Ultrafilter methods in combinatorics
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Discrete Mathematics and Foundations
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Invitation to quiver representation and Catalan combinatorics
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Representation theory is an area of mathematics that deals with abstract algebraic structures and has nu- merous applications across disciplines. In this snap- shot, 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.
From the dollar game to the Riemann-Roch Theorem
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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.
Formation Control and Rigidity Theory
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Formation 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 on-board sensing that ensures the desired formation is obtained.
Limits of graph sequences
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Graphs 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.
On Logic, Choices and Games
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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.
Sombrero Vueltiao –Weaving Mathematics
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This paper describes the weaving technique of a traditional Colombian hat called sombrero vueltiao and some of its patterns. The making of the hat–and weaving in general–involves interesting mathematical thoughts and concepts and offers many learning opportunities, for all of which we provide examples in this paper.
Snake graphs, perfect matchings and continued fractions
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A 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.
Computing with symmetries
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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.
Aperiodic Order and Spectral Properties
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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, ma- terials science, and physics. Let us catch a glimpse into the inherent beauty of aperiodic order!