Now showing items 7-26 of 123

• #### Billiards and flat surfaces ﻿

[SNAP-2015-001-ENSNAP-2015-001-DE] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
[also available in German] Billiards, the study of a ball bouncing around on a table, is a rich area of current mathematical research. We discuss questions and results on billiards, and on the related topic of flat surfaces.
• #### Biological shape analysis with geometric statistics and learning ﻿

[SNAP-2022-008-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2022-10-25)
The advances in biomedical imaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, shape data may hold ...
• #### $C^*$-algebras: structure and classification ﻿

[SNAP-2021-002-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021-02-23)
The theory of $C^*$-algebras traces its origins back to the development of quantum mechanics and it has evolved into a large and highly active field of mathematics. Much of the progress over the last couple of decades ...
• #### Chaos and chaotic fluid mixing ﻿

[SNAP-2015-005-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
Very simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with behavior that is sensitive to small deviations in the initial conditions. We illustrate this with a single recurrence ...
• #### Characterizations of intrinsic volumes on convex bodies and convex functions ﻿

[SNAP-2022-011-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2022-12-08)
If we want to express the size of a two-dimensional shape with a number, then we usually think about its area or circumference. But what makes these quantities so special? We give an answer to this question in terms of ...
• #### Closed geodesics on surfaces ﻿

[SNAP-2022-013-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2022-12-08)
We consider surfaces of three types: the sphere, the torus, and many-holed tori. These surfaces naturally admit geometries of positive, zero, and negative curvature, respectively. It is interesting to study straight line ...
• #### Closed geodesics on surfaces and Riemannian manifolds ﻿

[SNAP-2017-005-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-07)
Geodesics are special paths in surfaces and so-called Riemannian manifolds which connect close points in the shortest way. Closed geodesics are geodesics which go back to where they started. In this snapshot we talk ...
• #### The codimension ﻿

[SNAP-2018-009-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2018-06-19)
In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one.
• #### Computational Optimal Transport ﻿

[SNAP-2017-008-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-21)
Optimal transport is the mathematical discipline of matching supply to demand while minimizing shipping costs. This matching problem becomes extremely challenging as the quantity of supply and demand points increases; ...
• #### Computing the long term evolution of the solar system with geometric numerical integrators ﻿

[SNAP-2017-009-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2017-12-27)
Simulating the dynamics of the Sun–Earth–Moon system with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the ...
• #### Computing with symmetries ﻿

[SNAP-2018-003-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2018-03-06)
Group theory is the study of symmetry, and has many applications both within and outside mathematics. In this snapshot, we give a brief introduction to symmetries, and how to compute with them.
• #### Configuration spaces and braid groups ﻿

[SNAP-2019-011-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-10-08)
In this snapshot we introduce configuration spaces and explain how a mathematician studies their ‘shape’. This will lead us to consider paths of configurations and braid groups, and to explore how algebraic properties of ...
• #### Counting self-avoiding walks on the hexagonal lattice ﻿

[SNAP-2019-006-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-06-04)
In how many ways can you go for a walk along a lattice grid in such a way that you never meet your own trail? In this snapshot, we describe some combinatorial and statistical aspects of these so-called self-avoiding ...
• #### Curriculum development in university mathematics: where mathematicians and education collide ﻿

[SNAP-2015-011-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
This snapshot looks at educational aspects of the design of curricula in mathematics. In particular, we examine choices textbook authors have made when introducing the concept of the completness of the real numbers. Can ...
• #### Darcy's law and groundwater flow modelling ﻿

[SNAP-2015-007-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
Formulations of natural phenomena are derived, sometimes, from experimentation and observation. Mathematical methods can be applied to expand on these formulations, and develop them into better models. In the year 1856, ...
• #### Data assimilation: mathematics for merging models and data ﻿

[SNAP-2018-011-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2018-07-10)
When you describe a physical process, for example, the weather on Earth, or an engineered system, such as a self-driving car, you typically have two sources of information. The first is a mathematical model, and the ...
• #### Deep Learning and Inverse Problems ﻿

[SNAP-2019-015-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
Big data and deep learning are modern buzz words which presently infiltrate all fields of science and technology. These new concepts are impressive in terms of the stunning results they achieve for a large variety of ...
• #### Describing distance: from the plane to spectral triples ﻿

[SNAP-2021-009-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021-12-31)
Geometry draws its power from the abstract structures that govern the shapes found in the real world. These abstractions often provide deeper insights into the underlying mathematical objects. In this snapshot, we give a ...
• #### Determinacy versus indeterminacy ﻿

[SNAP-2020-004-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-22)
Can a continuous function on an interval be uniquely determined if we know all the integrals of the function against the natural powers of the variable? Following Weierstrass and Stieltjes, we show that the answer is yes ...
• #### Diophantine equations and why they are hard ﻿

[SNAP-2019-003-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-04-24)
Diophantine equations are polynomial equations whose solutions are required to be integer numbers. They have captured the attention of mathematicians during millennia and are at the center of much of contemporary research. ...