Termine

Abstract: The bottom homotopy group of a flat commutative algebra over the sphere spectrum inherits a delta-ring structure from the Tate-valued Frobenius of Nikolaus-Scholze, which plays a big role in the relationship between trace methods and prismatic cohomology. It is an interesting question how much of the E_infty ring structure is remembered by this delta-ring structure. In this talk, we present a modest step towards understanding this, by proving that there is a one-to-one correspondence between H_infinity ring structures on flat S-modules and delta-ring structures on their pi_0.

Markus Stroppel will give a lecture series on advanced topics in locally compact groups. The lectures take place on Monday at 14:15 in room SR 1D They will start on Monday, November 17. The planned topics are: * the scale function and tidy subgroups * automorphism groups of trees

Abstract: Tian's classical theorem shows that, for a polarized complex projective manifold, the Bergman kernel grows asymptotically like a monic polynomial of degree equal to the dimension. I will present a weighted extension of this result that includes, for instance, equivariant and divisor-adapted Bergman kernels. In contrast to the classical case, the leading coefficient now reflects the speed of a geodesic ray in Mabuchi geometry associated with the weight. In the equivariant case, this speed coincides with the Hamiltonian of the induced circle action, while in the divisor case, it arises via a resolution of the Monge-Ampère equation on the deformation to the normal cone. Finally, I will discuss how this equidistribution result connects to the theory of canonical metrics, and in particular, its implications for the Wess-Zumino-Witten equation.

We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $\omega^0_\varepsilon\in C^\infty_c(\mathbb{R}^2)$ converging to $\omega^0_{\mathrm{sing}}$ distributionally as ε $\to 0^+$, and show that the corresponding solutions $\omega_ε(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff--Rott system with initial datum $\omega^0_{\mathrm{sing}}$. Based on joint work with Alberto Enciso and Antonio Fernandez.

Xmas PDE Colloquium - coffee break after (14:45 - 15:15, SRZ Lounge)

Imagine agents moving according to simple rules that disfavor close crowding but favor a mutual intermediate range proximity. Systems of this type appear across the sciences, from molecular dynamics to microbiology and ecology. The competition between the underlying short-range repulsion and intermediate-range attraction can lead to a phase transition, where the preferred state changes from a "crystalline" equidistribution to the formation of clusters (or colonies) separated by vacuum regions. I will describe recent work that analyzes this transition (or bifurcation) in some detail, emphasizing a curious aspect that makes this transition "non-hysteretic" or "reversible" in an infinite system-size limit, thus allowing for easy switching from crystal to cluster -- and back. Results include predictions for sizes and shapes of vacuum regions, corrections due to noise, and expansions for finite system size corrections. This is based on joint work with Angela Stevens and on a summer REU project mentored by Olivia Clifton.

Xmas PDE Colloquium - coffee break before (14:45 - 15:15, SRZ Lounge)

Many linear and nonlinear partial differential equations (PDEs) arise from the minimization of an underlying energy functional. Examples are ubiquitous and arise, for instance, in the study of non-Newtonian fluids, minimal surfaces, or nonlinear mechanics. Whereas classical numerical methods and their analysis mostly focus entirely on the solution of the associated PDE/Euler-Lagrange equations, recent contributions have started to take the energetic structure into account. This is particularly relevant for engineering applications where the energetic behavior of minimizers can be more relevant than the solution itself. In addition, analysis in terms of energy often naturally fits the structure of the problem, which can be exploited to derive error bounds with explicit constants. We will introduce the method of flux-equilibration which yields highly effective a posteriori error bounds that can subsequently be used to steer adaptive mesh-refinement. After a general introduction to the concept, we highlight recent advances and generalizations to the nonlinear setting and present numerical results highlighting the efficiency of the method.

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