Browsing 2  Snapshots of Modern Mathematics from Oberwolfach by Title
Now showing items 2746 of 127

Diophantine equations and why they are hard
[SNAP2019003EN] (Mathematisches Forschungsinstitut Oberwolfach, 20190424)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. ... 
Dirichlet Series
[SNAP2014001EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)Mathematicians are very interested in prime numbers. In this snapshot, we will discuss some problems concerning the distribution of primes and introduce some special infinite series in order to study them. 
Domino tilings of the Aztec diamond
[SNAP2015016EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)Imagine you have a cutout from a piece of squared paper and a pile of dominoes, each of which can cover exactly two squares of the squared paper. How many different ways are there to cover the entire paper cutout with ... 
Drugs, herbicides, and numerical simulation
[SNAP2014010EN] (Mathematisches Forschungsinstitut Oberwolfach, 2014)[also available in German] The Colombian government sprays coca fields with herbicides in an effort to reduce drug production. Spray drifts at the EcuadorColombia border became an international issue. We developed a ... 
Emergence in biology and social sciences
[SNAP2022001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20220331)Mathematics is the key to linking scientific knowledge at different scales: from microscopic to macroscopic dynamics. This link gives us understanding on the emergence of observable patterns like flocking of birds, leaf ... 
The Enigma behind the Good–Turing formula
[SNAP2021008EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210716)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, ... 
Estimating the volume of a convex body
[SNAP2018015EN] (Mathematisches Forschungsinstitut Oberwolfach, 20181230)Sometimes the volume of a convex body needs to be estimated, if we cannot calculate it analytically. We explain how statistics can be used not only to approximate the volume of the convex body, but also its shape. 
Expander graphs and where to find them
[SNAP2019016EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191122)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 ... 
Fast Solvers for Highly Oscillatory Problems
[SNAP2018006EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180626)Waves of diverse types surround us. Sound, light and other waves, such as microwaves, are crucial for speech, mobile phones, and other communication technologies. Elastic waves propagating through the Earth bounce ... 
Felder und Räume: Symmetrie und Lokalität in Mathematik und theoretischen Wissenschaften
[SNAP2023003DE] (Mathematisches Forschungsinstitut Oberwolfach, 20230919)Wir werden einige grundlegende Ideen der Eichtheorie und der dazugehörigen Differentialtopologie erkunden. Damit kann sich die Leserin ein Bild des Modulraums flacher Zusammenhänge machen und ihn mit den physikalisch ... 
A few shades of interpolation
[SNAP2017007EN] (Mathematisches Forschungsinstitut Oberwolfach, 20171207)The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points ... 
Finite geometries: pure mathematics close to applications
[SNAP2021010EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210922)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 ... 
FokusErkennung bei Epilepsiepatienten mithilfe moderner Verfahren der Zeitreihenanalyse
[SNAP2016008DE] (Mathematisches Forschungsinstitut Oberwolfach, 2016)Viele epileptische Anfälle entstehen in einer begrenzten Region im Gehirn, dem sogenannten Anfallsursprung. Eine chirurgische Entfernung dieser Region kann in vielen Fällen zu Anfallsfreiheit führen. Aus diesem Grund ist ... 
Footballs and donuts in four dimensions
[SNAP2016012EN] (Mathematisches Forschungsinstitut Oberwolfach, 2016)In this snapshot, we explore connections between the mathematical areas of counting and geometry by studying objects called simplicial complexes. We begin by exploring many familiar objects in our three dimensional world ... 
Formation Control and Rigidity Theory
[SNAP2019017EN] (Mathematisches Forschungsinstitut Oberwolfach, 20191211)Formation control is one of the fundamental coordination tasks for teams of autonomous vehicles. Autonomous formations are used in applications ranging from searchandrescue operations to deep space exploration, with ... 
Friezes and tilings
[SNAP2015004EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)Friezes have occured as architectural ornaments for many centuries. In this snapshot, we consider the mathematical analogue of friezes as introduced in the 1970s by Conway and Coxeter. Recently, infinite versions of such ... 
From Betti numbers to ℓ²Betti numbers
[SNAP2020001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20200415)We provide a leisurely introduction to ℓ²Betti 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 ℓ²Betti numbers. 
From computer algorithms to quantum field theory: an introduction to operads
[SNAP2015017EN] (Mathematisches Forschungsinstitut Oberwolfach, 2015)An operad is an abstract mathematical tool encoding operations on specific mathematical structures. It finds applications in many areas of mathematics and related fields. This snapshot explains the concept of an operad and ... 
From the dollar game to the RiemannRoch Theorem
[SNAP2021001EN] (Mathematisches Forschungsinstitut Oberwolfach, 20210223)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 ... 
Geometry behind one of the Painlevé III differential equations
[SNAP2018010EN] (Mathematisches Forschungsinstitut Oberwolfach, 20180620)The Painlevé equations are second order differential equations, which were first studied more than 100 years ago. Nowadays they arise in many areas in mathematics and mathematical physics. This snapshot discusses the ...