Now showing items 95-114 of 120

• #### Solving inverse problems with Bayes' theorem ﻿

[SNAP-2022-006-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2022-09-05)
The goal of inverse problems is to find an unknown parameter based on noisy data. Such problems appear in a wide range of applications including geophysics, medicine, and chemistry. One method of solving them is known as ...
• #### Solving quadratic equations in many variables ﻿

[SNAP-2017-012-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-30)
Fields are number systems in which every linear equation has a solution, such as the set of all rational numbers $\mathbb{Q}$ or the set of all real numbers $\mathbb{R}$. All fields have the same properties in relation ...
• #### Spaces of Riemannian metrics ﻿

[SNAP-2017-010-ENSNAP-2017-010-ES] (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-28)
Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the ...
• #### Special values of zeta functions and areas of triangles ﻿

[SNAP-2015-010-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
In this snapshot we give a glimpse of the interplay of special values of zeta functions and volumes of triangles. Special values of zeta functions and their generalizations arise in the computation of volumes of moduli ...
• #### Statistics and dynamical phenomena ﻿

[SNAP-2014-006-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)
A friend of mine, an expert in statistical genomics, told me the following story: At a dinner party, an attractive lady asked him, "What do you do for a living?" He replied, "I model." As my friend is a handsome man, the ...
• #### A surprising connection between quantum mechanics and shallow water waves ﻿

[SNAP-2019-018-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-12-11)
We describe a connection between quantum mechanics and nonlinear wave equations and highlight a few problems at the forefront of modern research in the intersection of these areas.
• #### Swallowtail on the shore ﻿

[SNAP-2014-007-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)
Platonic solids, Felix Klein, H.S.M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not ...
• #### Swarming robots ﻿

[SNAP-2016-001-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)
When lots of robots come together to form shapes, spread in an area, or move in one direction, their motion has to be planned carefully. We discuss how mathematicians devise strategies to help swarms of robots behave like ...
• #### Symmetry and characters of finite groups ﻿

[SNAP-2016-005-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)
Over the last two centuries mathematicians have developed an elegant abstract framework to study the natural idea of symmetry. The aim of this snapshot is to gently guide the interested reader through these ideas. In ...
• #### A tale of three curves ﻿

[SNAP-2022-010-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2022-10-27)
In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th ...
• #### The ternary Goldbach problem ﻿

[SNAP-2014-003-ENSNAP-2014-003-DE] (Mathematisches Forschungsinstitut Oberwolfach, 2014)
[also available in German] Leonhard Euler (1707–1783) – one of the greatest mathematicians of the eighteenth century and of all times – often corresponded with a friend of his, Christian Goldbach (1690–1764), an amateur ...
• #### Topological Complexity, Robotics and Social Choice ﻿

[SNAP-2018-005-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2018-08-10)
Topological complexity is a number that measures how hard it is to plan motions (for robots, say) in terms of a particular space associated to the kind of motion to be planned. This is a burgeoning subject within the ...
• #### Topological recursion ﻿

[SNAP-2018-002-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2018-03-05)
In this snapshot we present the concept of topological recursion – a new, surprisingly powerful formalism at the border of mathematics and physics, which has been actively developed within the last decade. After introducing ...
• #### Touching the transcendentals: tractional motion from the bir th of calculus to future perspectives ﻿

[SNAP-2019-013-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
When the rigorous foundation of calculus was developed, it marked an epochal change in the approach of mathematicians to geometry. Tools from geometry had been one of the foundations of mathematics until the 17th century ...
• #### Towards a Mathematical Theory of Turbulence in Fluids ﻿

[SNAP-2016-015-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)
Fluid mechanics is the theory of how liquids and gases move around. For the most part, the basic physics are well understood and the mathematical models look relatively simple. Despite this, fluids display a dazzling mystery ...
• #### Tropical geometry ﻿

[SNAP-2018-007-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2018-07-19)
What kind of strange spaces hide behind the enigmatic name of tropical geometry? In the tropics, just as in other geometries, one of the simplest objects is a line. Therefore, we begin our exploration by considering tropical ...
• #### Ultrafilter methods in combinatorics ﻿

[SNAP-2021-006-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021-06-25)
Given a set X, ultrafilters determine which subsets of X should be considered as large. We illustrate the use of ultrafilter methods in combinatorics by discussing two cornerstone results in Ramsey theory, namely ...
• #### Vertex-to-self trajectories on the platonic solids ﻿

[SNAP-2020-003-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-15)
We consider the problem of walking in a straight line on the surface of a Platonic solid. While the tetrahedron, octahedron, cube, and icosahedron all exhibit the same behavior, we find a remarkable difference with the ...
• #### Visual analysis of Spanish male mortality ﻿

[SNAP-2015-012-ENSNAP-2015-012-DE] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
[also available in German] Statistical visualization uses graphical methods to gain insights from data. Here we show how a technique called principal component analysis is used to analyze mortality in Spain over about the ...
• #### What does ">" really mean? ﻿

[SNAP-2014-004-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)
This Snapshot is about the generalization of ">" from ordinary numbers to so-called fields. At the end, I will touch on some ideas in recent research.