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- 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.
- ItemSymmetry and characters of finite groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Giannelli, Eugenio; Taylor, JayOver the last two centuries mathematicians have developed an elegant abstract framework to study the natural idea of symmetry. The aim of this snapshot is to gently guide the interested reader through these ideas. In particular, we introduce finite groups and their representations and try to indicate their central role in understanding symmetry.
- ItemThe Robinson–Schensted algorithm(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Thomas, HughI am going to describe the Robinson–Schensted algorithm which transforms a permutation of the numbers from 1 to n into a pair of combinatorial objects called “standard Young tableaux”. I will then say a little bit about a few of the fascinating properties of this transformation, and how it connects to current research.
- ItemFinite geometries: pure mathematics close to applications(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Storme, LeoThe 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 structures are studied for their geometrical importance, they are also of great interest in other, more applied domains of mathematics. In this snapshot, finite vector spaces are introduced. We discuss the geometrical concept of partial t-spreads together with its implications for the “packing problem” and a recent application in the existence of “cooling codes”.
- 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.
- ItemProfinite groups(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Bartholdi, LaurentProfinite objects are mathematical constructions used to collect, in a uniform manner, facts about infinitely many finite objects. We shall review recent progress in the theory of profinite groups, due to Nikolov and Segal, and its implications for finite groups.
- ItemProny’s method: an old trick for new problems(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Sauer, TomasIn 1795, French mathematician Gaspard de Prony invented an ingenious trick to solve a recovery problem, aiming at reconstructing functions from their values at given points, which arose from a specific application in physical chemistry. His technique became later useful in many different areas, such as signal processing, and it relates to the concept of sparsity that gained a lot of well-deserved attention recently. Prony’s contribution, therefore, has developed into a very modern mathematical concept.
- ItemUltrafilter methods in combinatorics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Goldbring, IsaacGiven a set X, ultrafilters determine which subsets of X should be considered as large. We illustrate the use of ultrafilter methods in combinatorics by discussing two cornerstone results in Ramsey theory, namely Ramsey’s theorem itself and Hindman’s theorem. We then present a recent result in combinatorial number theory that verifies a conjecture of Erdos known as the “B + C conjecture”.
- ItemNumber theory in quantum computing(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Schönnenbeck, SebastianAlgorithms are mathematical procedures developed to solve a problem. When encoded on a computer, algorithms must be "translated" to a series of simple steps, each of which the computer knows how to do. This task is relatively easy to do on a classical computer and we witness the benefits of this success in our everyday life. Quantum mechanics, the physical theory of the very small, promises to enable completely novel architectures of our machines, which will provide specific tasks with higher computing power. Translating and implementing algorithms on quantum computers is hard. However, we will show that solutions to this problem can be found and yield surprising applications to number theory.
- 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.