Fellows Seminar - Two unifying principles in mathematics: duality and stability

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What is the relationship between Plato’s perfect solids and the Fourier transform? Do oceanic vortices and magnetic nanowires have anything in common? USIAS Fellows Raphaël Côte and Alexandru Oancea, both from Strasbourg University’s Institute for Advanced Mathematical Research (IRMA), intend to take the audience on a mathematical journey in order to explain, and illustrate, the fundamental concepts of duality and stability.

Instantiations of duality in mathematics, with applications

By Alexandru Oancea (2021 Fellow)

In mathematics, we speak of a duality whenever we are confronted with a correspondence between objects of some kind A and objects of some other kind B. Duality is a unifying principle in mathematics, and some dualities are more interesting than others.

The correspondence between lines and planes in Euclidean 3-space given by the orthogonality relation is the prototype of dualities of a geometric nature. A fancy version of this brings us to the five platonic solids: the cube and the octahedron are dual to each other, as are the icosahedron and the dodecahedron, whereas the tetrahedron is self-dual. The Fourier transform is the prototype of dualities of an analytic nature. It consists in representing a function as a trigonometric series, and it lies at the core of both modern analysis and engineering.  

Most dualities assume, in some form or another, finite dimensionality. The goal of the research project Applications of Poincaré duality in symplectic topology was to explore a new instance of duality in infinite dimensions.

Dynamics and pattern formation in ferromagnetism

By Raphaël Côte (2022 Fellow)

Magnetization occurs in a material when the spins attached to its particles share a common orientation; ferromagnetism is the ability that certain materials possess, to maintain a magnetization induced by a magnetic field, possibly an external one. It is the standard phenomenon that can be observed in everyday solid magnets (like a fridge magnet), and is remarkably stable at low temperatures.

During the presentation, I will show some numerical simulations on how magnetization evolves when geometry comes into play (not necessarily in a solid material): chains of spins, nanowires, etc. I will also describe, in comparison, what mathematics can prove about this.

Raphaël Côte’s USIAS project was entitled Dynamics and control in micromagnetism.