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.
Algebraic trees versus metric trees as states of stochastic processes
Donnerstag, 19.10.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
In this talk we are interested in limit objects of graph-theoretic trees\nas the number of vertices goes to infinity. Depending on which notion of\nconvergence we choose different objects are obtained.\n\nOne notion of convergence with several applications in different areas is\nbased on encoding trees as metric measure spaces and then using the\nGromov-weak topology. Apparently this notion is problematic in the\nconstruction of scaling limits of tree-valued Markov chains whenever the\nmetric and the measure have a different scaling regime. We therefore\nintroduce the notion of algebraic measure trees which capture only the tree\nstructure but not the metric distances.\nConvergence of algebraic measure trees will then rely on weak convergence\nof the random shape of a subtree spanned a sample of finite size.\nWe will be particularly interested in binary algebraic measure trees which\ncan be encoded by triangulations of the circle. We will show that in the\nsubspace of binary algebraic measure trees sample shape convergence is\nequivalent to Gromov-weak convergence when we equip the algebraic measure\ntree with an intrinsic metric coming from the branch point distribution.\nWe will illustrate this with the example of a Markov chain arising in\nphylogeny whose mixing behavior was studied in detail by Aldous (2000) and\nSchweinsberg (2001).\n\n (based on joint work with Wolfgang Löhr and Leonid Mytnik)\n
Donnerstag, 26.10.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 2.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 9.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 16.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 23.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 30.11.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 7.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 14.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Complete convex surfaces: intrinsic and extrinsic properties
Donnerstag, 21.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
We will examine convex surfaces which divide the ambient Euclidean 3-space into two parts. By the Theorem of Gauss-Bonnet such surfaces need to be of the intrinsic type of a cylinder, plane, or sphere. We will then discuss the extrinsic symmetry property of rotational invariance, and its infinitesimal version at a point of the surface. We outline the proof of a global extrinsic conjecture of Victor Andreevich Toponogov : "Any convex plane admits (at least) one point of infinitesimal symmetry, possibly at infinity". The proof, in collaboration with Brendan Guilfoyle, uses complex analysis and a parabolic curvature flow in the space of lines of Euclidean 3-space.
Donnerstag, 4.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Donnerstag, 11.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
INVERSE CURVATURE FLOWS AND GEOMETRIC INEQUALITIES
Donnerstag, 18.1.18, 16:00-17:00, Hörsaal II, Albertstr. 23b
In recent years curvature flows have played a crucial role in proving important geometric theorems. For instance, the Ricci flow lead to a proof of the Poincare conjecture, and the inverse mean curvature flow (IMCF) was crucial in the proof of the Riemannian Penrose inequality. In this talk we present further applications of the IMCF. First we review, how classical geometric inequalities, such as the Minkowski inequality for closed convex hypersurfaces, can be generalised to a wider class of hypersurfaces using curvature flows. Secondly, we present new estimates for a Willmore-type energy of hypersurfaces with boundary, satisfying a perpendicular Neumann-type condition on the unit sphere. The crucial ingredient is the IMCF with boundary conditions.
Homologie linearer Gruppen und die Vermutung von Quillen
Donnerstag, 18.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Invariance of closed convex cones for stochastic partial differential equations
Donnerstag, 25.1.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
The goal of this talk is to clarify when a closed convex cone is\ninvariant for a stochastic partial differential equation (SPDE) driven\nby a Wiener process and a Poisson random measure, and to provide\nconditions on the parameters of the SPDE, which are necessary and\nsufficient. As a particular example, we will show how the\nHeath-Jarrow-Morton-Musiela (HJMM) equation from Financial Mathematics,\nwhich models the evolution of interest rate curves, fits into the\npresent SPDE setting. Moreover, we will apply our result about the\ninvariance of closed convex cones in order to investigate when the HJMM\nequation produces nonnegative interest rate curves.
A multi-scale approach to reaction-diffusion processes in domains with microstructure
Donnerstag, 1.2.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Reaction-diffusion processes occur in many materials with microstructure such as biological cells, steel or concrete. The main difficulty in modelling and simulating accurately such processes is to account for the fine microstructure of the material. One method of upscaling multi-scale problems, which has proven reliable for obtaining feasible macroscopic models rigorously, is the method of periodic homogenisation. \n\nThe talk will give an introduction to multi-scale modelling of physico-chemical mechanisms in domains with microstructure as well as to the method of periodic homogenisation. Moreover, certain aspects particularly relevant in upscaling reaction-diffusion processes in biological cells will be discussed together with their applications.
Donnerstag, 8.2.18, 17:00-18:00, Hörsaal II, Albertstr. 23b