The `Harder-Narasimhan trace’ and unitarity of the KZ/Hitchin connection:...
Let a reductive group $G$ act on a projective variety $\mathcal{X}_+$, and suppose given a lift of the action to an ample line bundle $\hat{\theta}$. By definition, all $G$-invariant sections of...
View ArticleRegularity of flat level sets in phase transitions
We consider local minimizers of the Ginzburg-Landau energy functional \[\int \frac{1}{2}|\nabla u|^2 + \frac{1}{4}(1-u^2)^2dx\] and prove that, if the $0$ level set is included in a flat cylinder then,...
View ArticleExistence of Engel structures
We develop a construction of Engel structures on $4$-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable $4$-manifolds admit an Engel...
View ArticleOn Serre’s conjecture for 2-dimensional mod p representations of...
We prove the existence in many cases of minimally ramified $p$-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod $p$ representations $\overline{\rho}$ of the absolute Galois...
View ArticleHolomorphic curves into algebraic varieties
This paper establishes a defect relation for algebraically nondegenerate holomorphic mappings into an arbitrary nonsingular complex projective variety $V$ (rather than just the projective space)...
View ArticleFinite groups of symplectic automorphisms of K3 surfaces in positive...
We show that Mukai’s classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic $p$ under assumptions that (i) the order of the group is...
View ArticleFitting a $C^m$-smooth function to data, I
Suppose we are given a finite subset $E \subset \mathbb{R}^n$ and a function $f: E \rightarrow \mathbb{R}$. How to extend $f$ to a $C^m$ function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ with $C^m$...
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