Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Multiscale Coupled Models for Complex Media: From Analysis to Simulation in Geophysics and Medicine
No full textMany real-life applications require mathematical models at multiple scales, defined in domains with
complex structures, some of which having time dependent boundaries. Mathematical models of this
type are encountered in seemingly disparate areas e.g.,
flow and deformation in the subsurface or beneath the ocean
floor, and in processes of clinical relevance. While the areas are different, the structure
of the models and the challenges are shared: the analysis and simulation must account for the evolution
of the domain due to the many coupled processes in the multi-scale context. The key theme and focus of the workshop were novel ideas in the mathematical modeling, analysis, and numerical simulation, which are cross-cutting between the two application areas mentioned above. The talks have covered the mathematical treatment of such problems, as well as the development of efficent numerical discretization schemes and of solvers for large-scale problems
Hutchinson's Intervals and Entire Functions from the Laguerre-Pólya Class
No full textWe find the intervals such that if a univariate real polynomial or entire function with positive coefficients satisfy the conditions for all then belongs to the Laguerre-Pólya class. For instance, from J.I. Hutchinson's theorem, one can observe that belongs to the Laguerre-Pólya class (has only real zeros) when We are interested in finding those intervals which are not subsets of $[4, + \infty).
Multiscale Wave-Turbulence Dynamics in the Atmosphere and Ocean
No full textThe atmosphere and oceans present an ongoing first-rate
challenge to science and mathematics because they operate on
an extremely broad ranges of scales, from molecular to
planetary in length and from below seconds to millennia in
time. This is the reason why climate simulations still suffer
from leading-order uncertainties. Conceptual simplifications,
such as scale-separation assumptions and the neglect of many
physical processes, have enabled past progress in understanding
the interactions of the basic dynamic constituents,
i.e. large-scale mean flows, medium-scale waves and vortices,
and small-scale turbulence. But present-day research is
stretching the validity of this framework. For example, it is
recognized that intermediate-scale waves and vortices are key
elements linking all relevant players, and are often
characterized by nonlinear interactions on comparable scales
and also by additional physical nonlinearities due to effects
such as air moisture. Motivated by recent advances in
mathematical wave-vortex and wave-wave interaction theory,
turbulence theory, and the study of internal wave dynamics as
well as their numerical parametrization, the workshop
gathered leading experts in these fields to foster a synthesis of
new approaches and thereby a new level of understanding and
numerical treatment of climate dynamics
Population Dynamics and Statistical Physics in Synergy
No full textResearch at the interface between population dynamics and statistical physics has been developing rapidly, and represents a theme of growing interest worldwide. Population dynamics addresses fundamental questions about the cooperative behaviour controlling multi-type interacting populations subject to evolutionary forces in changing environments. Statistical physics is concerned with the macroscopic behaviour of systems with many interacting components, and with the role of emergent behaviour and phase transitions. Fundamental ideas, methods and techniques have gradually made their way from one field into the other, leading to new problems, new solutions, and new mathematics. This crossroad has developed into a very active research area. In the workshop the focus was on common mathematical concepts and tools, and on the surprising new connections that have become available recently
Jewellery from tessellations of hyperbolic space
No full textIn this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery
Heat Kernels, Stochastic Processes and Functional Inequalities
No full textThe workshop provided a forum for recent progress on a wide array of topics at the nexus of Analysis (elliptic, subelliptic and parabolic differential equations), Geometry (Riemannian and sub-Riemannian geometries, metric measure spaces, geometric analysis and curvature), and Probability Theory (Brownian motion, Dirichlet spaces, stochastic calculus and random media).
The workshop provides a unique opportunity to encourage and foster interactions between mathematicians who share some common interests but might use different research tools or work in different mathematical settings
Set Theory
No full textWhile set theory continues reaching out into various other fields of mathematics but also becomes more and more specialized, recent times have seen important results around holy grails of set theory which gave a new momentum to the whole field as a unit
Analytic Number Theory
No full textAnalytic number theory is a subject central to modern mathematics. There are many important unsolved problems which have stimulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions
The Laguerre-Pólya Class and Combinatorics
No full textThe talks at the workshop were focused on zero localization and zero finding of entire functions, with applications to analytic number theory and combinatorics.
The discussions included specific areas such as stable and hyperbolic polynomials, the Laguerre-Pólya class of entire functions, Pólya frequency sequences, total positivity for sequences and
functions, and zeros of generating functions arising in probability and combinatorics
Toric Geometry
No full textToric geometry is a vibrant subfield of algebraic geometry that
draws on strong connections to combinatorics. The 2022 workshop
brought together a broad group of mathematicians both in-person and
virtually to discuss aspects of the field, ranging from K-stability
to machine learning