Fachbereich Mathematik

Sommersemester 2018

Vortragsreihe - Geometric flows and singularity formation

Dozent: Prof. Dr. Simon Brendle



Beschreibung


In these lectures, I will discuss the formation of singularities for embedded, mean convex hypersurfaces evolving under mean curvature flow. Among other things, I will discuss the main a-priori estimates, and some recent results on the classification of singularity models. For example, for two-dimensional surfaces in R^3 (or for uniformly two-convex hypersurfaces in higher dimensions), the only possible singularity models are the shrinking spheres, shrinking cylinders, and the rotationally symmetric bowl soliton.






Geometrische Ungleichungen


Dozent: Prof. Dr. Gerhard Huisken 
Beginn: Freitag, den 20. April 2018
Ort: Hörsaal N15 / M2
Zeit: Freitags, 10 Uhr c.t. bis 12 Uhr


Beschreibung / Description


The course investigates geometric inequalities that arise from geometric variational problems and are important for the understanding of geometric partial differential equations. Inequalities that will be explored are estimates on the first eigenvalue of the Laplace operator under natural assumptions on the underlying domain, the proof of the fundamental gap conjecture concerning the first two eigenvalues of the Schrödinger operator on a convex domain, and the positivity of mass in General Relativity with its relation to conformal geometry.


Die Vorlesung untersucht geometrische Ungleichungen, die aus geometrischen Variationsproblemen entstehen und bei der Untersuchung von Lösungen geometrischer partieller Differentialgleichungen eine zentrale Rolle spielen. Insbesondere sollen Abschätzungen an den ersten Eigenwert des Laplace Operators bewiesen werden unter verschiedenen Annahmen an das zu Grunde liegende Gebiet, darüber hinaus der Beweis der "fundamental gap conjecture" sowie der Beweis der Positivität der Masse in der Allgemeinen Relativitätstheorie in Zusammenhang mit der konformen Geometrie Riemannscher Mannigfaltigkeiten.


Voraussetzungen / Prerequisites


Knowledge of differential equations and differential geometry comparable to one course each.


Grundlagen in partiellen Differentialgleichungen und Differentialgeometrie im Umfang von je einer Vorlesung.


Literatur


 


    

  • D. Gilbarg; N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 3rd ed 1998.
    L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
    • 

  • K. Ecker, Regularity theory of mean curvature flows, Birkhäuser Verlag Basel, 2004.
    • 

  • S. Brendle, Ricci Flow and the Sphere Theorem (Graduate Studies in Mathematics), American Mathematical Society, 2010.
    • 

  • G. Lieberman, Second order parabolic differential equations, World Scientific, 1996.
    D. Kinderlehrer; G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, SIAM, 2000.
    • 

  • R. Schoen; S.-T. Yau, Lectures in Differential Geometry, International Press, 1994.
    • 

  • M. Ritore; C. Sinestrari, Mean curvature flow and isoperimetric inequalities, Advanced courses in mathematics CRM Barcelona, Birkhäuser, 2000.
    • 

  • L. Simon, Lectures on Geometric Measure Theory, Australian National University, 1993.
    • 



 


Modulhandbuch


Modulcode: MAT6016V (Masterstudenten Mathematik and Math. Physics)


Prüfungsgebiet: Reine Mathematik


 


Studien- und Prüfungsleistungen / Type of exam


Written or oral exam depending on course size.


Je nach Größe der Veranstaltung gibt es eine Klausur oder mündliche Prüfung.


 


Mathematical Relativity


Professor: JProf. Dr. Carla Cederbaum
Time and place: Tue and Thu, 4.15 pm to 6 pm, N16; starting Apr 17


Teaching assistant: Sophia Jahns, jahns@math.uni-tuebingen.de


Description




After a short introduction to Special Relativity and its underlying Minkowskian geometry, we will study


general Lorentzian manifolds and the Einstein equations of General Relativity.


 


One part of the lecture course will focus on static solutions of the Einstein equation, describing space-


times that are in a state of equilibrium. These solutions are geometrically rather simple and therefore


suitable for a first approach to geometric, analytic, and physical questions about spacetimes and iso-


lated systems. In particular, we will prove the Bunting–Masood-ul-Alaam static black hole uniqueness


theorem.


In the second part, we will investigate causality, cosmological models, and the Big Bang, specifically


the Penrose–Hawking singularity theorems.


Requirements


Geometry in Physics or Differential Geometry or Mathematische Physik: Klassische Mechanik


Useful, but not required: Linear PDEs


Literature


    

  • R. M. Wald, General Relativity, The University of Chicago Press (1984)
    • 

  • H. Fischer und H. Kaul, Mathematik für Physiker, Band 3, Springer Spektrum, 3. Auflage (2013)
    • 

  • B. O’Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, Math. 103
    • 

  • S. W. Hawking und G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics (1973)
    • 



Exam


To be admitted to the exam, you will need to get 50% of all points on the exercise sheets (including the project theses, see below). Depending on the number of participants, the exam will be written or oral.


Project theses


In the week of May 14 – 18, the participants will be asked to write little project theses about classicals result in GR instead of solving exercises. The project theses will count like two exercise sheets.


Tutorials


Time and place to be determined in the first lecture.