The ‘What is…?’ seminar series is about an idea, concept or method that the speaker has found surprising, useful or intriguing, and which they would like to share with colleagues and students. The talk answers a question of the form “What is…?” and is directed at a broad audience of non-experts and experts alike. There is ample time for discussion, comments and questions. This talk may also serve as a prelude to a more technical talk in a specialised seminar series. Recordings can be found in the YouTube playlist.
‘What is…?’ Seminar ‘What is a cactus group?’
Iva Halacheva, Northeastern University
Thursday 3 August 2023
Abstract: The braid group is a classical algebraic object with an intuitively natural presentation via stacking pictures of braided strands. One point of view which makes them interesting to topologists is the interpretation of braid groups as the fundamental groups of certain configuration spaces. Braid groups also play a central role in representation theory through the Yang-Baxter equation, where they capture the symmetries of quantum group representations. The cactus group is a close relative of the braid group whose properties are yet to be fully explored. Cactus groups can also be viewed as the fundamental groups of interesting topological spaces and have recently been linked to combinatorial structures associated with quantum groups. I will describe the construction of the cactus group and outline some of the settings in which it appears.
‘What is…?’ Seminar ‘What is the parametrization method in dynamical systems?’
Bob Rink, Vrije Universiteit Amsterdam
Thursday 11 May 2023
Abstract: The parametrization method is a tool to compute invariant manifolds in dynamical systems, such as periodic orbits, (un-)stable manifolds, slow manifolds and invariant tori. The idea behind the method is simple: it works by (algorithmically) finding an embedding of the invariant manifold together with a representation of its dynamics in a coordinate chart. De La Llave et al realized that the method can nicely be combined with ideas from rigorous numerics, to provide computer-assisted proofs for the existence of invariant manifolds. Others, including myself, have used the method to compute high-precision approximations of the dynamics on the invariant manifolds. I will discuss both approaches, starting with the basics and finishing with an unpublished result on high-order phase reduction. Watch the recording.
Speaker bio: Bob Rink is a professor of mathematical analysis at the Vrije Universiteit Amsterdam, The Netherlands. He obtained his PhD in 2003 from Utrecht University, after which he was an EPSRC postdoctoral fellow at Imperial College London. He came to the Vrije Universiteit Amsterdam in 2007, where he became full professor in 2016. His research is in various directions of dynamical systems, with a focus on network dynamics and bifurcation theory.
‘What is…?’ Seminar ‘What is the parametrization method in dynamical systems?’
Sándor Kovács, University of Washington
Thursday 20 April 2023
Abstract: Max Noether said that algebraic curves were created by God and algebraic surfaces by the Devil. Unfortunately, that description seems to be also valid for the moduli theory of these objects respectively. I will recall one of the first obstacles one faces when trying to extend the basic results of the moduli theory of curves to that of surfaces and then discuss how one may resolve the arising issue. Time permitting I will also explain the various stability conditions this problem and its resolution led to.
Speaker bio: Sándor Kovács is a Professor of Mathematics at the University of Washington. He received his BS degree at Eötvös University in his native country of Hungary, and his PhD at the University of Utah. He held positions at MIT and the University of Chicago before moving to the University of Washington. He received the National Science Foundation’s Faculty Early Career Development Award, the American Mathematical Society’s Centennial Research Fellowship, an Alfred P. Sloan Foundation Research Fellowship, and two Simons Foundation Fellowships, one of which he is currently holding. He is a Fellow of the American Mathematical Society.