Abstract:We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most . As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most .

Abstract:We study the topology of a space parametrizing stable tropical curves of genus with volume , showing that its reduced rational homology is canonically identified with both the top weight cohomology of and also with the genus part of the homology of Kontsevich's graph complex. Using a theorem of Willwacher relating this graph complex to the Grothendieck-Teichmüller Lie algebra, we deduce that

Abstract:We introduce the notion of a categorical join, which can be thought of as a categorification of the classical join of two projective varieties. This notion is in the spirit of homological projective duality, which categorifies classical projective duality. Our main theorem says that the homological projective dual category of the categorical join is naturally equivalent to the categorical

Abstract:After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (i.e., they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift

Abstract:Given a henselian pair of commutative rings, we show that the relative -theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace . This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod coefficients, with invertible in ) and McCarthy's theorem on relative -theory (when is nilpotent). We deduce

Abstract:We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito's theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebe

Abstract:A new approach to stable motivic homotopy theory is given. It is based on Voevodsky's theory of framed correspondences. Using the theory, framed motives of algebraic varieties are introduced and studied in the paper. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension -spectrum of any smooth scheme . Moreover, it is shown that

Abstract:We show that the chromatic number of is not concentrated on fewer than consecutive values. This addresses a longstanding question raised by Erdős and several other authors.

Abstract:We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.

Abstract:We prove the test function conjecture of Kottwitz and the first-named author for local models of Shimura varieties with parahoric level structure, and their analogues in equal characteristic.

Abstract:We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of Maulik and Okounkov [Astérisque 408 (2019), ix+209]. We apply them to the computation of the monodromy of -difference equations arising in the enumerative K-theory of rational curves in Nakajima varieties, including the quantum

Abstract:We construct and study a -structure on -typical cyclotomic spectra and explain how to recover crystalline cohomology of smooth schemes over perfect fields using this -structure. Our main tool is a new approach to -typical cyclotomic spectra via objects we call -typical topological Cartier modules. Using these, we prove that the heart of the cyclotomic -structure is the full subcategory of

Abstract:We prove the functoriality for a proper push-forward of the characteristic cycles of constructible complexes by morphisms of smooth projective schemes over a perfect field, under the assumption that the direct image of the singular support has the dimension at most that of the target of the morphism. The functoriality is deduced from a conductor formula which is a special case for morphisms

Abstract:In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.

Abstract:We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a nonempty phase in which there are infinite light clusters, which implies the existence of a nonempty phase in which there are infinitely many infinite clusters. That

Abstract:We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields (the trivial bound being coming from the bound on the entire class group). This yields corresponding improvements to: (1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves, (2) bounds of Helfgott and Venkatesh

Abstract:We give new examples of compact, negatively curved Einstein manifolds of dimension . These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of four-manifolds previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1-12). The construction begins with a certain sequence

Abstract:In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume of a stratum indexed by a partition is , as tends to . This confirms a prediction of Eskin-Zorich and generalizes some of

Abstract:In this paper we prove the following results: We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. We prove that the period map associated to any pure polarized

Abstract:In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of ``Pacman Renormalization Theory'' that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points

Abstract:For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no -torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in and . In manifolds with -torsion, one surface can be put into a normal

Abstract:In this article we give a new proof of Ngô's geometric stabilisation theorem, which implies the fundamental lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme to the cohomology of Hitchin fibres for the endoscopy groups . Our proof avoids the decomposition and support theorem, instead the argument is based on results for -adic

Abstract:We study the symplectic topology of certain surfaces (including the ``mirror quartic'' and ``mirror double plane''), equipped with certain Kähler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence

Abstract:Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of Špenko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more

Abstract:Soit  une compactification lisse d'un espace homogène d'un groupe algébrique linéaire  sur un corps de nombres . Nous établissons la conjecture de Colliot-Thélène, Sansuc, Kato et Saito sur l'image du groupe de Chow des zéro-cycles de  dans le produit des mêmes groupes sur tous les complétés de . Lorsque  est semi-simple et simplement connexe et que le stabilisateur géométrique est fini et