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- ItemFormation Control and Rigidity Theory(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Zelazo, Daniel; Zhao, ShiyuFormation 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 benefits including increased robustness to failures and risk mitigation for human operators. The challenge of formation control is to develop distributed control strategies using vehicle onboard sensing that ensures the desired formation is obtained. This snapshot describes how the mathematical theory of rigidity has emerged as an important tool in the study of formation control problems.
- ItemConfiguration spaces and braid groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Jiménez Rolland, Rita; Xicoténcatl, Miguel A.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 these groups determine features of the spaces.
- ItemCounting self-avoiding walks on the hexagonal lattice(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Duminil-Copin, HugoIn 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 walks. In particular, we discuss a recent result concerning the number of self-avoiding walks on the hexagonal (“honeycomb”) lattice. In the last part, we briefly hint at the connection to the geometry of long random self-avoiding walks.
- 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.
- 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.
- ItemExpander graphs and where to find them(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Khukhro, AnaGraphs 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 properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours.
- ItemAlgebra, matrices, and computers(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Detinko, Alla; Flannery, Dane; Hulpke, AlexanderWhat 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 rely on the theory of matrix groups and new methods for handling matrix groups in a computer.
- ItemOn Logic, Choices and Games(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Oliva, PauloCan we always mathematically formalise our taste and preferences? We discuss how this has been done historically in the field of game theory, and how recent ideas from logic and computer science have brought an interesting twist to this beautiful theory.
- ItemDiophantine equations and why they are hard(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Pasten, HectorDiophantine 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. Some Diophantine equations are easy, while some others are truly difficult. After some time spent with these equations, it might seem that no matter what powerful methods we learn or develop, there will always be a Diophantine equation immune to them, which requires a new trick, a better idea, or a refined technique. In this snapshot we explain why.
- ItemThe Mathematics of Fluids and Solids(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Kaltenbacher, Barbara; Kukavica, Igor; Lasiecka, Irena; Triggiani, Roberto; Tuffaha, Amjad; Webster, JustinFluid-structure interaction is a rich and active field of mathematics that studies the interaction between fluids and solid objects. In this short article, we give a glimpse into this exciting field, as well as a sample of the most significant questions that mathematicians try to answer.
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