Martin’s Conjecture is a proposed classification of the definable functions on the Turing degrees. It is usually divided into two parts, the first of which classifies functions which are not above the identity and the second of which classifies functions which are above the identity. Slaman and Steel proved the second part of the conjecture for Borel functions which are order-preserving (i.e. which

We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable ∞ \infty -category of non- A 1 \mathbb {A}^1 -invariant motivic spectra, which turns out to be equivalent to the ∞ \infty -category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We

We prove a purity property in telescopically localized algebraic K K -theory of ring spectra: For n ≥ 1 n\geq 1 , the T ( n ) T(n) -localization of K ( R ) K(R) only depends on the T ( 0 ) ⊕ ⋯ ⊕ T ( n ) T(0)\oplus \dots \oplus T(n) -localization of R R . This complements a classical result of Waldhausen in rational K K -theory. Combining our result with work of Clausen–Mathew–Naumann–Noel, one finds

Let A A be drawn uniformly at random from the set of all n × n n\times n symmetric matrices with entries in { − 1 , 1 } \{-1,1\} . We show that \[ P ( det ( A ) = 0 ) ⩽ e − c n , \mathbb {P}( \det (A) = 0 ) \leqslant e^{-cn}, \] where c > 0 c>0 is an absolute constant, thereby resolving a long-standing conjecture.

The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the n-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent

We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the Kardar-Parisi-Zhang (KPZ) equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of Wu [Tightness and local fluctuation

This work concerns a map φ : R → S \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D n ( S / R ; − ) = 0 \mathrm {D}_n(S/R;-)=0 for some n ≥ 1 n\ge 1 , then D i ( S / R ; − ) = 0 \mathrm {D}_i(S/R;-)=0 for all i ≥ 2 i\ge 2 and φ {\varphi } is locally complete intersection. This extends

We give a fully faithful integral model for simply connected finite complexes in terms of E ∞ \mathbb {E}_{\infty } -ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of p p -complete E ∞ \mathbb {E}_{\infty } -rings for each prime p p . Using this, we show that the data of a simply connected

Abstract:In this article we prove a conjecture of Braverman-Kazhdan in [Geom. Funct. Anal. Special Volume (2000), pp. 237–278] on acyclicity of $\rho$-Bessel sheaves on reductive groups. We do so by proving a vanishing conjecture proposed in our previous work [A vanishing conjecture: the GLn case, arXiv:1902.11190]. As a corollary, we obtain a geometric construction of the non-linear Fourier kernel

Motivated by the problem of constructing fast matrix multiplication algorithms, Strassen (FOCS 1986, Crelle 1987–1991) introduced and developed the theory of asymptotic spectra of tensors. For any sub-semiring X \mathcal {X} of tensors (under direct sum and tensor product), the duality theorem that is at the core of this theory characterizes basic asymptotic properties of the elements of X \mathcal

Abstract:A sorting network (also known as a reduced decomposition of the reverse permutation) is a shortest path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of the symmetric group $S_n$ generated by adjacent transpositions. We prove that in a uniform random $n$-element sorting network $\sigma ^n$, all particle trajectories are close to sine curves with high probability. We also find the

Abstract:We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over $\mathbb {Q}$ when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a Taylor-Wiles patching argument for the completed homology in this case. As a key input, we generalize the work of Skinner-Wiles in the ordinary

Abstract:We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted

Abstract:We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact $1$-forms $\lambda _1^*$ on rational homology spheres which admit irreducible solutions together with a version of the

Abstract:We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \leqslant 3$. For $d > 1$ the Gibbs measure is supported on distributions

Abstract:Let $X$ be an affine spherical variety, possibly singular, and $\mathsf L^+X$ its arc space. The intersection complex of $\mathsf L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified $L$-functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient

Abstract:We show that the totally nonnegative part of a partial flag variety $G/P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.

Abstract:We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary

Abstract:We derive $C^\infty$ a priori estimates for solutions of the inhomogeneous Boltzmann equation without cut-off, conditional to pointwise bounds on their mass, energy and entropy densities. We also establish decay estimates for large velocities, for all derivatives of the solution.

Abstract:The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak {g}$ defines a semigroup of triples of weights. Namely, given $\lambda$ in the set $P_+$ of dominant integral weights, $V(\lambda )$ denotes the irreducible representation of $\mathfrak {g}$ with highest weight $\lambda$. We are interested

Abstract:We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. More generally, we prove that it can be solved in the field of cardinality $p^n$ in expected time $(pn)^{2\log _2(n) + O(1)}$.

Abstract:We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every $n$, all but countably many reals are $n$-random for such a measure, where $n$ indicates the arithmetical complexity of the Martin-Löf tests allowed. The proof rests upon an application of Borel determinacy. Therefore

Abstract:We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions

Abstract:We obtain new upper bounds on the minimal density $\Theta _{n, \mathcal {K}}$ of lattice coverings of ${\mathbb {R}}^n$ by dilates of a convex body $\mathcal {K}$. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice $L$ satisfies $L+\mathcal {K}= {\mathbb {R}}^n$. As a step in the proof, we utilize

Abstract:We prove that the conformal group of a closed, simply connected, real analytic Lorentzian manifold is compact. D’Ambra proved in 1988 that the isometry group of such a manifold is compact [Invent. Math. 92 (1988), pp. 555–565]. Our result implies the Lorentzian Lichnerowicz Conjecture for real analytic Lorentzian manifolds with finite fundamental group.

Abstract:We exhibit families of Ricci-flat Kähler metrics on the K3 surface which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the $K3$ surface to the interval, with regular fibers diffeomorphic to either $3$-tori or Heisenberg nilmanifolds.

Abstract:We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago–Ivanov–Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which

Abstract:In this paper, we apply our previous estimates in Chen and Cheng [On the constant scalar curvature Kähler metrics (I): a priori estimates, Preprint] to study the existence of cscK metrics on compact Kähler manifolds. First we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies the existence of cscK metrics. We also show

Abstract:In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the Kähler potential.

Abstract:Let $X$ be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base $Q$. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space $Y$, which can be considered as a variant of the $T$-dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians

Abstract:We consider a natural class of ${\mathbf {R}}^d$-valued one-dimensional stochastic partial differential equations (PDEs) driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on ${\mathbf {R}}^d$. This class contains in particular the Kardar-Parisi-Zhang (KPZ) equation, the multiplicative stochastic heat equation, the additive stochastic heat

Abstract:We axiomatise the theory of $(\infty ,n)$-categories. We prove that the space of theories of $(\infty ,n)$-categories is a $B(\mathbb {Z}/2)^n$. We prove that Rezk’s complete Segal $\Theta _n$ spaces, Simpson and Tamsamani’s Segal $n$-categories, the first author’s $n$-fold complete Segal spaces, Kan and the first author’s $n$-relative categories, and complete Segal space objects in any model

Abstract: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

Abstract:We prove two new results on the $K$-polystability of $\mathbb {Q}$-Fano varieties based on purely algebro-geometric arguments. The first one says that any $K$-semistable log Fano cone has a special degeneration to a uniquely determined $K$-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which

Abstract:We give a variational proof of a version of the Yau–Tian–Donaldson conjecture for twisted Kähler–Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve a continuity method or the Cheeger–Colding–Tian theory, and uses instead pluripotential theory and valuations. Along the

Abstract:This paper lays the foundation for Plancherel theory on real spherical spaces $Z=G/H$, namely it provides the decomposition of $L^2(Z)$ into different series of representations via Bernstein morphisms. These series are parametrized by subsets of spherical roots which determine the fine geometry of $Z$ at infinity. In particular, we obtain a generalization of the Maass-Selberg relations. As

Abstract:We study the spectral measures of conservative mixing flows on the $2$-torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients: (1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows

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

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 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