Snapshots of Modern Mathematics from Oberwolfach
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- ItemOperator theory and the singular value decomposition(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Knese, GregThis is a snapshot about operator theory and one of its fundamental tools: the singular value decomposition (SVD). The SVD breaks up linear transformations into simpler mappings, thus unveiling their geometric properties. This tool has become important in many areas of applied mathematics for its ability to organize information. We discuss the SVD in the concrete situation of linear transformations of the plane (such as rotations, reflections, etc.).
- ItemArrangements of lines(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Harbourne, Brian; Szemberg, TomaszWe discuss certain open problems in the context of arrangements of lines in the plane.
- ItemSwallowtail on the shore(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Buchweitz, Ragnar-Olaf; Faber, EleonorePlatonic 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 only for their sheer beauty but also because of their many symmetry properties. In this snapshot we will start from these symmetries, move on to groups, singularities, and finally find the connection between a tetrahedron and a “swallowtail”. Our running example is the tetrahedron, but every construction can be carried out with any other of the Platonic solids.
- ItemWhat does ">" really mean?(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Reznick, BruceThis 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.
- ItemDirichlet Series(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) McCarthy, John E.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.
- ItemDrugs, herbicides, and numerical simulation(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Benner, Peter; Mena, Hermann; Schneider, RenéThe Colombian government sprays coca fields with herbicides in an effort to reduce drug production. Spray drifts at the Ecuador-Colombia border became an international issue. We developed a mathematical model for the herbicide aerial spray drift, enabling simulations of the phenomenon.
- ItemStatistics and dynamical phenomena(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Tong, HowellA 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 lady did not question his statement and continued, "What do you model?" "Genes." She then looked at him up and down and said, "Mh, you must be very much in demand." "Yes, very much so, especially after I helped discover a new culprit gene for a common childhood disease." The lady looked puzzled. In this snapshot, I will give you an insight into Statistics, the field that fascinated my friend (and myself) so much. I will concentrate on phenomena that change over time, in other words, dynamical events.
- ItemMatrixfaktorisierungen(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Lerche, WolfgangIm Folgenden soll ein kurzer Abriss des Themas Matrixfaktorisierungen gegeben werden. Wir werden darlegen, warum dieses recht simple Konzept zu erstaunlich tiefen mathematischen Gedankengängen führt und auch in der modernen theoretischen Physik wichtige Anwendungen hat.
- ItemThe Kadison-Singer problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Valette, AlainIn quantum mechanics, unlike in classical mechanics, one cannot make precise predictions about how a system will behave. Instead, one is concerned with mere probabilities. Consequently, it is a very important task to determine the basic probabilities associated with a given system. In this snapshot we will present a recent uniqueness result concerning these probabilities.
- ItemThe ternary Goldbach problem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Helfgott, HaraldLeonhard 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 and polymath who lived and worked in Russia, just like Euler himself. In a letter written in June 1742, Goldbach made a conjecture – that is, an educated guess – on prime numbers: "Es scheinet wenigstens, dass eine jede Zahl, die größer ist als 2, ein aggregatum trium numerorum primorum sey. (It seems (...) that every positive integer greater than 2 can be written as the sum of three prime numbers.)" In this snapshot, we will describe to what extent the mathematical community has resolved Goldbach's conjecture, with some emphasis on recent progress.
- ItemQuantum diffusion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Knowles, AnttiIf you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see something that is never observed in the real world. Such diffusive and irreversible behaviour is ubiquitous in nature. Nevertheless, the fundamental equations that describe the motion of individual particles – Newton’s and Schrödinger’s equations – are reversible in time: a film depicting the motion of just a few particles looks as realistic when played forwards as when played backwards. In this snapshot, we discuss how one may try to understand the origin of diffusion starting from the fundamental laws of quantum mechanics.
- ItemHow to choose a winner: the mathematics of social choice(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Powers, Victoria AnnSuppose 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 group might ask: what is the best method for choosing a winner, in the sense that it best reflects the individual preferences of the group members? We will see some examples showing that many voting methods in use around the world can lead to paradoxes and bad outcomes, and we will look at a mathematical model of group decision making. We will discuss Arrow’s impossibility theorem, which says that if there are more than two choices, there is, in a very precise sense, no good method for choosing a winner.
- ItemIdeas of Newton-Okounkov bodies(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Kiritchenko, Valentina; Timorin, Vladlen; Smirnov, EvgenyIn this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion.
- ItemZero-dimensional symmetry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Willis, GeorgeThis snapshot is about zero-dimensional symmetry. Thanks to recent discoveries we now understand such symmetry better than previously imagined possible. While still far from complete, a picture of zero-dimensional symmetry is beginning to emerge.
- ItemFriezes and tilings(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Holm, ThorstenFriezes 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 friezes have appeared in current research. We are going to describe them and explain how they can be classified using some nice geometric pictures.
- ItemFrom computer algorithms to quantum field theory: an introduction to operads(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Krähmer, UlrichAn 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 of an algebra over an operad, with a view towards a conjecture formulated by the mathematician Pierre Deligne. Deligne’s (by now proven) conjecture also gives deep inights into mathematical physics.
- ItemSpecial values of zeta functions and areas of triangles(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Kramer, Jürg; Pippich, Anna-Maria vonIn 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 spaces (for example of Abelian varieties) and their universal spaces.
- ItemModelling the spread of brain tumours(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Swan, Amanda; Murtha, AlbertThe study of mathematical biology attempts to use mathematical models to draw useful conclusions about biological systems. Here, we consider the modelling of brain tumour spread with the ultimate goal of improving treatment outcomes.
- ItemThe mystery of sleeping sickness – why does it keep waking up?(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Funk, SebastianSleeping sickness is a neglected tropical disease that affects rural populations in Africa. Deadly when untreated, it is being targeted for elimination through case finding and treatment. Yet, fundamental questions about its transmission cycle remain unanswered. One of them is whether transmission is limited to humans, or whether other species play a role in maintaining circulation of the disease. In this snapshot, we introduce a mathematical model for the spread of Trypanosoma brucei, the parasite responsible for causing sleeping sickness, and present some results based on data collected in Cameroon. Understanding how important animals are in harbouring Trypanosoma brucei that can infect humans is important for assessing whether the disease could be reintroduced in human populations even after all infected people have been successfully treated.
- ItemChaos and chaotic fluid mixing(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Solomon, TomVery 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 equation that can be easily simulated, and with mixing in simple fluid flows.