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 15:00 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.
Vom Jungbrunnen zur Regeneration : Nummelin-Splitting für Harris-rekurrente Markovprozesse in stetiger Zeit und Grenzwertsätze für additive Funktionale
Donnerstag, 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
Trends in Nichtlinearer Gemischt-Ganzzahliger Optimalsteuerung
Donnerstag, 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.
Donnerstag, 21.1.10, 17:00-18:00, Hörsaal II, Albertstr. 23b
Harmonic maps and the Bernstein problem
Donnerstag, 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