In this talk we will investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation, a well-known model for the three-dimensional Euler equations.

We will see that these equations can also be realised as geodesic equations on groups of diffeomorphisms. We will then illustrate precisely when the corresponding exponential map is non-linear Fredholm of index 0. Finally, we will examine the distribution of conjugate points in these settings via a Morse theoretic argument.

]]>*In recent years, there has been an explosion of mathematical activity tied to the formation of singularities in solutions to two of the most revered physical systems of PDEs: Einstein’s equations of general relativity and Euler’s equations of fluid mechanics. In this talk, I will provide an overview of the progress with an eye towards describing profound mathematical connections between these two systems and the role that geometry has played in the analysis. I will also highlight some key challenges for the future.*

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The talk will be structured into two main parts: the first part centers on an almost sure convergence proof for a broader formulation of such schemes, while the second delves into the expressive capabilities of a single step within these iterative methods. We will delve into properties such as invariance, equivariance, and Lipschitz properties associated with the slice-matching operator, which yield recovery and stability outcomes for these approximations. We will also explore various associated affine registration problems and their relationship with slice-matching’s ability to incorporate transformations like shifts and scaling in the initial step. The talk is based on joint work with Caroline Moosmueller.

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According to the citation, “The authors are leading researchers in group theory as well as allied areas of topology and geometry. Their expertise shines through with masterful and clear expositions of the combinatorial, algebraic, geometric and analytic viewpoints that mapping class groups enjoy. Many of the classical theorems, for example, the work of Dehn, are presented from a modern perspective and in particular through the work of Thurston, which was introduced and developed decisively in the short time since the primer was published.”

The full announcement from the American Mathematical Society can be found here: https://www.ams.org/news?news_id=7251

]]>** **** **“Classically”, Mackey and Tambara functors are equivariant generalizations of abelian groups and commutative rings, respectively. What this means is that, in equivariant homotopy theory, Mackey functors appear wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find commutative rings. More recently, work by Bachmann and Hoyois has garnered interest in related structures which appear in motivic homotopy theory – these Motivic Mackey Functors and Motivic Tambara Functors do not have anything to do with group-equivariance, but have the same axiomatics. In this talk, I’ll introduce a general context for interpreting the notions of Mackey and Tambara Functors, which subsumes both the equivariant and motivic notions.

The aim of this approach is to translate theorems between contexts, enriching the theory and providing cleaner proofs of essential facts. To this end, I’ll discuss recent progress in boosting a foundational result about norms from equivariant algebra to this more general context. (contact person: Anna Marie Bohmann)

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