Snapshots of Modern Mathematics from Oberwolfach
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Browsing Snapshots of Modern Mathematics from Oberwolfach by Subject "Discrete Mathematics and Foundations"
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- ItemAperiodic Order and Spectral Properties(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Baake, Michael; Damanik, David; Grimm, UwePeriodic structures like a typical tiled kitchen floor or the arrangement of carbon atoms in a diamond crystal certainly possess a high degree of order. But what is order without periodicity? In this snapshot, we are going to explore highly ordered structures that are substantially nonperiodic, or aperiodic. As we construct such structures, we will discover surprising connections to various branches of mathematics, materials science, and physics. Let us catch a glimpse into the inherent beauty of aperiodic order!
- ItemComputing with symmetries(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Roney-Dougal, Colva M.Group theory is the study of symmetry, and has many applications both within and outside mathematics. In this snapshot, we give a brief introduction to symmetries, and how to compute with them.
- 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.
- 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”.
- 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.
- 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.
- 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 the dollar game to the Riemann-Roch Theorem(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Lamboglia, Sara; Ulirsch, MartinWhat 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 a new perspective on a century-old theorem.
- ItemGeproci Sets: a New Perspective in Algebraic Geometry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Chiantini, Luca; Harbourne, BrianGeproci sets arise from applying the perspective of inverse scattering problems to algebraic geometry. Analogous to the reconstruction of an object from multiple X-ray images, we aim at a classification of sets with certain algebraic properties under multiple projections.
- 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.
- ItemInvitation to quiver representation and Catalan combinatorics(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Rognerud, BaptisteRepresentation theory is an area of mathematics that deals with abstract algebraic structures and has numerous applications across disciplines. In this snapshot, we will talk about the representation theory of a class of objects called quivers and relate them to the fantastic combinatorics of the Catalan numbers.
- ItemLimits of graph sequences(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Klimošová, TerezaGraphs are simple mathematical structures used to model a wide variety of real-life objects. With the rise of computers, the size of the graphs used for these models has grown enormously. The need to efficiently represent and study properties of extremely large graphs led to the development of the theory of graph limits.
- 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.
- ItemDas Problem der Kugelpackung(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Dostert, Maria; Krupp, Stefan; Rolfes, Jan HendrikWie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3-dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen Kommunikation hat, hört sich einfach an, ist jedoch für Kugeln in höheren Dimensionen noch immer ungelöst. Sogar die Berechnung guter Näherungslösungen ist für die meisten Dimensionen schwierig.
- ItemRandom sampling of domino and lozenge tilings(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2016) Fusy, ÉricA grid region is (roughly speaking) a collection of “elementary cells” (squares, for example, or triangles) in the plane. One can “tile” these grid regions by arranging the cells in pairs. In this snapshot we review different strategies to generate random tilings of large grid regions in the plane. This makes it possible to observe the behaviour of large random tilings, in particular the occurrence of boundary phenomena that have been the subject of intensive recent research.
- 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.
- 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.
- ItemThe Geometry of Fair Division(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Frick, FlorianHow can we fairly divide a necklace with various types of beads? We use this problem as a motivating example to explain how geometry naturally appears in solutions of non-geometric problems. The strategy we develop to solve this problem has been used in several other contexts.
- ItemTropical geometry(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Brugallé, Erwan; Itenberg, Ilia; Shaw, Kristin; Viro, OlegWhat kind of strange spaces hide behind the enigmatic name of tropical geometry? In the tropics, just as in other geometries, one of the simplest objects is a line. Therefore, we begin our exploration by considering tropical lines. Afterwards, we take a look at tropical arithmetic and algebra, and describe how to define tropical curves using tropical polynomials.
- 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”.