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

Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative

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-Kebekus-Kov\'acs-Peternell

After H\"older 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 operator

We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of arXiv:1211.1287. We apply them to the computation of the monodromy of $q$-difference equations arising the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik-Zamolodchikov equations.

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.

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.

Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives $SH_{S^1}^{fr}(k)$ is introduced and studied. To any smooth algebraic variety $X\in Sm/k$, the framed motive $M_{fr}(X)$ is associated in the category $SH_{S^1}^{fr}(k)$. Also, for any smooth scheme $X\in Sm/k$ an explicit quasi-fibrant motivic replacement of its suspension P^1-spectrum is given

We show that the chromatic number of $G_{n, \frac 12}$ is not concentrated on fewer than $n^{\frac 14 - \varepsilon}$ consecutive values. This addresses a longstanding question raised by Erdős and several other authors.

We prove the functoriality for 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 to curves

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

Let $G$ be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of $G$, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of $\operatorname{Bun}_G$. It is independent of the work of Lafforgue on the

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized

In this paper, we derive apriori estimates for constant scalar curvature K\"ahler metrics on a compact K\"ahler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the K\"ahler potential. We also discuss some local versions of these estimates which can be of independent interest.

We prove an isoperimetric inequality which holds for minimal submanifolds in Euclidean space of arbitrary dimension and codimension. Our estimate is sharp if the codimension is at most $2$.

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 $\nu_1 \big( \mathcal{H}_1 (m) \big)$ of a stratum indexed by a partition $m = (m_1, m_2, \ldots , m_n)$ is $\big( 4 + o(1) \big) \prod_{i

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 non-empty phase in which there are infinite light clusters, which implies the existence of a non-empty phase in which there are infinitely many infinite clusters. That is, we

We give a new proof of a uniform version of the Yau-Tian-Donaldson conjecture for Fano manifolds with finite automorphism group, and of the semistable case of the conjecture. Our approach does not involve the continuity method or Cheeger-Colding-Tian theory. Instead, the proof is variational and uses pluripotential theory and certain non-Archimedean considerations.

We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of $(S, \mathbb{V})$ is a countable

We give new examples of compact, negatively curved Einstein manifolds of dimension $4$. These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of 4-manifolds $(X_k)$ previously considered by Gromov and Thurston. The construction begins with a certain sequence $(M_k)$ of hyperbolic 4-manifolds, each containing a totally geodesic surface $\Sigma_k$

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 join of the

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 $K$ (the trivial bound being $O_{\epsilon}(|{\rm Disc}(K)|^{1/2+\epsilon})$ by Brauer--Siegel). 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

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.

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. Then we prove

In this article we give a new proof of Ng\^o'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 $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_{\kappa}$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results

For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-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 $S^4$ and $\pi_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain Kahler 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 group of

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko 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 complicated

Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Th\'el\`ene, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence

We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of "homological conjectures" in commutative algebra [H1][HH2]. Namely, for any local homomorphism $ R\to R'$ of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay $R$-algebra and some Cohen-Macaulay $R'$-algebra. When $R$ contains a field, this is already known [[3

We compute the p-adic geometric etale cohomology of the coverings of Drinfeld's half-plane, and we show that, if the base field is Q_p, this cohomology encodes the p-adic local Langlands correspondence for 2-dimensional de Rham representations (of weight 0 and 1).

We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input size)}^{1+o(1)}$. This improves upon the previously known $\text{(input size)}^{\frac32 +o(1)}$ bound. The new algorithm relies on numerical continuation along

The Remodeling Conjecture proposed by Bouchard-Klemm-Mari\~{n}o-Pasquetti (BKMP) [arXiv:0709.1453, arXiv:0807.0597] relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for

We prove that a complete K\"ahler manifold with holomorphic curvature bounded between two negative constants admits a unique complete K\"ahler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background K\"ahler metric. Furthermore, all three metrics are shown to be uniformly equivalent to the Bergman metric, if the complete K\"ahler manifold

We prove a quantitative equidistribution statement for adelic homogeneous subsets whose stabilizer is maximal and semisimple. Fixing the ambient space, the statement is uniform in all parameters. We explain how this implies certain equidistribution theorems which, even in a qualitative form, are not accessible to measure-classification theorems. As another application, we describe another proof of

We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a $K$-subalgebra of $R$ generated by $B \leq {}^\eta\mathcal{B}(n,d)$ forms $G_1,..., G_B$ of degree $\leq d$, where ${}^\eta\mathcal{B}(n,d)$ does not depend on $N$ or $K$,

Tsirelson's problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to Connes embedding

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory.

We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic

We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked

We show that over any uncountable field of characteristic different from two, a very general hypersurface of dimension $n > 2$ and degree at least $log_2 (n) + 2$ is not stably rational. This significantly improves earlier results of Kollar and Totaro. As a byproduct of our proof, we obtain new counterexamples to the integral Hodge conjecture, answering a question of Voisin and Colliot-Thelene - Voisin

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}={\rm Lie}(G)$. Specifically, we show that one of the following holds: $\mathfrak{m}={\rm Lie}(M)$ for some maximal connected subgroup $M$ of

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these

We prove that any finite-degree polynomial functor is topologically Noetherian. This theorem is motivated by the recent resolution of Stillman's conjecture and a recent Noetherianity proof for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman's conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators

We give a construction of contact homology in the sense of Eliashberg--Givental--Hofer. Specifically, we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo-holomorphic curves.

We prove a conjecture of Boucksom-Demailly-Puaun-Peternell, namely that on a projective manifold $X$ the cone of pseudoeffective classes in $H^{1,1}_{\mathbb{R}}(X)$ is dual to the cone of movable classes in $H^{n-1,n-1}_{\mathbb{R}}(X)$ via the Poincar\'e pairing. This is done by establishing a conjectured transcendental Morse inequality for the volume of the difference of two nef classes on a projective

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems. The results

We study the category of G(O)-equivariant perverse coherent sheaves on the affine Grassmannian of G. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized r-matrices. These are canonical nonzero maps between convolution products

We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.

The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion

We establish geometric criteria to decide whether a contact manifold is over- twisted. Starting with the original denition, we rst relate the dierent overtwisted disks (D 2n ; ot) in each dimension and show that a manifold is overtwisted if the Legendrian un- knot is loose. Then we characterize overtwistedness in terms of open book decompositions and provide several applications.

Let $X$ be a locally finite irreducible affine building of dimension $\geq 2$ and $\Gamma \leq \mathrm{Aut}(X)$ be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is $\Gamma$ linear? More generally, when does $\Gamma$ admit a finite-dimensional representation with infinite image over a commutative unital ring? If $X$ is the Bruhat--Tits building

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing

We construct a smooth, area preserving, mixing flow with finitely many non-degenerate fixed points and no saddle connections on a closed surface of genus 5. This resolves a problem that has been open for four decades.

—The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VPws and VNP. Mulmuley and Sohoni [32] suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed

We prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of degree $k$ is locally equivalent to (Spec $A,\omega'$) for Spec $A$ an affine derived scheme whose cdga $A$ has Darboux-like coordinates in which the symplectic form

Let $G$ be a compact Lie group and $\mathbf{k}$ be a field of characteristic $p \geq 0$ such that $H^* (G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category $\mathcal{F}(X; \mathbf{k})$ if and only if it represents a non-zero object of that

We prove that every Besicovitch set in ℝ^3 must have Hausdorff dimension at least 5/2 + ϵ_0 for some small constant ϵ_0 > 0. This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the SL_2 example; this object resembles a Besicovitch set that has Minkowski