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DESCRIPTION
In many areas of mathematics there are important quantifiable characteristic values (and is perhaps a real number giving useful information about the problem). In analysis, this might be the dimension of a set. In geometry, this might be curvature at some point. In dynamical systems, particularly in applications to number theory, this might be a Lyapunov exponent.
Of course, an important aspect of the problem is to find methods to estimate and compute which are both efficient and verifiably accurate. There are interesting connections with dynamical defined determinants and early work of Grothendieck. This is the theme of some recent work where, in particular, we use this method to numerically compute the Hausdorff Dimension of E2. i.e., the set of points whose continued fraction expansion contains only the digits 1 and 2. However, there are other methods and techniques which work more effectively on different problems, such as the so-called finite section method. For example, the dimension of the Apollonian circle packing was computed by McMullen using this method. Important contributions have been made in recent years by Urbanski, Bandtlow, Nussbaum and Falk. Interesting progress on the computation of diffusion coefficients using the Ulam method was made by Bahsoun, Galatolo and others. Within the French community there have been relevant related work by Seuret, Fan and his coauthors. A recent pioneer in the field of validated numerics (and the author of a book of the same name) is Tucker. This workshop will be focused on Thermodynamic Formalism and its applications. It will deal both with recent developments in the field and also its applications. It is a feature of this area that it has many applications to related fields, such as hyperbolic geometry and fractal geometry. |
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