 27.10.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 3.11.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 10.11.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 17.11.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Aspherical manifolds, what we know and what we do not know
Thursday, 24.11.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Aspherical closed manifolds arise very often in topology, for\ninstance in low dimensional topology,\nclosed Riemannian manifolds with non-positive sectional curvature and so on. We\nwant to give a survey about open problems (and their status) such as the\nBorel Conjecture about topological rigidity, the Novikov Conjecture about the\ntopological invariance of higher signatures,\nthe Singer Conjecture about the distribution of L^2-Betti numbers,\napproximation of L^2-torsion,\nand the realizablility of Poincare duality groups as fundamental groups of\naspherical closed manifolds. Moreover, we present results about the rational\nhomotopy groups of\nthe group of diffeomorphisms and homeomorphisms of aspherical closed manifolds\nand the problem which hyperbolic groups have the standard sphere as boundary.\n\n\n\n\n\n\n
Thursday, 1.12.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Analysis on Riemannian singular and noncompact spaces and Lie algebroids
Thursday, 8.12.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
After reviewing the definition of a Lie algebroid and a related\nSerre-Swan theorem, I will explain how Lie algebroids can be used to\nmodel simple singularities starting with conical and edge\nsingularities. Then I will explain how the structural algebroid, which\nplays the role of the tangent space, leads to a natural class of\nRiemannian metrics, called "compatible metrics." One of the main\nresults gives a connection between the structure of the Lie algebroid\nand the analysis of the geometric operators associated to a compatible\nmetric (Laplace, Dirac, ... ). This results expresses Fredholm\ncriteria in terms of operators invariant with respect to suitable\ngroups, which allows to use tools from harmonic analysis. These\nresults are part of joint works with B. Ammann, R. Lauter,\nB. Monthubert, and others
Thursday, 15.12.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 22.12.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 29.12.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 5.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 12.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Discrete Alexandroff estimate and pointwise rates of convergence for FEMs
Thursday, 19.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
We derive an Alexandroff estimate for continuous piecewise linear functions which states that the max-norm of their negative part is controlled by the Lebesgue measure of the sub-differential of their convex envelope at the contact nodes. We develop a discrete Alexandroff-Bakelman-Pucci estimate which controls the Lebesgue measure of the sub-differential in terms of the discrete Laplacian via gradient jumps. We further apply these estimates in the analysis of three finite element methods (FEMs).\n\nWe first discretize the Monge-Ampere equation (MA) with a FEM based on the geometric interpretation of MA. We next discretize MA with a two-scale FEM which exploits an eigenvalue representation of the determinant of SPD matrices. We finally present a two-scale FEM for linear elliptic PDEs in non-divergence form. We prove rates of convergence in the max-norm for all three FEMs, study their optimality, and check it computationally.\n\nThis is joint work with D. Ntogkas and W. Zhang.\n
Thursday, 26.1.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Effective behavior of random media: From an error analysis to elliptic regularity theory
Thursday, 2.2.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Heterogeneous media, like a sediment, are often naturally described in statistical terms. \nHow to extract their effective behavior on large scales, like the permeability in Darcy's law, from the\nstatistical specifications? A practioners numerical approach is to sample the medium \naccording to these specifications and to determine\nthe permeability in the Cartesian directions by imposing simple boundary conditions.\nWhat is the error made in terms of the size of this "representative volume element''?\nOur interest in what is called "stochastic homogenization'' grew out of this error analysis.\n\nIn the course of developing such an error analysis, connections with the classical\nregularity theory for elliptic operators have emerged. It turns out that the\nrandomness, in conjunction with statistical homogeneity, of the coefficient field (which can be seen as a Riemannian metric)\ngenerates large-scale regularity of harmonic functions (w.r. t.the corresponding Laplace-Beltrami operator). \nThis is embodied by a hierarchy of Liouville properties: \nAlmost surely, the space of harmonic functions of given but arbitrary growth rate\nhas the same dimension as in the flat (i.e. Euclidean) case. \nClassical examples show that from a deterministic point of view, this Liouville property fails \nalready for a small growth rate:\nThere are (smooth) coefficient fields, which correspond to the geometry of a cone at infinity,\nthat allow for sublinearly growing but non-constant harmonic functions
Thursday, 9.2.17, 17:00-18:00, Hörsaal II, Albertstr. 23b