Thursday, 1.1.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
Thursday, 8.1.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
The Calabi conjecture for QAC geometries
Monday, 12.1.15, 16:15-17:15, Raum 404, Eckerstr. 1
Cardinal characteristics at supercompact kappa in the small u(kappa), large 2^kappa model
Wednesday, 14.1.15, 16:30-17:30, Raum 404, Eckerstr. 1
When generalising arguments about cardinal characteristics of the continuum to cardinals kappa greater than omega, one frequently comes up against the problem of how to ensure that a filter built up through an iterated forcing remains kappa complete at limit stages of small cofinality. A technique of Dzamonja and Shelah is useful for overcoming this problem; in particular, there is a natural application of this technique to obtain a model in which 2^kappa is large but the ultrafilter number u(kappa) is kappa^+. After introducing this model, I will talk about joint work with Vera Fischer (Technical University of Vienna) and Diana Montoya (University of Vienna) calculating many other cardinal characteristics at kappa in the model and its variants.
Thursday, 15.1.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
Good Reduction of K3 Surfaces
Friday, 16.1.15, 10:15-11:15, Raum 404, Eckerstr. 1
By a classical theorem of Serre and Tate, extending previous results of Néron, Ogg, and Shafarevich, an Abelian variety over a p-adic field has good reduction if and only if the Galois action on its first l-adic cohomology is unramified. In this talk, we show that if the Galois action on second l-adic cohomology of a K3 surface over a p-adic field is unramified, then the surface has admits an ``RDP model'' over the that field, and good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction for K3's.) Moreover, we give examples where such an unramified extension is really needed. On our way, we establish existence existence and termination of certain semistable flops, and study group actions of models of varieties. This is joint work with Yuya Matsumoto.\n
Rigidity results for metric measure spaces
Monday, 19.1.15, 16:15-17:15, Raum 404, Eckerstr. 1
Discrete ABP estimate and rates of convergence for linear elliptic PDEs in non-divergence form.
Tuesday, 20.1.15, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We design a two-scale finite element method (FEM) for linear elliptic\nPDEs in non-divergence form. Besides the meshsize, a second larger scale\nis dictated by an integro-differential approximation of the PDE. We show\nthat the FEM satisfies the discrete maximum principle (DMP) provided\nthat the mesh is weakly acute. Combining the DMP and weak operator\nconsistency of the FEM, we establish convergence of the numerical\nsolution to the viscosity solution of the PDE.\n\nWe develop a discrete Alexandroff-Bakelman-Pucci (ABP) estimate which is\nsuitable for finite element analysis. Its proof relies on a geometric\ninterpretation of the Alexandroff estimate and control of the measure of\nthe sub-differential of piecewise linear functions in terms of jumps,\nand thus of the discrete PDE. The discrete ABP estimate leads to optimal\nrates of convergence for our finite element method under natural\nregularity assumptions on the solution and coefficient matrix.
Some Aspects of the Dynamic of V=H−H
Tuesday, 20.1.15, 16:00-17:00, Raum 404, Eckerstr. 1
We consider the evolution of a surface Γ(t) according to the equation V=H−H, where V is the normal velocity of Γ(t), H is the sum of the two principal curvatures and H is the average of H on Γ(t). We study the case where Γ(t) intersects orthogonally a fixed surface Σ and discuss some aspects of the dynamics of Γ(t) under the assumption that the volume of the region enclosed between Γ(t) and Σ is small. We show that, in this case, if Γ(0) is near a hemisphere, Γ(t) keeps its almost hemispherical shape and slides on Σ crawling approximately along orbits of the tangential gradient ∇HΣ of the sum HΣ of the two principal curvatures of Σ. We also show that, if p∈Σ is a nondegenerate zero of ∇HΣ and a>0 is sufficiently small, then there is a surface of constant mean curvature which is near a hemisphere of radius a with center near p and intersects Σ orthogonally.
On the theory of universal specializations of Zariski structures
Wednesday, 21.1.15, 16:30-17:30, Raum 404, Eckerstr. 1
Thursday, 22.1.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
Statistical analysis of modern sequencing data – quality control, modelling and interpretation
Friday, 23.1.15, 11:30-12:30, Raum 404, Eckerstr. 1
Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces
Monday, 26.1.15, 08:45-09:45, Raum 404, Eckerstr. 1
I will present some results on nontriviality of Lipschitz homotopy groups in metric spaces\nsuch as the Heisenberg group and discuss their implications on density of Lipschitz mappings\nin the Sobolev space.
(Spinorielle) Yamabe-artige Konstanten und Bordismenschranken
Monday, 26.1.15, 11:00-12:00, Raum 404, Eckerstr. 1
Wir geben eine Einführung in die Yamabekonstante und deren Umfeld. Die \nYamabekonstante misst, ob eine geschlossene Riemannsche Mannigfaltigkeit eine\nMetrik mit positiver Skalarkrümmung besitzt. Leider kennt man ihren expliziten\nWert nur für wenige Mannigfaltigkeiten. Insbesondere kennt man in Dimension\ngrößer gleich 5 keine nicht einfach zusammenhängende Mannigfaltigkeit mit\nYamabekonstante ungleich Null oder die der Standardsphäre. Ein wichtiger erster\nSchritt um herauszufinden, ob es solche Mannigfaltigkeiten geben kann, ist es\nAbschätzungen für die Yamabekonstante zu finden. Dabei helfen ein spinorieller\nGeschwister der Yamabekonstante und eine Bordismusungleichung für die\nYamabekonstante. Letztere enthält jedoch Threshold-Konstanten –- Yamabe-artige\nKonstanten von nichtkompakten Modellräumen. Wir untersuchen diese\n(spinoriellen) Yamabe-artigen Konstanten und geben Anwendungen für geschlossene\nMannigfaltigkeiten. Das ist zum größten Teil ein gemeinsames Projekt mit Bernd\nAmmann, Regensburg.
Positive Krümmung
Monday, 26.1.15, 14:15-15:15, Raum 404, Eckerstr. 1
Die Beschreibung von Mannigfaltigkeiten positiver Schnittkrümmung ist ein\nausgesprochen klassisches Gebiet der Riemannschen Geometrie. Viele bekannte\nBeispiele, wie Sphären oder projektive Räume, gehören zu den uns vertrautesten\nObjekten in Geometrie und Topologie. Umso mehr mag es verwundern, dass bis\nheute nur vergleichsweise wenige allgemeine Aussagen über die Eigenschaften von\nMannigfaltigkeiten, die eine solche Metrik positiver Krümmung zulassen, bekannt\nsind. Vielmehr bedarf es hierzu häufig Symmetrieannahmen, wie isometrisch\noperierender Lie Gruppen. In diesem Vortrag will ich versuchen, zum einen einen\ngroben Überblick über meine Forschungsinteressen zu geben, um dann zum anderen\nexemplarisch einige Techniken und Resultate am Beispiel des Studiums von\npositiv gekrümmten Mannigfaltigkeiten zu illustrieren. Konkret soll dabei die\nFrage, wie rigide die Struktur von positiv gekrümmten Mannigfaltigkeiten unter\ngewisser Symmetrie sein sollte, aus verschiedenen Blickwinkeln beleuchtet\nwerden. Vorgestellte Ergebnisse entstammen gemeinsamen Projekten mit Lee\nKennard und Wolfgang Ziller.\n
Some analytic and geometric properties of metric measure spaces with lower Ricci curvature bounds
Monday, 26.1.15, 16:30-17:30, Raum 404, Eckerstr. 1
Infinitesimally Hilbertian metric measure spaces with lower Ricci curvature\nbounds, RCD ∗ (K, N)-spaces for short (where K ∈ R stands for the lower bound\non the Ricci curvature and N ∈ [1, +∞] for the upper bound on the dimension)\nconstitute a natural abstract framework where to study Gromov–Hausdorff limits\nof Riemannian manifolds with Ricci lower bounds. After a brief introduction to\nthe topic, in the talk I will report on some recent geometric and analytic\nproperties of these spaces.\n
Wie kann man gegen Minimalflächen fließen
Tuesday, 27.1.15, 08:45-09:45, Raum 404, Eckerstr. 1
Das Problem unter einer gegebenen Nebenbedingung, z.B. zu gegebener Randkurve, eine\nFläche mit minimal möglichem Flächeninhalt zu finden ist ein klassisches Problem der Differentialgeometrie und der Variationsrechnung.\nIn diesem Vortrag untersuchen wir die Frage, wie man mit Hilfe eines geometrischen Flusses\neine gegebene Anfangsfläche so modifizieren kann, dass sie gegen einen solchen Minimierer\noder allgemeiner gegen einen kritischen Punkt des Flächenfunktionals konvergiert.\nFür diesen neuen geometrischen Fluss, den sogenannte Teichmüller harmonic map flow, werden\nwir insbesondere sehen, dass für geschlossene Flächen in nicht positiv gekrümmten Mannigfaltigkeiten Singularitäten ausgeschlossen sind und die globalen glatten Lösungen des Flusses beliebige Anfangsflächen in Minimalflächen, eventuell mit tieferem Genus, ändern oder allgemeiner zerlegen.
Flat superconnections and the loop space
Tuesday, 27.1.15, 12:30-13:30, Raum 404, Eckerstr. 1
The holonomies of a vector bundle with connection give rise to interesting structures on the based and the free loop space, respectively. I will explain how these structures generalize to flat superconnections.\nThe talk is based on ongoing joint work with Camilo Arias Abad (University of Toronto).\n
Q-curvature and GJMS operators on Riemannian manifolds
Tuesday, 27.1.15, 14:45-15:45, Raum 404, Eckerstr. 1
In Riemannian geometry a popular generalisation of the Gaussian curvature to \nmanifolds of dimension greater than 2 is the scalar curvature, but in fact one\ncan define other curvatures on a manifold. Of special interest are those\ncurvatures which behave well under a conformal change of metric, in particular\nwhat is known as Q-curvature, which turns out to be a natural substitute of the\nGaussian curvature in a higher-dimensional version of the Gauss-Bonnet formula.\nSimilarly, conformally covariant differential operators of order greater than 2\nsuch as the GJMS operators are useful generalisations of the conformal\nLaplacian.\n\nIn this context I will state some geometrically relevant model problems related\nto the Q- curvature, discussing its relation to nonlocal equations and an\napplication to the Adams–Moser–Trudinger embedding.\n
On first integrals of the geodesic flow on the Heisenberg Lie group
Thursday, 29.1.15, 14:15-15:15, Raum 404, Eckerstr. 1
Abstract: In the first part we recall the definition of the symplectic structure on nilpotent Lie groups. We apply the information to the Heisenberg Lie group and its quotients. The goal is to find first integrals for the geodesic flow.\n
Über die frühe Geschichte des Mathematischen Forschungsinstituts Oberwolfach
Thursday, 29.1.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
Motives, nearby cycles and Milnor fibers
Friday, 30.1.15, 10:15-11:15, Raum 404, Eckerstr. 1
Let k be a field of characteristic zero, and f a regular function on a smooth quasi-projective algebraic k-variety. By analogy to the work of Igusa, Denef and Loeser have associated with the function f a Zeta function which is a power series with coefficients in a Grothendieck ring of varieties. Using motivic integration, they have shown that this power series is rational and defined, in the Grothendieck ring, an element viewed as a motivic version of the Milnor fiber. An analytic avatar in rigid geometry of the Milnor fiber has also been introduced by Nicaise and Sebag. \n\n In this talk I will explain how the theory of motives and stable homotopy theory may be used to recover these Milnor fibers and relate them. These results are joint work with J. Ayoub and J. Sebag. I will also discuss and illustrate the advantages obtained by working with motives instead of Grothendieck rings via some open questions in birational geometry.
Parameter selection for nonlinear modeling using L1 regularization
Friday, 30.1.15, 11:30-12:30, Raum 404, Eckerstr. 1
Verbindungen zwischen Semiparametrik und Robustheit
Friday, 30.1.15, 11:30-12:30, Raum 404, Eckerstr. 1
Discrete dualities in string theory and automorphic forms
Friday, 30.1.15, 14:15-15:15, Raum 404, Eckerstr. 1
String theory is a candidate quantum completion of Einstein's non-renormalisable theory of general relativity. In flat space, string theory contains a finite number of massless modes (corresponding to the standard gravitational degrees of freedom) and an infinite number of massive excitations. These massive excitations modify the standard gravitational scattering amplitudes of general relativity and are thought to be responsible for the improved high energy behaviour of string theory. However, computing these modifications from first principles in a perturbative fashion is not easy. I will present a different approach to computing the modifications that is based on exploiting so-called discrete duality symmetries of string theory. The method has fascinating links to the theory of automorphic forms and representation theory and also gives a handle on non-perturbative effects.
On maximal inequalities for random processes
Friday, 30.1.15, 16:15-17:15, Hörsaal II, Albertstr. 23b
In the talk we give methods and results to the problem of estimation the expectation of the maximum of a random process until a Markov time. We consider cases of continuous time (standard Brownian motion, skew Brownian motion, Bessel \nprocesses). Also we consider case of discrete time (Bernoulli random walk, its module and others).
On maximal inequalities for random processes
Friday, 30.1.15, 16:15-17:15, Hörsaal II, Albertstr. 23b
In the talk we give methods and results to the problem of estimation the expectation of the maximum of a random process until a Markov time. We consider cases of continuous time (standard Brownian motion, skew Brownian motion, Bessel processes). Also we consider case of discrete time (Bernoulli random walk, its module and others).