11 results
Search Results
Now showing 1 - 10 of 11
- 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”.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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”.
- 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.
- 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.