Snapshots of Modern Mathematics from Oberwolfach

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A tale of three curves
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Balakrishnan, Jennifer S.
In this snapshot, we give a survey of some problems in the study of rational points on higher genus curves, discussing questions ranging from the era of the ancient Greeks to a few posed by mathematicians of the 20th century. To answer these questions, we describe a selection of techniques in modern number theory that can be used to determine the set of rational points on a curve.
Algebra, matrices, and computers
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Detinko, Alla; Flannery, Dane; Hulpke, Alexander
What 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.
The Algebraic Statistics of an Oberwolfach Workshop
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Seigal, Anna
Algebraic Statistics builds on the idea that statistical models can be understood via polynomials. Many statistical models are parameterized by polynomials in the model parameters; others are described implicitly by polynomial equalities and inequalities. We explore the connection between algebra and statistics for some small statistical models.
Aperiodic Order and Spectral Properties
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Baake, Michael; Damanik, David; Grimm, Uwe
Periodic 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!
Arrangements of lines
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2014) Harbourne, Brian; Szemberg, Tomasz
We discuss certain open problems in the context of arrangements of lines in the plane.
Computing 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.
Diophantine equations and why they are hard
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Pasten, Hector
Diophantine 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.
Expander graphs and where to find them
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Khukhro, Ana
Graphs 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.
A few shades of interpolation
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Szpond, Justyna
The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points from given data points. The new points usually lie in between the already-known points. The purpose of this snapshot is to introduce a particular type of interpolation, namely, polynomial interpolation. This will be explained starting from basic ideas that go back to the ancient Babylonians and Greeks, and will arrive at subjects of current research activity.
Finite geometries: pure mathematics close to applications
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Storme, Leo
The 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”.
Friezes and tilings
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Holm, Thorsten
Friezes 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.
From computer algorithms to quantum field theory: an introduction to operads
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Krähmer, Ulrich
An 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.
From the dollar game to the Riemann-Roch Theorem
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Lamboglia, Sara; Ulirsch, Martin
What 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.
Geproci Sets: a New Perspective in Algebraic Geometry
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2023) Chiantini, Luca; Harbourne, Brian
Geproci 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.
Ideas of Newton-Okounkov bodies
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2015) Kiritchenko, Valentina; Timorin, Vladlen; Smirnov, Evgeny
In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion.
Invitation to quiver representation and Catalan combinatorics
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2021) Rognerud, Baptiste
Representation 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.
Jewellery from tessellations of hyperbolic space
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Gangl, Herbert
In this snapshot, we will first give an introduction to hyperbolic geometry and we will then show how certain matrix groups of a number-theoretic origin give rise to a large variety of interesting tessellations of 3-dimensional hyperbolic space. Many of the building blocks of these tessellations exhibit beautiful symmetry and have inspired the design of 3D printed jewellery.
News on quadratic polynomials
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2017) Pottmeyer, Lukas
Many problems in mathematics have remained unsolved because of missing links between mathematical disciplines, such as algebra, geometry, analysis, or number theory. Here we introduce a recently discovered result concerning quadratic polynomials, which uses a bridge between algebra and analysis. We study the iterations of quadratic polynomials, obtained by computing the value of a polynomial for a given number and feeding the outcome into the exact same polynomial again. These iterations of polynomials have interesting applications, such as in fractal theory.
Number theory in quantum computing
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2018) Schönnenbeck, Sebastian
Algorithms 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.
On Logic, Choices and Games
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2019) Oliva, Paulo
Can 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.

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Browsing Snapshots of Modern Mathematics from Oberwolfach by Subject "Algebra and Number Theory"

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