Dr. Blaz Mramor
Oktober 2014 - September 2015
Freiburg Institute for Advanced Studies
79104 Freiburg im Breisgau
In 2003-2004 I did the first year of Mathematics studies at the Faculty of Mathematics and Physics, University of Ljubljana, Slovenia. I continued my BSc studies in 2004-2006 with the program of Mathematics with Physics at Faculty of Mathematics and Physics, Eberhard Karls University, Tübingen, Germany. In 2006-2008 I was a part of the MSc program in Mathematics at the KdV Institute, University of Amsterdam, The Netherlands. I graduated "cum laude" with the MSc-thesis on the topic of unbounded hyperbolic strange attractors, under the supervision of Prof. Dr. A. J. Homburg. I did my PhD in Mathematics in 2008-2012 at FEW, VU University Amsterdam, The Netherlands under the supervision of Dr. B.W. Rink and Prof. Dr. R.C.A.M van der Vorst. The title of my PhD thesis is Monotone Variational Recurrence Relations. During my PhD studies I spent one semester at Boston University in Boston, USA. In the Fall of 2012 I continued my research at FEW, VU University Amsterdam with a Postdoc position in Mathematics. Since April 2013 I have been a postdoc in Mathematics at the Institute for Mathematics, University of Freiburg, Germany under the supervision of Prof. Dr. V. Bangert.
My main research interests are Calculus of variation on Riemannian manifolds, Variational problems on metric groups, Aubry-Mather Theory and Hamiltonian dynamical systems.
- V. Knibbeler, B. Mramor, B. Rink; The laminations of a crystal near an anti-continuum limit. Nonlinearity 27 (2014), no. 5, 927-952.
- B. Mramor, B. Rink; On the destruction of minimal foliations. Proc. Lond. Math. Soc. (3) 108(2014), no. 3, 704-737.
- B. Mramor, B. Rink; A dichotomy theorem for minimizers of monotone recurrence relations. Ergodic Theory and Dynamical Systems, available on CJO2013.
- B. Mramor, B. Rink; Continuity of the Peierls barrier and robustness of laminations . Ergodic Theory and Dynamical Systems, available on CJO2014.
- B. Mramor, B. Rink; Ghost circles in lattice Aubry-Mather theory. J. Differential Equations 252(2012), no. 4, 3163-3208.
Minimizers of nonlinear elliptic PDEs on hyperbolic manifolds
Within the proposed research project, we plan to study partial differential equations (PDEs) on a class of Riemannian manifolds. The differential equations we consider are nonlinear second order elliptic PDEs that arise in a convex variational setting. Specifically, we intend to investigate minimal solutions of such equations, which are functions on the manifold. The direction of the proposed research is twofold. On one hand we plan to further develop Aubry-Mather Theory in this setting, building on the work of Moser, de la Llave, Valdinocci and others. Aubry-Mather Theory establishes the existence of foliation and laminations of minimal solutions. We are especially interested in the question of when laminations and when foliation occur. On the other hand, we are interested in the setting where the underlying manifold is a Cartan-Hadamard manifold. In this setting we plan to investigate the existence of a different type of minimal solutions, which cannot be obtained by the methods of Aubry-Mather theory.