Now showing items 1-20 of 47

• #### Algebra, matrices, and computers ﻿

[SNAP-2019-005-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-05-03)
What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that ...
• #### Analogue mathematical instruments: Examples from the “theoretical dynamics” group (France, 1948–1964) ﻿

[SNAP-2019-012-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-21)
Throughout the history of dynamical systems, instruments have been used to calculate and visualize (approximate) solutions of differential equations. Here we describe the approach of a group of physicists and engineers ...
• #### $C^*$-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 ...
• #### 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 ...
• #### 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 ...
• #### 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 ...
• #### 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 ...
• #### 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. ...
• #### The Enigma behind the Good–Turing formula ﻿

[SNAP-2021-008-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021-07-16)
Finding the total number of species in a population based on a finite sample is a difficult but practically important problem. In this snapshot, we will attempt to shed light on how during World War II, two cryptanalysts, ...
• #### Expander graphs and where to find them ﻿

[SNAP-2019-016-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-11-22)
Graphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical ...
• #### Finite geometries: pure mathematics close to applications ﻿

[SNAP-2021-010-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021-09-22)
The research field of finite geometries investigates structures with a finite number of objects. Classical examples include vector spaces, projective spaces, and affine spaces over finite fields. Although many of these ...
• #### Formation Control and Rigidity Theory ﻿

[SNAP-2019-017-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-12-11)
Formation control is one of the fundamental coordination tasks for teams of autonomous vehicles. Autonomous formations are used in applications ranging from search-and-rescue operations to deep space exploration, with ...
• #### From Betti numbers to ℓ²-Betti numbers ﻿

[SNAP-2020-001-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-04-15)
We provide a leisurely introduction to ℓ²-Betti numbers, which are topological invariants, by relating them to their much older cousins, Betti numbers. In the end we present an open research problem about ℓ²-Betti numbers.
• #### From the dollar game to the Riemann-Roch Theorem ﻿

[SNAP-2021-001-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021-02-23)
What is the dollar game? What can you do to win it? Can you always win it? In this snapshot you will find answers to these questions as well as several of the mathematical surprises that lurk in the background, including ...
• #### Higgs bundles without geometry ﻿

[SNAP-2020-008-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2020-09-29)
Higgs bundles appeared a few decades ago as solutions to certain equations from physics and have attracted much attention in geometry as well as other areas of mathematics and physics. Here, we take a very informal ...
• #### How to choose a winner: the mathematics of social choice ﻿

[SNAP-2015-009-ENSNAP-2015-009-DE] (Mathematisches Forschungsinstitut Oberwolfach, 2015)
[also available in German] Suppose a group of individuals wish to choose among several options, for example electing one of several candidates to a political office or choosing the best contestant in a skating competition. ...
• #### The Interaction of Curvature and Topology ﻿

[SNAP-2019-020-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2019-12-18)
In this snapshot we will outline the mathematical notion of curvature by means of comparison geometry. We will then try to address questions as the ways in which curvature might influence the topology of a space, and ...
• #### Invitation to quiver representation and Catalan combinatorics ﻿

[SNAP-2021-004-EN] (Mathematisches Forschungsinstitut Oberwolfach, 2021-04-08)
Representation theory is an area of mathematics that deals with abstract algebraic structures and has numerous applications across disciplines. In this snapshot, we will talk about the representation theory of a class ...