The L-space conjecture makes a prediction about which rational homology spheres can admit a taut foliation. But where could the predicted taut foliations "come from"? Must they be compatible with “natural” geometric structures on the 3-manifold? In this talk, I'll discuss forthcoming work with John Baldwin and Matt Hedden, where we address a type of geography problem for taut foliations. In particular, we show that when K is a fibered strongly quasipositive knot, large surgeries along K can never admit a taut foliation which is ‘’compatible’’ with the natural flow on the Dehn surgered manifold. I'll explain why this is surprising, and if time permits, sketch the proof. No background will be assumed — all are welcome!

Abstract: When a spatially homogeneous state destabilizes, localized perturbations can grow into large amplitude spatial patterns, which spread into the bulk, invading the unstable state. The nature and properties of the patterns which appear, such as wavelength, orientation, and amplitude, are frequently determined by the behavior in the leading edge of the spreading process. We consider such pattern-forming fronts in the FitzHugh–Nagumo PDE in the so-called oscillatory regime. The pattern is selected from a family of periodic traveling wave train solutions by an invasion front. Using geometric singular perturbation techniques, we construct “pushed” and “pulled” pattern-forming fronts as heteroclinic orbits between the unstable steady state and a periodic orbit representing the wave train in the wake. In the case of pushed fronts, the wave train necessarily passes near a pair of nonhyperbolic fold points on the associated critical manifold. We also discuss implications for the stability of the pattern-forming fronts and the challenges introduced by the fold points in the corresponding spectral stability problem.

We define essential dimension for finite étale covers as the minimal dimension of a space W to which the cover compresses over a dense open, recalling both the scheme-theoretic and rigid-analytic versions. We then introduce perfectoid essential dimension: the same compression problem, but allowing W to be perfectoid and measuring its dimension in the spectral sense. The main theorem is tilting invariance: for a finite étale cover Y→X of perfectoid spaces, ed(Y/X)=ed(Y♭/X♭), using the tilting equivalence on perfectoid spaces and finite étale sites together with compatibility properties of base change and dimension.

In this talk, I will present recent advances on the Calderon problem for nonlocal wave equations, with particular emphasis on models incorporating a weak viscosity term. I will begin by reviewing inverse problems for viscous and damped nonlocal wave equations, and discuss the analytical tools used to establish unique determination results in each setting. I will then highlight the new challenges posed by weakly viscous equations and explain how these difficulties can be overcome.

A central part of the talk is devoted to a space–time Runge approximation theorem for such equations, which relies on the existence of very weak solutions to linear nonlocal wave equations with irregular sources. This approximation result plays a crucial role in the analysis of inverse problems for nonlinear weakly viscous wave models. At the end, I will present several illustrative applications. The results presented in this talk are based on joint work with Yi-Hsuan Lin, Teemu Tyni, and Katya Krupchyk.