We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of , where . The result is expressed in terms of Lie graph complex homology.
]]>Let G be a permutation group on a finite set . The base size of G is the minimal size of a subset of with trivial pointwise stabiliser in G. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.
]]>We use the theory of trianguline -modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at p, including those with characteristic p coefficients. The use of pseudorigid spaces lets us construct integral models of the trianguline varieties of [BHS17], [Che13] after bounding the slope, and we carry out a Taylor–Wiles patching argument for families of overconvergent modular forms. This permits us to construct a patched quaternionic eigenvariety and deduce our modularity results.
]]>We show that the measure of maximal entropy of every complex Hénon map is exponentially mixing of all orders for Hölder observables. As a consequence, the Central Limit Theorem holds for all Hölder observables.
]]>For a finite-type surface , we study a preferred basis for the commutative algebra of regular functions on the -character variety, introduced by Sikora–Westbury. These basis elements come from the trace functions associated to certain trivalent graphs embedded in the surface . We show that this basis can be naturally indexed by nonnegative integer coordinates, defined by Knutson–Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.
]]>The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains.
We consider the symmetric group, , one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on , which are functions whose -norm remains small when restricting coordinates of the input, and assert that low-degree, global functions have small q-norms, for .
As applications, we show the following:
1. An analog of the level-d inequality on the hypercube, asserting that the mass of a global function on low degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group .
2. Isoperimetric inequalities on the transposition Cayley graph of for global functions that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube.
3. Hypercontractive inequalities on the multi-slice and stability versions of the Kruskal–Katona Theorem in some regimes of parameters.
We introduce Chern classes in -equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the -cohomology of . For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the -completion theorem of Greenlees–May and La Vecchia.
]]>Let be a sequence of natural numbers. For a graph G, let denote the number of colourings of the edges of G with colours such that, for every , the edges of colour c contain no clique of order . Write to denote the maximum of over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for :
(i) A sufficient condition on which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results.
(ii) Addressing the original question of Erdős and Rothschild, in the case of length , the unique extremal graph is the complete balanced -partite graph, with colourings coming from Hadamard matrices of order .
(iii) In the case , for which the sufficient condition in (i) does not hold, for , the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as possible.
In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.
]]>We exhibit the Hodge degeneration from nonabelian Hodge theory as a -fold delooping of the filtered loop space -groupoid in formal moduli problems. This is an iterated groupoid object which in degree recovers the filtered circle of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an -cogroupoid object in the -category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on , as well as the Todd class of the Lie algebroid ; this is an invariant of a smooth and proper scheme X that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.
]]>The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.
]]>A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing.
The () is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let be the statement that in all (set) generic extensions there is a model of whose Suslin, co-Suslin sets are the universally Baire sets.
We show that over some mild large cardinal theory, is equiconsistent with . In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that is weaker than the theory ‘ there is a Woodin cardinal which is a limit of Woodin cardinals’.
A variation of , called , is also shown to be equiconsistent with over the same large cardinal theory.
The result is proven via Woodin’s technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of as explained in the paper.
]]>We establish the asymptotic expansion in matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model () as well as orthogonal () and skew-orthogonal () polynomials outside the bulk.
]]>We consider manifold-knot pairs , where Y is a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface in a homology ball X, such that can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from to any knot in can be arbitrarily large. The proof relies on Heegaard Floer homology.
]]>We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals.
We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem - of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.
]]>This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.
The following summarizes the main results proved under suitable partition hypotheses.
• If is a cardinal, , , and , then satisfies the almost everywhere short length continuity property: There is a club and a so that for all , if and , then .
• If is a cardinal, is countable, holds and , then satisfies the strong almost everywhere short length continuity property: There is a club and finitely many ordinals so that for all , if for all , , then .
• If satisfies , and , then satisfies the almost everywhere monotonicity property: There is a club so that for all , if for all , , then .
• Suppose dependent choice (), and the almost everywhere short length club uniformization principle for hold. Then every function satisfies a finite continuity property with respect to closure points: Let be the club of so that . There is a club and finitely many functions so that for all , for all , if and for all , , then .
• Suppose satisfies for all . For all , does not inject into , the class of -length sequences of ordinals, and therefore, . As a consequence, under the axiom of determinacy , these two cardinality results hold when is one of the following weak or strong partition cardinals of determinacy: , , (for all ) and (assuming in addition ).
We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.
]]>We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools.
After this general definition, we focus on quiver gauge theories attached to a quiver . The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra on category for these Coulomb branch algebras. When is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.
To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.
]]>The -Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a -Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field and partitioning the -Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at .
]]>We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of étale, smooth or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin–Tyomkin.
]]>We prove the compatibility of local and global Langlands correspondences for up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation of . We show that the restriction of to the decomposition group of a place of F corresponds up to semisimplification to , the image of under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of is ‘more nilpotent’ than the monodromy of .
]]>The K-theoretic Schur P- and Q-functions and may be concretely defined as weight-generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of stable Grothendieck polynomials and were introduced by Ikeda and Naruse for applications in geometry. Nakagawa and Naruse specified families of dual K-theoretic Schur P- and Q-functions and via a Cauchy identity involving and . They conjectured that the dual power series are weight-generating functions for certain shifted plane partitions. We prove this conjecture. We also derive a related generating function formula for the images of and under the involution of the ring of symmetric functions. This confirms a conjecture of Chiu and the second author. Using these results, we verify a conjecture of Ikeda and Naruse that the -functions are a basis for a ring.
]]>For certain quasismooth Calabi–Yau hypersurfaces in weighted projective space, the Berglund-Hübsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror. We prove that the minimal log discrepancy of the quotient of such a hypersurface by its toric automorphism group is closely related to the weights and degree of the BHK mirror. As an application, we exhibit klt Calabi–Yau varieties with the smallest known minimal log discrepancy. We conjecture that these examples are optimal in every dimension.
]]>We show that the independence number of is concentrated on two values if . This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in general, concentrated on two values for . The extent of concentration of the independence number of for remains an interesting open question.
]]>We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on .
Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.
]]>We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti and De Lellis is generic in some sense. On this map, we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of -valued harmonic maps.
]]>Dirac rings are commutative algebras in the symmetric monoidal category of -graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger -category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to and in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.
]]>We prove a decomposition theorem for the nef cone of smooth fiber products over curves, subject to the necessary condition that their Néron–Severi space decomposes. We apply it to describe the nef cone of so-called Schoen varieties, which are the higher-dimensional analogues of the Calabi–Yau threefolds constructed by Schoen. Schoen varieties give rise to Calabi–Yau pairs, and in each dimension at least three, there exist Schoen varieties with nonpolyhedral nef cone. We prove the Kawamata–Morrison–Totaro cone conjecture for the nef cones of Schoen varieties, which generalizes the work by Grassi and Morrison.
]]>We introduce a new concept of rank – relative rank associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case, we obtain a new proof of the results in [21]). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is of high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry, and algebra. For example, we prove that any collection of polynomials of degrees in a polynomial ring over an algebraically closed field of characteristic is contained in an ideal , generated by a collection of polynomials of degrees which form a regular sequence, and is of size , where is independent of the number of variables.
]]>We give an elementary symmetric function expansion for the expressions and when in terms of what we call -parking functions and lattice -parking functions. Here, and are certain eigenoperators of the modified Macdonald basis and . Our main results, in turn, give an elementary basis expansion at for symmetric functions of the form whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.
]]>In the article [CEGS20b], we introduced various moduli stacks of two-dimensional tamely potentially Barsotti–Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil–Kisin modules with descent data. We study the irreducible components of these stacks, establishing, in particular, that the components of the former are naturally indexed by certain Serre weights.
]]>We prove that, for any countable acylindrically hyperbolic group G, there exists a generating set S of G such that the corresponding Cayley graph is hyperbolic, , the natural action of G on is acylindrical and the natural action of G on the Gromov boundary is hyperfinite. This result broadens the class of groups that admit a non-elementary acylindrical action on a hyperbolic space with a hyperfinite boundary action.
]]>In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topology of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures concerning tame fundamental groups of curves over algebraically closed fields of characteristic from the point of view of moduli spaces. The conjectures are generalized versions of the Weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus .
]]>Let denote Hassett’s moduli space of weighted pointed stable curves of genus g for the heavy/light weight data
and let be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which computes the generating function for -equivariant Hodge–Deligne polynomials of these spaces in terms of the generating functions for -equivariant Hodge–Deligne polynomials of and .
]]>We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures.
Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.
]]>We study the distribution of the length of longest increasing subsequences in random permutations of n integers as n grows large and establish an asymptotic expansion in powers of . Whilst the limit law was already shown by Baik, Deift and Johansson to be the GUE Tracy–Widom distribution F, we find explicit analytic expressions of the first few finite-size correction terms as linear combinations of higher order derivatives of F with rational polynomial coefficients. Our proof replaces Johansson’s de-Poissonization, which is based on monotonicity as a Tauberian condition, by analytic de-Poissonization of Jacquet and Szpankowski, which is based on growth conditions in the complex plane; it is subject to a tameness hypothesis concerning complex zeros of the analytically continued Poissonized length distribution. In a preparatory step an expansion of the hard-to-soft edge transition law of LUE is studied, which is lifted to an expansion of the Poissonized length distribution for large intensities. Finally, expansions of Stirling-type approximations and of the expected value and variance of the length distribution are given.
]]>In 1964, Erdős proposed the problem of estimating the Turán number of the d-dimensional hypercube . Since is a bipartite graph with maximum degree d, it follows from results of Füredi and Alon, Krivelevich, Sudakov that . A recent general result of Sudakov and Tomon implies the slightly stronger bound . We obtain the first power-improvement for this old problem by showing that . This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes.
We use a similar method to prove that any n-vertex, properly edge-coloured graph without a rainbow cycle has at most edges, improving the previous best bound of by Tomon. Furthermore, we show that any properly edge-coloured n-vertex graph with edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight.
]]>The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if is nilpotent.
Given a finite presentation of the structure skew brace associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.
]]>Many problems and conjectures in extremal combinatorics concern polynomial inequalities between homomorphism densities of graphs where we allow edges to have real weights. Using the theory of graph limits, we can equivalently evaluate polynomial expressions in homomorphism densities on kernels W, that is, symmetric, bounded and measurable functions W from . In 2011, Hatami and Norin proved a fundamental result that it is undecidable to determine the validity of polynomial inequalities in homomorphism densities for graphons (i.e., the case where the range of W is , which corresponds to unweighted graphs or, equivalently, to graphs with edge weights between and ). The corresponding problem for more general sets of kernels, for example, for all kernels or for kernels with range , remains open. For any , we show undecidability of polynomial inequalities for any set of kernels which contains all kernels with range . This result also answers a question raised by Lovász about finding computationally effective certificates for the validity of homomorphism density inequalities in kernels.
]]>We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of -polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of -polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.
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