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- ItemMixed volumes and mixed integrals(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Rotem, LiranIn recent years, mathematicians have developed new approaches to study convex sets: instead of considering convex sets themselves, they explore certain functions or measures that are related to them. Problems from convex geometry become thereby accessible to analytic and probabilistic tools, and we can use these tools to make progress on very difficult open problems. We discuss in this Snapshot such a functional extension of some “volumes” which measure how “big” a set is. We recall the construction of “intrinsic volumes”, discuss the fundamental inequalities between them, and explain the functional extensions of these results.
- ItemThe Interaction of Curvature and Topology(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Kordaß, Jan-BernhardIn 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 vice versa.
- ItemSnake graphs, perfect matchings and continued fractions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Schiffler, RalfA continued fraction is a way of representing a real number by a sequence of integers. We present a new way to think about these continued fractions using snake graphs, which are sequences of squares in the plane. You start with one square, add another to the right or to the top, then another to the right or the top of the previous one, and so on. Each continued fraction corresponds to a snake graph and vice versa, via “perfect matchings” of the snake graph. We explain what this means and why a mathematician would call this a combinatorial realization of continued fractions.
- 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.
- 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.
- ItemVisual analysis of Spanish male mortality(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Marron, J. S.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 last hundred years. This data decomposition both reflects expected historical events and reveals some perhaps less expected trends in mortality over the years.
- ItemFootballs and donuts in four dimensions(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Klee, StevenIn 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 and then discuss the ways one may generalize these ideas into higher dimensions.
- ItemTouching the transcendentals: tractional motion from the bir th of calculus to future perspectives(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Milici, PietroWhen 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 but today, mainstream conception relegates geometry to be merely a tool of visualization. In this snapshot, however, we consider geometric and constructive components of calculus. We reinterpret “tractional motion”, a late 17th century method to draw transcendental curves, in order to reintroduce “ideal machines” in math foundation for a constructive approach to calculus that avoids the concept of infinity.
- ItemTopological recursion(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Sułkowski, PiotrIn 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 necessary ingredients – expectation values, random matrices, quantum theories, recursion relations, and topology – we explain how they get combined together in one unifying picture.
- ItemDomino tilings of the Aztec diamond(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Rué, JuanjoImagine 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 dominoes? One specific paper cutout can be mathematically described as the so-called Aztec Diamond, and a way to cover it with dominoes is a domino tiling. In this snapshot we revisit some of the seminal combinatorial ideas used to enumerate the number of domino tilings of the Aztec Diamond. The existing connection with the study of the so-called alternating-sign matrices is also explored.