Title: Triple junction solution for the Allen-Cahn systemAbstract: In this talk, I will discuss recent results on the vector-valued Allen-Cahn system with a triple-well potential. We establish the existence of an entire minimizing solution that asymptotically converges to a triple junction, corresponding to a planar minimal cone, at infinity. Furthermore, we prove the uniqueness of […]
Title: Hodge symmetries of singular varieties.Abstract: The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge–Du Bois diamond. For example, the failure […]
Title: Higher Modularity of Elliptic Curves Abstract: Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), […]
Title: Spectral algebraic geometry and topological modular formsAbstract: An important area of study in homotopy theory is the study of elliptic cohomology. In this talk we briefly review the context of elliptic cohomology and its interpretation in terms of spectral algebraic geometry. Furthermore, we construct the Deligne-Mumford compactification of the stack of oriented elliptic curves. If time […]
Title - Variations of the Yang-Mills Lagrangian in high dimensionAbstract - In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under L^2-control of their curvature. We will start from the classical theory in critical […]
Title: Motives in geometric representation theoryAbstract: The spherical and Iwahori-Hecke algebras of a reductive group are of great importance in the Langlands program. They are categorified by equivariant sheaves on the affine Grassmannian and affine flag variety respectively, as introduced in the previous talk. Similarly, the Satake and Bernstein isomorphisms are categorified by the Satake equivalence and […]
Title: Relative Langlands duality. Abstract: We will review those aspects of the conjectures of Ben-Zvi–Sakellaridis–Venkatesh, that are relevant to the preprint https://arxiv.org/abs/2504.00275 by Liu and Wang. Reference: https://arxiv.org/abs/2409.04677.
Title: Stable maps to quotient stacks with a properly stable point.Abstract: I will present a compactification of the moduli space of maps from families of curves, to certain moduli spaces M, via the example of M being the GIT moduli space of binary forms of degree 2n. The main application of our results is the construction of […]