The good pants homology and the Ehrenpreis Conjecture
We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces. (Good pants are pairs of pants whose cuffs have the length nearly equal to some...
View ArticlePositivity for cluster algebras
We prove the positivity conjecture for all skew-symmetric cluster algebras.
View ArticleAnomalous dissipation for $1/5$-Hölder Euler flows
Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager’s...
View ArticleMutations of puzzles and equivariant cohomology of two-step flag varieties
We introduce a mutation algorithm for puzzles that is a three-direction analogue of the classical jeu de taquin algorithm for semistandard tableaux. We apply this algorithm to prove our conjectured...
View ArticleUniqueness of blowups and Łojasiewicz inequalities
Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly...
View ArticleChern slopes of simply connected complex surfaces of general type are dense...
We prove that for any number $r \in [2,3]$, there are spin (resp. nonspin and minimal) simply connected complex surfaces of general type $X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$. In...
View ArticleInterlacing families I: Bipartite Ramanujan graphs of all degrees
We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than $2$. We do this by proving a variant of a conjecture of Bilu and Linial about the existence...
View ArticleInterlacing families II: Mixed characteristic polynomials and the...
We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the Kadison–Singer problem. The first is Weaver’s conjecture...
View ArticleThe proof of the $l^2$ Decoupling Conjecture
We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is...
View ArticleThe period-index problem for fields of transcendence degree $2$
Using geometric methods we prove the standard period-index conjecture for the Brauer group of a field of transcendence degree $2$ over $\mathbf{F}_p$.
View ArticleAlmost contact 5-manifolds are contact
The existence of a contact structure is proved in any homotopy class of almost contact structures on a closed $5$-dimensional manifold.
View ArticleA polynomial upper bound on Reidemeister moves
We prove that any diagram of the unknot with $c$ crossings may be reduced to the trivial diagram using at most $(236 \,c)^{11}$ Reidemeister moves.
View ArticleRationality of $W$-algebras: principal nilpotent cases
We prove the rationality of all the minimal series principal $W$-algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and $C_2$-cofinite vertex operator algebras. A...
View ArticleA proof of Demailly’s strong openness conjecture
In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.
View ArticleThe circle method and bounds for $L$-functions – IV: Subconvexity for twists...
Let $\pi$ be an $\mathrm{SL}(3,\mathbb Z)$ Hecke-Maass cusp form satisfying the Ramanujan conjecture and the Selberg-Ramanujan conjecture, and let $\chi$ be a primitive Dirichlet character modulo $M$,...
View ArticleIsolation, equidistribution, and orbit closures for the...
We prove results about orbit closures and equidistribution for the $\mathrm{SL}(2,\mathbb{R})$ action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent...
View ArticleTightness is preserved by Legendrian surgery
This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is...
View ArticleOn Schmidt and Summerer parametric geometry of numbers
Recently, W. M. Schmidt and L. Summerer introduced a new theory that allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in...
View ArticleHidden symmetries and decay for the wave equation on the Kerr spacetime
Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use...
View ArticleKontsevich’s graph complex, GRT, and the deformation complex of the sheaf of...
We generalize Kontsevich’s construction of $L_{\infty}$-derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the...
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