Mathematical structures in biology: Jürgen Jost

• Date: Friday, 13:30 - 15:00 h
• Location: MPI MIS, room A 01
• First lecture: October 12th, 2012
• Description: I shall develop methods from information theory and other fields in order to build and analyze models of gene transmission and regulation.
  This course is mainly for mathematics and physics students interested in topics like information geometry, but also for biologists with a strong background in mathematics.

• Date: Tuesday, 9:00 - 11:00 h
  • in case the lecture cannot take place due to travel etc.:
  • Thursday, 9:00 - 11:00 h
• Location: MPI MiS, room A 01
• First lecture: October 9th
• Description: We start with an introduction into elasticity theory, in particular the notion of strain (in the geometrically linear setting). We then introduce the notion of shape memory alloys, with their transition from a high symmetry Austenite phase to a low symmetry Martensite phase; this will involve a bit of symmetry considerations (point group etc).
  We then will address Martensitic twins and their compatibility with Austenite, establishing "rigidity" for certain symmetries (like square-to-rectangular in 2-d and cubic-to-tetragonal in 3-d) and non-rigidity in other cases (like hexagonal-to-rhombic in 2-d). The non-rigidity results will be obtained via the tool of "convex integration". We then will introduce interfacial energy as a means to select physically relevent microstructure. In the rigid cases, the optimal microstructure is a branched Martensitic twin. This involves establishing matching upper and lower bounds for the minimal energy (that is a sum of elastic and interfacial energy); obtaining the lower bounds involves suitable interpolation inequalities. We also plan to address nucleation barriers for Martensite in Austenite in that context.

• Date: Wednesday, 11:00 - 12:30 h
• Location: MPI MiS, room A 02
• First lecture: October 10th
• Description: In this course, I shall concentrate on information sources. Fundamental notions of entropy will be introduced, including Shannon entropy, Kolmogorov-Sinai entropy, and the relative entropy, also known as Kullback-Leibler information. Ergodic information sources will be discussed in more detail, and the Shannon-McMillan-Breiman theorem will be presented. This course will be continued during the summer term 2013 where information channels and the fundamental coding theorems of information theory will constitute the main subject. I shall assume basic knowledge from measure and probability theory, and functional analysis.
• References
  [1] Y. Kakihara. Abstract Methods in Information Theory. Singapore: World Scientific, 1999.
  [2] A. I. Khinchin. Mathematical Foundations of Information Theory. New York: Dover, 1957.
  [3] T. M. Cover, J. A. Thomas. Elements of Information Theory. New York: Wiley, 1991.

• Date: Tuesday, 11:00 - 12:30 h
• Location: MPI MiS, room A 02
• First lecture: October 16th
• Description: Hadamard spaces are metric spaces of nonpositive curvature, where curvature is defined as a very simple feature of the metric, and unlike in Riemannian geometry, no differentiable structure is needed. Even though Hadamard space theory has become classical part of geometry, it is still a very active area of mathematical research.
  The aim of this course is to systematically develop the theory of Hadamard spaces with a special emphasis on convexity. It turns out that Hadamard spaces are a natural nonlinear setting for convex sets and convex functions. We will see that many of the convexity results in linear spaces can be extended to Hadamard spaces. Although we will be primarily concerned with analytical aspects of convexity, there will be a strong interplay with geometry and probability as well. After building the rudiments of Hadamard space theory, we will get to more advanced topics such as resolvents and gradient flow semigroups, convex optimization, martingale theory, Markov processes, etc. A nice application to biology will be presented in the end of the course.
  The course is intended to be rather elementary. No preliminary knowledge is expected.

• Date: Wednesday, 13:15 - 14:45 h
• Location: MPI MiS, room A 02
• First lecture: October 10th
• Classification: Informatik, 5th semester Bachelor or 1st to 3rd semester Master
• Lecture materials (exercises and additional material)
• Description: Einführung in die funktionale Programmierung anhand der Programmiersprachen Scheme und Haskell. Sowohl die theoretischen Grundlagen (Lambda-Kalkül) als auch praktischen Anwendungen werden behandelt (z.B. Lazy evaluation, Parser-Kombinatoren). Darüberhinaus werden Beziehungen zur Objekt-orientierten Programmierung (Design Patterns) als auch Vorteile fuer nebenläufige Programme diskutiert.

• Date: Wednesday, 10:30 - 12:00 h
• Location: MPI MiS, room A 01
• First lecture: October 10th
• Description: The topic of this course is the classical theory of linear elliptic partial differential equations of second order. We will concentrate mainly on the most fundamental results: L²-estimates, Schauder estimates including Campanato theory, and L^p estimates.

• Date: Monday, 13:30 - 15:00 h
• Location: MPI MiS, room A 02
• First lecture: October 8th
• Description: In this course, we shall study some basic facts about manifolds with lower Ricci (sectional) curvature bound in the comparison geometry such as Cheng-Myers' theorem, Bishop-Gromov volume comparison, the Laplacian comparison and Cheeger-Gromoll's splitting theorem. Topological obstructions of manifolds with nonnegative Ricci curvature will be investigated. To understand the structure theory of the limit space of a sequence of manifolds with Ricci curvature bounded below (by Cheeger-Colding) is our main purpose. In the end, we survey the harmonic function theory on manifolds with nonnegative Ricci curvature (Cheng-Yau, Colding-Minicozzi, Li).

• Date: Thursday, 14:00 - 15:30 h
• Location: MPI MiS, room A 02
• First lecture: October 11th
• Description: Introduction to ergodic theory and its application in dynamical systems. Topics to be covered include: existence of invariant measures for continuous transformations. Measure preserving transformations. Ergodicity and mixing. Recurrence and ergodic theorems. Topological dynamics. Isomorphism problem, metric and topological entropies. Thermodynamical formalism.

• Date: Thursday, 11:00 - 12:30 h
• Location: MPI MiS, room A 01
• First lecture: October 11th
• Description: In the classical mathematical theory of percolation, the edges (or vertices) of an infinite lattice are deleted independently with probability 1-p, and properties of the remaining components are studied. Despite its simple description, this model captures a variety of phenomena, including structural phase transition, self-similarity, universality. It has been used in studies of materials, social and computer networks, epidemic spreading. This course will provide an introduction to the subject of percolation, focusing on basic results and techniques.

• Date: Thursday, 14:00 - 15:30 h
• Location: MPI MiS, room G 10
• First lecture: October 11th
• Description: In this lecture we present the classical theory of interpolation of Banach spaces. Starting with the theorem of Riesz-Thorin and its immediate application to some famous operators, we concentrate then on the real methods, the so-called $K$- and $J$-methods. As a major application we show the interpolation of Sobolev- and Besov spaces with a short introduction to function spaces. If time allows, we present a further application to quantitative approximation theory in terms of entropy- and approximation numbers. This lecture will be a 2 SWS course held in english and is recommended for PhD students or students of higher semesters with interests in higher analysis.

• Date: Tuesday, 13:30 - 15:00 h
• Location: MPI MiS, room G 10
• First lecture: October 9th
• see all dates, titles and lecturers

• Organizators: Antoine Choffrut, Peter Hornung, Bernd Kirchheim, Stephan Luckhaus, Felix Otto, Emanuele Spadaro, Laszlo Szekelyhidi
• Date: Wednesday, 13:00 - 15:00 h
• Location: University Leipzig, Neues Augusteum, room A314
• First lecture: October 10th