Mathematisches Kolloquium
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.
Anstehende Vorträge
Vorträge
Zeit und Ort
Donnerstag, 18.6.26, 15:00–16:30, Hörsaal 2
Zusammenfassung
Consider an unknown random vector X, taking values in R^d. Is it possible to"guess" its mean accurately if the only information one is given consists of N independent copies of X? More accurately, given an arbitrary norm on R^d, the goal is to find a mean estimation procedure: upon receiving a wanted confidence parameter \delta and N independent copies X1,...,XN of an unknown random vector X - that has a finite mean and covariance -, the procedure returns \hat{\mu} for which the error | \hat{\mu} - E X| is as small as possible with probability at least 1-\delta (with respect to the product measure). The mean estimation problem has been studied extensively over the years and I will present some of the ideas that have led to its solution (and to the solution of other questions of a similar flavour that I will outline). Two surprising facts are that in all these problems the obvious choices fail miserably (for mean estimation, that choice is N^{-1}\sum{i=1}^N Xi); and, that the solution behaves as if the (arbitrary) random vector X were gaussian.
Zeit und Ort
Donnerstag, 25.6.26, 15:00–16:30, Hörsaal 2
Zusammenfassung
Partial Differential Equations (PDEs) are often described as the language of Physics as they describe a wide array of physical phenomena over a vast range of scales. Despite their remarkable success over many decades, numerical methods for approximating PDEs can incur a very high computational cost. This limitation has provided the impetus for the design of fast and accurate Machine Learning/AI based neural PDE surrogates which can learn the PDE solution operator from data. In this talk, we review some latest developments in the field of Neural Operators, which are widely used as an ML paradigm for PDEs and discuss state of the art neural operators based on convolutions or attention. We will discuss graph and transformer based architectures for PDEs on arbitrary domains and conditional Diffusion models for PDEs with chaotic multiscale solutions. Finally, the issue of sample complexity is addressed by the design of general purpose Foundation models for PDEs.
Zeit und Ort
Donnerstag, 23.7.26, 15:00–16:30, Hörsaal 2