Page 601 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 601
A JOURNEY FROM PURE TO APPLIED MATHEMATICS (MS-53)
Hilbert’s Trip to the Casino
Stefanie Petermichl, stefanie.petermichl@mathematik.uni-wuerzburg.de
Universitaet Wuerzburg, Germany
We discuss a central object in harmonic analysis - the Hilbert transform and its generalisations,
the Riesz transforms. The Hilbert transform turns sinus waves into cosinus waves and is even
used (mechanically) in antennas. In this talk, we discuss various probabilistic models and tools
that can be used to precisely describe or control these objects in a variety of settings. We present
results spanning 1999 to 2019. The talk targets general audiences.
The fast p-Laplacian evolution equation. Global Harnack principle and
fine asymptotic behaviour
Diana Stan, diana.stan@unican.es
Universidad de Cantabria, Spain
Coauthors: Matteo Bonforte, Nikita Simonov
We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast
p-Laplacian evolution equation on the whole Euclidean space, in the so-called "good fast dif-
fusion range". It is well-known that non-negative solutions behave for large times as B, the
Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called
Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates
of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the
celebrated Gaussian estimates for the linear heat equation. To the best of our knowledge, anal-
ogous issues for the linear heat equation, do not possess such clear answers, only partial results
are known. Also, we characterize the maximal (hence optimal) class of initial data such that
the GHP holds, by means of an integral tail condition, easy to check. Finally, we derive sharp
global quantitative upper bounds of the modulus of the gradient of the solution, and, when data
are radially decreasing, we show uniform convergence in relative error for the gradients.
On New Approaches for Nonsmooth Optimization
Andrea Walther, andrea.walther@math.hu-berlin.de
Humboldt Universität zu Berlin, Germany
Numerous optimization tasks exhibit a nonsmooth behavior. In contrast to the classical smooth
case, where optimality conditions are well studied and understood, criteria to determine whether
a given point is optimal or even just stationary are still the subject of ongoing research for
nonsmooth functions to be minimized. In this presentation, first we discuss new optimality
conditions for a large class of piecewise smooth functions using so-called kink qualifications.
Here, also the computational complexity to verify the new criteria is covered. Next, we present
optimization algorithms resulting from these findings. Finally, we present some applications
that fit into the considered problem class.
599
Hilbert’s Trip to the Casino
Stefanie Petermichl, stefanie.petermichl@mathematik.uni-wuerzburg.de
Universitaet Wuerzburg, Germany
We discuss a central object in harmonic analysis - the Hilbert transform and its generalisations,
the Riesz transforms. The Hilbert transform turns sinus waves into cosinus waves and is even
used (mechanically) in antennas. In this talk, we discuss various probabilistic models and tools
that can be used to precisely describe or control these objects in a variety of settings. We present
results spanning 1999 to 2019. The talk targets general audiences.
The fast p-Laplacian evolution equation. Global Harnack principle and
fine asymptotic behaviour
Diana Stan, diana.stan@unican.es
Universidad de Cantabria, Spain
Coauthors: Matteo Bonforte, Nikita Simonov
We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast
p-Laplacian evolution equation on the whole Euclidean space, in the so-called "good fast dif-
fusion range". It is well-known that non-negative solutions behave for large times as B, the
Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called
Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates
of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the
celebrated Gaussian estimates for the linear heat equation. To the best of our knowledge, anal-
ogous issues for the linear heat equation, do not possess such clear answers, only partial results
are known. Also, we characterize the maximal (hence optimal) class of initial data such that
the GHP holds, by means of an integral tail condition, easy to check. Finally, we derive sharp
global quantitative upper bounds of the modulus of the gradient of the solution, and, when data
are radially decreasing, we show uniform convergence in relative error for the gradients.
On New Approaches for Nonsmooth Optimization
Andrea Walther, andrea.walther@math.hu-berlin.de
Humboldt Universität zu Berlin, Germany
Numerous optimization tasks exhibit a nonsmooth behavior. In contrast to the classical smooth
case, where optimality conditions are well studied and understood, criteria to determine whether
a given point is optimal or even just stationary are still the subject of ongoing research for
nonsmooth functions to be minimized. In this presentation, first we discuss new optimality
conditions for a large class of piecewise smooth functions using so-called kink qualifications.
Here, also the computational complexity to verify the new criteria is covered. Next, we present
optimization algorithms resulting from these findings. Finally, we present some applications
that fit into the considered problem class.
599
