Beschränkte Familien kohärenter Garben auf Kähler-Mannigfaltigkeiten
Freitag, 1.7.11, 10:00-11:00, Raum 404, Eckerstr. 1
Freitag, 1.7.11, 13:30-14:30, Karlsruhe
Geometry and Physics of K3 surfaces
Montag, 4.7.11, 16:15-17:15, Raum 404, Eckerstr. 1
I talk about two insights from string theory on the issue how to think about the geometry of K3 surfaces.
Legendrian knots and nonalgebraic contact Anosov flows on 3-manifolds
Mittwoch, 6.7.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
We describe a surgery construction in a neighborhood of a transverse Legendrian knot that gives rise to new contact structures preserved by Anosov flows. In particular, this includes examples on many hyperbolic 3-manifolds, and it gives contact Anosov flows that are not quasigeodesic. As a byproduct, this also yields quasigeodesic pseudo-Anosov flows.
Die Wittendeformation für singuläre Räume
Donnerstag, 7.7.11, 15:00-16:00, Hörsaal II, Albertstr. 23b
Motiviert durch Fragestellungen aus der Quantenfeldtheorie schlug Witten in seinem Artikel Supersymmetry and Morse theory (J.Diff.Geom 1982) einen neuen analytischen Beweis der Morseungleichungen vor. Die Wittendeformation spielt eine wichtige Rolle in dem Beweis der Vergleichssätze von Bismut und Zhang zwischen geometrischer und analytischer Torsion. \nIn meinen Arbeiten, die ich in diesem Vortrag vorstellen möchte, habe ich mich mit der Verallgemeinerung von Wittens Ideen für singuläre Räume mit konischen Singularitäten beschäftigt. \nIch werde in meinem Vortrag zuerst die grundlegenden Ideen der Wittendeformation im glatten Kontext erklären. Danach werde ich ihre Verallgemeinerung für den Modellfall von singulären Kurven und stratifizierten Morsefunktionen im Sinne der Theorie von Goresky und MacPherson erklären. Schon in diesem einfachen Modellfall treten im Vergleich zur glatten Theorie neue Phänomene auf. Im letzten Teil des Vortrags werde ich Verallgemeinerungen für höher dimensionale singuläre Räume besprechen. \n
Integral invariants in complex differential geometry
Donnerstag, 7.7.11, 17:00-18:00, Hörsaal II, Albertstr. 23b
I will discuss on some integral invariants which are\ncharacters of the Lie algebras of holomorphic vector fields.\nThere are three interesting sub-cases. 1) obstructions for a\nKähler class to admit a constant scalar curvature metric; \n2) obstruction for a polarized manifold to be asymptotically \nsemi-stable; and 3) invariant for possibly non-Kähler complex \nmanifolds. At the end of this talk, the case 3) is discussed for \ncoverings of compact complex manifolds which typically occur \nfor locally conformally Kähler manifolds
Lagrangian fibrations on hyperkähler manifolds
Freitag, 8.7.11, 10:00-11:00, Raum 404, Eckerstr. 1
Hyperkähler (also called irreducible holomorphic symplectic) manifolds form an important class of manifolds with trivial canonical bundle. One fundamental aspect of their structure theory is the question whether a given hyperkähler manifold admits a Lagrangian fibration. I\nwill report on a joint project with Christian Lehn and Sönke Rollenske investigating the following question of Beauville: if a hyperkähler manifold contains a complex torus T as a Lagrangian submanifold, does it\nadmit a (meromorphic) Lagrangian fibration with fibre T ?
TBA
Freitag, 8.7.11, 10:15-11:15, Raum 404, Eckerstr. 1
Alumni-Tag 2011
Freitag, 8.7.11, 15:00-16:00, Hörsaal Rundbau, Albertstr. 21a
Programm (vorläufig):\n\nBegrüßung durch den Dekan\n\nVerleihung der Alumni-Preise\n\nÜberreichung der Examenszeugnisse, Diplom- und Doktorurkunden\n\nFestvortrag\n\nab 17 Uhr gemütliches Beisammensein im Rahmen des Sommerfestes der Fakultät für Mathematik und Physik im Innenhof des Physikalischen Instituts\n
Rational homology of loop spaces
Montag, 11.7.11, 16:15-17:15, Raum 404, Eckerstr. 1
In the talk, I will explain the main ideas behind the proof of the theorem of Sullivan and Vigue-Poirrier: the Betti numbers of the free loop space of a simply-connected manifold M are unbounded if and only if the cohomology algebra of M needs at least two generators. More precisely, the main goal of the talk is the description of minimal Sullivan models for based and free loop spaces.
Central-Upwind Schemes for Shallow Water Models.
Dienstag, 12.7.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
I will first give a brief review on simple and robust central-upwind\nschemes for hyperbolic conservation laws.\n\nI will then discuss their application to the Saint-Venant system of\nshallow water equations. This can be done in a straightforward manner, but\nthen the resulting scheme may suffer from the lack of balance between the\nfluxes and (possibly singular) geometric source term, which may lead to a\nso-called numerical storm, and from appearance of negative values of the\nwater height, which may destroy the entire computed solution. To\ncircumvent these difficulties, we have developed a special technique,\nwhich guarantees that the designed second-order central-upwind scheme is\nboth well-balanced and positivity preserving.\n\nFinally, I will show how the scheme can be extended to the two-layer\nshallow water equations and to the Savage-Hutter type model of submarine\nlandslides and generated tsunami waves, which, in addition to the\ngeometric source term, contain nonconservative interlayer exchange terms.\nIt is well-known that such terms, which arise in many different multiphase\nmodels, are extremely sensitive to a particular choice their numerical\ndiscretization. To circumvent this difficulty, we rewrite the studied\nsystems in a different way so that the nonconservative terms are\nmultiplied by a quantity, which is, in all practically meaningful cases,\nvery small. We then apply the central-upwind scheme to the rewritten\nsystem and demonstrate robustness and superb performance of the proposed\nmethod on a number numerical examples.
Vollständige Ricci-flache Kählermetriken auf offenen komplexen Flächen
Dienstag, 12.7.11, 16:15-17:15, Raum 127, Eckerstr. 1
Ich werde eine neue Konstruktion von nichtkompakten vollständigen Ricci-flachen 4-Mannigfaltigkeiten vorstellen, die auf der Geometrie elliptisch gefaserter algebraischer Flächen beruht. Eine weitere Zutat ist die Asymptotik uniform beschränkter Lösungen der komplexen Monge-Ampere-Gleichung. Die resultierenden Geometrien beinhalten beispielsweise Familien von sogenannten ALG-Räumen, deren Existenz von Physikern vorhergesagt wurde. Ich möchte auch kurz besprechen, inwieweit diese Konstruktion - zusammen mit anderen, schon bekannten - vermutlich alle 4-dimensionalen Beispiele abdeckt.
Einblicke in die Geschichte der Mathematik im islamischen Orient
Dienstag, 12.7.11, 19:30-20:30, Hörsaal II, Albertstr. 23b
Im islamischen Orient erfolgte die Aneignung der\nvorislamischen mathematischen Traditionen in Form eines langwierigen und\nüber weite Strecken auch problematischen Prozesses. Die wichtigsten Etappen\ndieses Prozesses sollen im ersten Teil des Vortrages beschrieben werden. Der\nzweite Teil wird sich mit zwei ganz verschiedenen arabischen Rechentexten\n(beide 10. Jh.) befassen und anhand einer kursorischen Wiedergabe der\nInhalte den komplexen Charakter der Rolle der Mathematik im Orient\nherausarbeiten. Der dritte Teil soll mit einigen vergnüglichen und\nunterhaltsamen, immer aber ernstzunehmenden und z.T. ganz unbekannten\nRechenproblemen unterschiedlicher Herkunft den Grad der Herausbildung einer\ngenuinen und produktiven vormodernen islamischen Wissenschaft der Mathematik\nbelegen.
Numerical schemes for short wave long wave interaction equations
Mittwoch, 13.7.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
We prove convergence of some finite volume type numerical schemes for short wave long wave interaction equations. These are systems of coupled nonlinear PDEs consisting of a Schroedinger equation for the short waves and a hyperbolic conservation law modeling the long waves. We use compensated compactness along with energy estimates. We present some computations and a numerical study of some open problems. This is joint work with M. Figueira of CMAF-UL.\n
Donnerstag, 14.7.11, 18:00-19:00, Hörsaal II, Albertstr. 23b
Sharp bounds on the denominators of the moduli part in the canonical bundle formula
Freitag, 15.7.11, 10:00-11:00, Raum 404, Eckerstr. 1
The canonical bundle formula for a fibration f from (X,B)\nto Y consists in writing KX+B as the pullback of a sum of\nQ-divisors on Y, more precisely KX+B is the pullback of KY+D+M where KY is the canonical divisor, D contains some informations on the singular fibres and M is called moduli part.\n\nIt has been conjectured by Prokhorov and Shokurov that if\nthe fibres of f are curves then 12rM is base point free, where r is the Cartier index of the fibre. The conjecture in particular implies that 12rM has integer coefficients.\nIn this talk we will give a counterexample to the conjecture and we will give a sharp bound depending on r for the integer m such that mM has integer coefficients.
Rational homology of loop spaces II
Montag, 18.7.11, 16:15-17:15, Raum 404, Eckerstr. 1
The Gradient Flow Of O’Hara’s Knot Energies
Mittwoch, 20.7.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
All of us know how hard it can be to decide whether the cable spaghetti lying in front of us is really knotted or whether the knot vanishes into thin air after pushing and pulling at the right strings. In this talk we approach this problem using gradient flows of a family of energies introduced by O’Hara in 1991. We will see that this allows\nus to transform any closed curve into a special set of representatives - the stationary points of these energies - without changing the type of knot. We prove longtime existence and smooth convergence to stationary points for\nthese evolution equations.\n
Clopen graphs and their finite induced subgraphs.
Mittwoch, 20.7.11, 16:15-17:15, SR 318, Eckerstr. 1
Donnerstag, 21.7.11, 17:00-18:00, Hörsaal II, Albertstr. 23b
TBA
Freitag, 22.7.11, 10:00-11:00, Raum 404, Eckerstr. 1
Ricci Curvature Comparison for Lorentzian Manifolds
Freitag, 22.7.11, 14:15-15:15, Hörsaal Weismannhaus
Ricci curvature (of a Lorentzian manifold) is the central curvature quantity in General Relativity. For instance, in the form of so-called Energy Conditions it is a main ingredient of the famous Penrose-Hawking Singularity Theorems.\n\nIn Riemannian geometry, on the other hand, it is well-known that Ricci curvature controls volume growth of geodesic balls. This is the content of the famous Bishop-Gromov theorem.\n\nIn this talk, I will explain similar volume comparison results for Lorentzian manifolds and show how they can be used to prove Hawking's singularity theorem.
Yano-Obata conjecture for holomorph-projective transformations
Montag, 25.7.11, 16:15-17:15, Raum 404, Eckerstr. 1
The basic geometric structure in holomorph-projective geometry is the family of holomorphically-planar curves that are associated to a given Kaehler metric. Such curves can be viewed as some generalisation of geodesics on Kaehler manifolds. In this context, one question of interest is if the group of all holomorph-projective transformations (i.e. all bi-holomorphic mappings that preserve the set of all holomorphically-planar curves) is really bigger than the group of holomorphic isometries of the Kaehler metric. Together with V. S. Matveev, we have proven a classical conjecture attributed to Yano and Obata: On a compact Kaehler manifold which cannot be covered by the complex projective space with (a multible of the) Fubini-Study metric, the connected components of the group of holomorph-projective transformations and holomorphic isometries coincide.
Hybrid DG Methods for Incompressible Flow.
Dienstag, 26.7.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We propose a class of hybrid discontinuous Galerkin methods for the numerical\n solution of incompressible flow problems. Our approach yields consistent and locally\n conservative discretizations, and allows to treat non-conforming h and p-refinements\n in a natural way.\n\nBy explicit construction of a Fortin-operator, we show that the proposed methods\n are inf-sup stable with a constant which is independent of the meshsize, and only\n slightly dependent on the polynomial degree. This allows to derive a-priori error\n estimates for the Stokes problem, which are optimal in the mesh size and only\n slightly sub-optimal in the polynomial degree.\n\nWe also discuss the extension to stationary Oseen and Navier-Stokes equations, and illustrate our theoretical results with numerical tests.\n\n
The Newtonian Limit of Geometrostatics
Dienstag, 26.7.11, 16:15-17:15, Raum 127, Eckerstr. 1
Geometrostatics is the geometric theory of static isolated relativistic systems modeling for example static stars or black holes in our universe. It is governed by a system of (mainly) elliptic PDEs relating a 3-dimensional Riemannian metric and a positive function describing the geometry of space at any given point of time and the passage of time, respectively. While these PDEs have been intensively studied in the last century from both analytic and physical viewpoints, their geometric nature had not been exploited to its full extent. We will present a number of new insights into geometrostatics that mainly rely on a geometric approach but are also interrelated with analytic and physical aspects of the theory. For example, we will discuss geodesics of the corresponding Lorentzian spacetimes, uniqueness questions, mass and center of mass of geometrostatic systems, and last but not least their Newtonian limit -- the question of whether and how these\nrelativistic systems have Newtonian counterparts.
Ein Vergleich lokal analytischer Gruppenkohomologie mit Lie algebren Kohomologie für p-adische Lie Gruppen
Mittwoch, 27.7.11, 14:00-15:00, Raum 404, Eckerstr. 1
Definierbarkeit in abstrakter Kummertheorie
Mittwoch, 27.7.11, 16:15-17:15, Raum 318, Eckerstr. 1
Donnerstag, 28.7.11, 17:00-18:00, Hörsaal II, Albertstr. 23b
Calculating the arithmetic volume of certain Shimura varieties using Borcherds theory.
Freitag, 29.7.11, 10:00-11:00, Raum 404, Eckerstr. 1
The talk will be about the verification of parts of Kudla's\nconjectures on arithmetic theta functions, in particular of\ntheir relation to derivatives of Eisenstein series in\narbitrary dimensions. This approach uses an extended Arakelov theory, the theory of Borcherds products, and a functorial theory of integral canonical models of toroidal compactifications of Shimura varieties. Kudla's conjectures arose to conceptually understand the mechanism in Gross- and Zagier's approach to the Birch and Swinnerton-Dyer conjecture.