Vom Jungbrunnen zur Regeneration : Nummelin-Splitting für Harris-rekurrente Markovprozesse in stetiger Zeit und Grenzwertsätze für additive Funktionale
Thursday, 7.1.10, 17:00-18:00, Hörsaal II, Albertstr. 23b
Die Technik des "Nummelin-Splittings" ist 1978 von Nummelin und\nAthreya-Ney\nfür Harris-rekurrente Markov-Ketten eingeführt worden.\nSie erlaubt, Regenerationszeiten zu definieren und die Trajektorie der\nKette in i.i.d.-Zyklen zu zerlegen,\nso dass Grenzwertsätze wie Ergodensatz, Gesetz des iterierten Logarithmus\netc. Folgerungen der bekannten Sätze im i.i.d. Fall sind.\nWir verallgemeinern diese Technik auf den zeitstetigen Fall und stellen im\nVortrag die Konstruktion eines "größeren" Markovprozesses vor,\nder rekurrente Atome (Regenerationszeiten) besitzt und dessen erste\nKomponente eine Version des ursprünglichen Prozesses ist.\nAls Anwendungen betrachten wir das Gesetz des iterierten Logarithmus für\nadditive Funktionale und Konzentrationsungleichungen.\n
Higher direct images of the structure sheaf in positive characteristic
Friday, 8.1.10, 11:15-12:15, SR 127, Eckerstr. 1
Minimal curvature trajectories: Riemannian geometry concepts for computing slow attracting manifolds in chemical kinetics
Monday, 11.1.10, 16:15-17:15, Raum 404, Eckerstr. 1
Trends in Nichtlinearer Gemischt-Ganzzahliger Optimalsteuerung
Thursday, 14.1.10, 17:00-18:00, Hörsaal II, Albertstr. 23b
In model-based nonlinear optimal control switching decisions that can be \noptimized often play an important role. Prominent examples of such hybrid \nsystems are gear switches for transport vehicles, traffic lights, or on/off \nvalves in engineering. Optimization algorithms need to take the discrete \nnature of the variables that model these switching decisions into account.\n
\nMixed-integer optimal control problems (MIOCPs) include features related to \ndifferent mathematical disciplines. Hence, it is not surprising that distinct \napproaches have been proposed to analyze and solve them. There are at least \nthree generic approaches to solve model-based optimal control problems: first, \nsolution of the Hamilton-Jacobi-Bellman equation and in a discrete setting \nDynamic Programming, second indirect methods, also known as the first \noptimize, then discretize approach, and third direct methods (first optimize, \nthen discretize) and in particular all--at--once approaches that solve the \nsimulation and the optimization task simultaneously. The combination with the \ncombinatorial restrictions on control functions comes at different levels: for \nfree in dynamic programming, as the full control space is explored anyhow, by \nmeans of an enumeration in the inner optimization problem of the necessary \nconditions of optimality in Pontryagin's maximum principle, or by various \nmethods from integer programming in the direct methods. We will survey some of \nthese approaches.\n
\nWe will mention several extensions that have been made possible by recent \nadvances. They include an extension to multiple objective optimization, \nnonlinear model predictive control in real time, and the efficient treatment \nof switching constraints.\n
\nWe conclude by pointing out future challenges for process control with \nswitching decisions, among them the availability of well-defined test \ninstances for algorithm developpers.
Defect via logarithmic differential forms
Friday, 15.1.10, 11:15-12:15, SR 127
Quickest detection problems
Monday, 18.1.10, 15:15-16:15, Raum 232, Eckerstr. 1
Naturally reductive pseudo-Riemannian nilpotent Lie groups
Monday, 18.1.10, 16:00-17:00, Raum 404, Eckerstr. 1
Exotic function spaces and (their use in the theory of) integration by compensation : Lect 5
Tuesday, 19.1.10, 16:15-17:15, Raum 404, Eckerstr. 1
Kompetenzorientierte und technologiegestützte Konzepte im Mathematikunterricht
Tuesday, 19.1.10, 19:30-20:30, Hörsaal II, Albertstr. 23b
Im Zentrum des Vortrages stehen theoretisch begründete Vorstellungen zum langfristigen mathematischen Kompetenzaufbau in Verbindung mit Technologieeinsatz in den Sekundarstufen, die derzeit in den Projekten CAliMERO und MABIKOM in Niedersachsen materialgestützt umgesetzt und erprobt werden. Berichtet werden Beispiele zu den einzelnen Elementen des Unterrichtskonzeptes sowie empirische Ergebnisse, beispielsweise zur Entwicklung der rechnerfreien Fähigkeiten über mehrere Schuljahre mit CAS-Einsatz (ab Klassenstufe7). Das sogenannte "Rechnerpotenzial" von einzelnen Aufgaben wird diskutiert in Verbindung mit dem potenziellen Mehrwert des Technologieeinsatzes für die Kompetenzentwicklung und den aktuellen Schwierigkeiten bei der Umsetzung. \n
Regularity of optimal transportation maps on compact locally nearly spherical manifolds
Wednesday, 20.1.10, 16:15-17:15, Hörsaal II, Albertstr. 23b
Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold M, we look for a smooth optimal transportation map G, pushing one measure to the other at a least total squared\ndistance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge-Ampere type satisfied by the\npotential function u, such that G=exp(grad u). This approach boils down to proving an a priori upper bound on the Hessian of u. In this talk, based\non the recent local C^2 estimate of MaTrudinger-Wang, we treat the case\nof manifolds with curvature sufficiently close to 1 in C^2 norm.
Model completion of varieties of Heyting algebras
Thursday, 21.1.10, 09:00-10:00, Raum 318, Eckerstr. 1
Optimal rebalancing a portfolio in the disordered Black-Sholes model
Thursday, 21.1.10, 11:00-12:00, Raum 232, Eckerstr. 1
Thursday, 21.1.10, 17:00-18:00, Hörsaal II, Albertstr. 23b
Gorenstein Liaison and determinantal scheme
Friday, 22.1.10, 11:15-12:15, SR 125, Eckerstr. 1
The theory of liaison or linkage formally started in the \nseventies, although it had been used before in an hoc manner. Roughly \nspeaking, liaison aims at understanding the class of projective schemes, \nby partitioning it into families of schemes (the liaison classes) that \ncan all be ultimately ``linked'' to the same scheme. A linkage step \nconsists of taking the union of the scheme that we study with another \none, so that the union belongs to a well-studied family of schemes \n(complete intersections or arithmetically Gorenstein schemes). In an \nideal situation, the scheme that we study is linked to one that we \nunderstand better, and their union is simpler than each of the two \nparts. In this talk, we will introduce the concept of liaison and \ndiscuss its relevance. Many varieties which are classically studied in \nalgebraic geometry are defined by determinantal equations. We will give \nan overview of some results about the linkage class of schemes cut out \nby minors and their applications.
Hamiltonian group actions in complex geometry
Friday, 22.1.10, 14:00-15:00, HS II, Physik-Hochhaus
Symplectic reduction is one of the main principles of Hamiltonian mechanics in the presence of group actions: given a Hamiltonian system with symmetries it produces a new, smaller system (a so-called reduced phase space and a reduced Hamiltonian) by factoring out the symmetries. This theory becomes especially rich in the context of complex geometry, i.e., for Hamiltonian actions of Lie groups on Kaehler manifolds, and it has many important consequences both in mathematics and physics. I will explain how to endow reduced phase spaces with compatible complex stuctures that make the reduced symplectic structure into\na Kaehler structure. I will demonstrate this process with the help of examples.\nFurthermore, I will also look at the theory from a slightly di erent point of view focussing on holmorphic symmetries of complex manifolds and algebraic varieties. If time permits I will try to explain how this ts into schemes of so-called "Geometric Quantisation", discuss the "Quantisation commutes with Reduction"-Conjecture and indicate how all this can be used to easily evaluate certain oscillatory integrals using localisation formulas.
Hamiltonian group actions in Complex Geometry
Friday, 22.1.10, 14:15-15:15, Hörsaal II, Physikhochhaus
Symplectic reduction is one of the main principles of\nHamiltonian mechanics in the presence of group actions:\ngiven a Hamiltonian system with symmetries it produces a\nnew, smaller system (a so-called reduced phase space and a\nreduced Hamiltonian) by factoring out the symmetries. This\ntheory becomes especially rich in the context of\ncomplex geometry, i.e., for Hamiltonian actions of Lie\ngroups on Kaehler manifolds, and it has many important\nconsequences both in mathematics and physics. I will\nexplain how to endow reduced phase spaces with compatible\ncomplex stuctures that make the reduced symplectic\nstructure into a Kaehler structure. I will demonstrate this\nprocess with the help of examples. Furthermore, I will also\nlook at the theory from a slightly different point of view\nfocussing on holmorphic symmetries of complex manifolds and\nalgebraic varieties. If time permits I will try to explain\nhow this fits into schemes of so-called Geometric\nQuantisation, discuss the Quantisation commutes\nwith Reduction-Conjecture and indicate how all this can be\nused to easily evaluate certain oscillatory integrals using\nlocalisation formulas.
tba
Monday, 25.1.10, 16:15-17:15, Raum 404, Eckerstr. 1
Erhaltungsgleichungen auf Mannigfaltigkeiten.
Tuesday, 26.1.10, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
T.b.a
Tuesday, 26.1.10, 16:15-17:15, Raum 404, Eckerstr. 1
Model completion of varieties of Heyting algebras II
Thursday, 28.1.10, 09:00-10:00, Raum 318, Eckerstr. 1
Harmonic maps and the Bernstein problem
Thursday, 28.1.10, 17:00-18:00, Hörsaal II, Albertstr. 23b
The minimal surface equation is one of the classical\nequations of mathematics, and it constitutes an important model in\nmany applications. In this regard, the Bernstein problem, i.e., to\nshow that there is no other minimal graph over all of Euclidean space\nthan a hyperplane, has been one of the guiding problems of geometric\nanalysis. In this talk, I shall explain the approach to this problem\nvia Gauss maps, and I shall present new results obtained in\ncooperation with Ling Yang and Yuanlong Xin. These results involve\nconvexity properties of spheres and the regularity theory for\nnonlinear partial differential equations.\n
Differenzen-abgeschlossene und pseudo-endliche Körper
Friday, 29.1.10, 11:00-12:00, SR 125
Robust partial correlation graphs
Friday, 29.1.10, 11:15-12:15, Raum 404, Eckerstr. 1
Graphical models allow a simple graphical visualization of the (conditional) dependence structure among multiple variables: each variable is represented by a vertex, and conditional dependence between a pair of variables given all the other variables is illustrated by connecting the corresponding pair of vertices by an edge. Within the Gaussian framework, conditional independence is equivalent to zero partial correlation, i.e. we just need to estimate the partial correlation between each pair of variables and test whether it is zero or not in order to decide on the inclusion of edges in the graph. The arising diagram can thus be called a partial correlation graph.\n\nIn this talk we treat two extensions of Gaussian graphical models. Brillinger (1996) and Dahlhaus (2000) suggest to explore the linear dependence structure among multivariate time series by analyzing the partial spectral coherences between the component processes. These are a natural generalization of the partial correlations to the frequency domain. Fried and Didelez (2003) show how to perform stepwise model selection in this context by estimating the partial spectral coherences from suitably chosen subsets of the component processes.\n\nAnother generalization of Gaussian graphical models are elliptical graphical models, that is, we allow the population distribution to be elliptical instead of normal. We examine the class of affine equivariant scatter estimators and show how they can be used to derive generalizations of classical Gaussian graphical modelling tools derived from the empirical covariance matrix and the adjusted deviance tests. We demonstrate the feasibility of our approach by a simulation study, using, among others, Tyler's scatter estimator (Tyler, 1987), which is distribution-free within the elliptical model. This technique is in particular suited to robustify the established, likelihood-based Gaussian graphical modelling methods, which are known to be very sensitive to model misspecifications and outlying observations. Some of the results are summarized in Vogel and Fried (2009).\n\nThe robust fitting of partial correlation graphs to multivariate time series data, e.g. by extending the ideas of elliptical graphical modelling to the time series context, is the scope of future research.\n\n \nReferences\n\n[1] D. R. Brillinger. Remarks concerning graphical models for time series and point processes. Revista de Econometria, 16:1-23, 1996.\n\n[2] R. Dahlhaus. Graphical interaction models for multivariate time series. Metrika, 51:157-172, 2000.\n\n[3] R. Fried and V. Didelez. Decomposability and selection of graphical models for multivariate time series. Biometrika, 90: 251-267, 2003.\n\n[4] D. E. Tyler. A distribution-free Mestimator of multivariate scatter. Annals of Statistics, 15:234-251, 1987.\n\n[5] D. Vogel and R. Fried. On robust Gaussian graphical modelling. Discussion Paper 36/2009, SFB 823, Technische UniversitÄat Dortmund, 2009.