The Soliton Resolution Conjecture predicts that finite-energy solutions to nonlinear dispersive PDEs asymptotically decouple into a sum of stationary solutions, called solitons, and free radiation, with an error that goes to zero in the energy norm. In this talk, we discuss the conjecture for the energy-critical nonlinear heat equation in dimension $d\geq 3$ and present its proof in the radial case. If time permits, we will also discuss recent progress in the non-radial setting, with potential applications to other geometric flows such as the Yang-Mills heat flow.

Poisson sigma models sit at the intersection of deformation theory, geometry, and quantum field theory; specifically, the perturbative expansion of the two-dimensional Poisson sigma model with boundary is known to recover Kontsevich’s deformation quantization formula. In this talk, we introduce a higher-dimensional holomorphic–topological generalization of Poisson sigma models and explain their connections to the deformation quantization (or obstruction) of holomorphic–topological factorization algebras. This construction can be viewed as a field-theoretic incarnation of the higher Deligne conjecture. We also explain how these models relate to the construction of Hopf-type algebras via Koszul duality, and explore examples related to the quantization of Lie bialgebras, W-algebras, and Yangians.