Prof. Dr. Julian Scheuer
Goethe Universität Frankfurt
Institut für MathematikRobert-Mayer-Str. 10, Raum 803
60325 Frankfurt
E-Mail: scheuer@math.uni-frankfurt.de
Phone: +49-69-798-22568
Partial Differential Equations
Prof. Dr. Julian ScheuerGoethe Universität FrankfurtInstitut für MathematikRobert-Mayer-Str. 10, Raum 803 60325 Frankfurt E-Mail: scheuer@math.uni-frankfurt.de Phone: +49-69-798-22568 |
Geometric Analysis and Partial Differential Equations |
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Home Vita Research Teaching Funding Conferences
Programme and Abstracts are found here
Timeline and Speakers:
Title: The gradient flow of the elastic and p-elastic energy for curves and fractional versions thereof
Abstract: During the last couple of years a surprising large number of new research papers appeared dealing with the elastic energy of curves and similar geometric equations.
In this survey talk we give an overview over new and old results in the context of gradient flows of the p-elastic energy for curves and fractional versions thereof. We discuss both, $L^2$ gradient flows and the more recently studied $H^2$-gradient flows for these elastic energies and try to highlight the main differences between the resulting evolution equations.
Title: The Lévy flight foraging hypothesis and high-risk/high-gain strategies
Abstract: We investigate the problem of the Lévy flight foraging hypothesis both in an ecological niche described by a bounded region of space and in the whole R^n.
To this end, we consider a forager diffusing according to a suitable fractional heat equation and we define several efficiency functionals whose optimality is discussed in relation to the fractional exponent s\in (0,1) of the diffusive equation.
We analyze the biological scenarios in which a target is close to the forager or far from it. In particular, for all the efficiency functionals considered here, we show that if the target is either close or far enough to the forager, then the most rewarding search strategy will be in a small neighborhood of s=0.
Interestingly, we show that s=0 is a global pessimizer for some of the efficiency functionals. From this, together with the aforementioned optimality results, we deduce that the most rewarding strategy can be unsafe or unreliable in practice, given its proximity with the pessimizing exponent, thus the forager may opt for a less performant, but safer, hunting method.
However, the biological literature has already collected several pieces of evidence of foragers diffusing with very low Lévy exponents, often in relation with a high energetic content of the prey. It is thereby suggestive to relate these patterns, which are induced by distributions with a very fat tail, with a high-risk/high-gain strategy, in which the forager adopts a potentially very profitable, but also potentially completely unrewarding, strategy due to the high value of the possible outcome.
Title: Towards geodesic completeness of spaces of embedded curves
joint work with Elias Döhrer and Henrik Schumacher (Chemnitz University of Technology)
Abstract: In pursuit of finding optimal paths in the manifold of closed embedded space curves we study a Riemannian metric which is inspired by the tangent-point energy. The latter is a self-repulsive functional defined on the space of curves which blows up if an embedding degenerates. Thereby it erects infinite barriers between distinct isotopy classes.
For finite-dimensional Riemannian manifolds the Hopf—Rinow theorem states that the Heine—Borel property (bounded sets are relatively compact), geodesic completeness (long-time existence of geodesic shooting), and metric completeness of the geodesic distance are equivalent. Moreover, it states that existence of length-minimizing geodesics follows from each of these statements. Albeit the Hopf—Rinow theorem does not hold true in this generality for infinite-dimensional Riemannian manifolds, we can prove all its four assertions for a suitably chosen Riemannian metric on the space of closed embedded curves.
Title: Quantitative stability for metrics of almost-constant fractional scalar curvature
Abstract
Title: div-curl estimates and harmonic maps: local and nonlocal
Abstract: I will present definitions and applications of a notion of
fractional div-curl structures. I will talk about their role in the
theory of fractional harmonic maps, such as regularity theory and
conservation laws.
