Authors:Stoll; Robin First page: 1 Abstract: 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. PubDate: 2024-01-05 DOI: 10.1017/fms.2023.113

Authors:Huang; Hong Yi First page: 2 Abstract: 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. PubDate: 2024-01-04 DOI: 10.1017/fms.2023.121

Authors:Bellovin; Rebecca First page: 3 Abstract: 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. PubDate: 2024-01-05 DOI: 10.1017/fms.2023.116

Authors:Bianchi; Fabrizio, Dinh, Tien-Cuong First page: 4 Abstract: 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. PubDate: 2024-01-05 DOI: 10.1017/fms.2023.110

Authors:Douglas; Daniel C., Sun, Zhe First page: 5 Abstract: 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. PubDate: 2024-01-05 DOI: 10.1017/fms.2023.120

Authors:Filmus; Yuval, Kindler, Guy, Lifshitz, Noam, Minzer, Dor First page: 6 Abstract: 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. PubDate: 2024-01-08 DOI: 10.1017/fms.2023.118

Authors:Schwede; Stefan First page: 7 Abstract: 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. PubDate: 2024-01-05 DOI: 10.1017/fms.2023.124

Authors:Pikhurko; Oleg, Staden, Katherine First page: 8 Abstract: 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 fr... PubDate: 2024-01-08 DOI: 10.1017/fms.2023.117

Authors:Wan; Xueyuan First page: 9 Abstract: 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. PubDate: 2024-01-08 DOI: 10.1017/fms.2023.125

Authors:Moulinos; Tasos First page: 10 Abstract: 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. PubDate: 2024-01-15 DOI: 10.1017/fms.2023.122

Authors:Blomer; Valentin, Brüdern, Jörg, Derenthal, Ulrich, Gagliardi, Giuliano First page: 11 Abstract: 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. PubDate: 2024-01-18 DOI: 10.1017/fms.2023.123

Authors:Sargsyan; Grigor, Trang, Nam First page: 12 Abstract: 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 c... PubDate: 2024-01-18 DOI: 10.1017/fms.2023.127

Authors:Borot; Gaëtan, Guionnet, Alice First page: 13 Abstract: 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. PubDate: 2024-01-24 DOI: 10.1017/fms.2023.129

Authors:Hom; Jennifer, Stoffregen, Matthew, Zhou, Hugo First page: 14 Abstract: 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. PubDate: 2024-01-25 DOI: 10.1017/fms.2023.126

Authors:Aguilera; J. P., Pakhomov, F., Weiermann, A. First page: 15 Abstract: 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. PubDate: 2024-01-26 DOI: 10.1017/fms.2023.128

Authors:Chan; William, Jackson, Stephen, Trang, Nam First page: 16 Abstract: 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 PubDate: 2024-01-29 DOI: 10.1017/fms.2023.130

Authors:Huang; Daoji, Striker, Jessica First page: 17 Abstract: 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. PubDate: 2024-01-29 DOI: 10.1017/fms.2023.131

Authors:Kamnitzer; Joel, Webster, Ben, Weekes, Alex, Yacobi, Oded First page: 18 Abstract: 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. PubDate: 2024-01-31 DOI: 10.1017/fms.2024.3

Authors:Griffin; Sean T. First page: 19 Abstract: 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 . PubDate: 2024-01-31 DOI: 10.1017/fms.2024.1

Authors:Alper; Jarod, Hall, Jack, Halpern-Leistner, Daniel, Rydh, David First page: 20 Abstract: 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. PubDate: 2024-02-01 DOI: 10.1017/fms.2023.60

Authors:Varma; Ila First page: 21 Abstract: 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 . PubDate: 2024-02-12 DOI: 10.1017/fms.2024.7

Authors:Lewis; Joel, Marberg, Eric First page: 22 Abstract: 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. PubDate: 2024-02-13 DOI: 10.1017/fms.2024.8

Authors:Esser; Louis First page: 23 Abstract: 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. PubDate: 2024-02-19 DOI: 10.1017/fms.2024.10

Authors:Bohman; Tom, Hofstad, Jakob First page: 24 Abstract: 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. PubDate: 2024-02-23 DOI: 10.1017/fms.2024.6

Authors:Kian; Yavar, Liimatainen, Tony, Lin, Yi-Hsuan First page: 25 Abstract: 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. PubDate: 2024-02-26 DOI: 10.1017/fms.2024.18

Authors:Barchiesi; Marco, Henao, Duvan, Mora-Corral, Carlos, Rodiac, Rémy First page: 26 Abstract: 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. PubDate: 2024-02-26 DOI: 10.1017/fms.2024.9

Authors:Hesselholt; Lars, Pstrągowski, Piotr First page: 27 Abstract: 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. PubDate: 2024-02-27 DOI: 10.1017/fms.2024.2

Authors:Gachet; Cécile, Lin, Hsueh-Yung, Wang, Long First page: 28 Abstract: 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. PubDate: 2024-03-05 DOI: 10.1017/fms.2024.22

Authors:Lampert; Amichai, Ziegler, Tamar First page: 29 Abstract: 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. PubDate: 2024-03-06 DOI: 10.1017/fms.2024.15

Authors:Iraci; Alessandro, Romero, Marino First page: 30 Abstract: 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. PubDate: 2024-03-07 DOI: 10.1017/fms.2024.14

Authors:Caraiani; Ana, Emerton, Matthew, Gee, Toby, Savitt, David First page: 31 Abstract: 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. PubDate: 2024-03-11 DOI: 10.1017/fms.2024.4

Authors:Oyakawa; Koichi First page: 32 Abstract: 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. PubDate: 2024-03-11 DOI: 10.1017/fms.2024.24

Authors:Hu; Zhi, Yang, Yu, Zong, Runhong First page: 33 Abstract: 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 . PubDate: 2024-03-14 DOI: 10.1017/fms.2024.12

Authors:Kannan; Siddarth, Serpente, Stefano, Yun, Claudia He First page: 34 Abstract: 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 . PubDate: 2024-03-14 DOI: 10.1017/fms.2024.20

Authors:Chen; Ruiyuan First page: 35 Abstract: 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. PubDate: 2024-03-14 DOI: 10.1017/fms.2024.25

Authors:Bornemann; Folkmar First page: 36 Abstract: 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. PubDate: 2024-03-15 DOI: 10.1017/fms.2024.13

Authors:Chen; Chih-Whi, Cheng, Shun-Jen First page: 37 Abstract: We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over with a triangular decomposition and a character of the nilpotent radical, the associated Backelin functor sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category . In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor and its target category, respectively, categorify a q-symmetrizing map and the corresponding q-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type . PubDate: 2024-04-02 DOI: 10.1017/fms.2024.17

Authors:Janzer; Oliver, Sudakov, Benny First page: 38 Abstract: 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. PubDate: 2024-03-15 DOI: 10.1017/fms.2024.27

Authors:Ballester-Bolinches; A., Esteban-Romero, R., Ferrara, M., Pérez-Calabuig, V., Trombetti, M. First page: 39 Abstract: 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. PubDate: 2024-03-18 DOI: 10.1017/fms.2024.29

Authors:Blekherman; Grigoriy, Raymond, Annie, Wei, Fan First page: 40 Abstract: 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. PubDate: 2024-03-18 DOI: 10.1017/fms.2024.19

Authors:Cheltsov; Ivan, Denisova, Elena, Fujita, Kento First page: 41 Abstract: We prove that all smooth Fano threefolds in the families are K-stable, and we also prove that smooth Fano threefolds in the family PubDate: 2024-03-20 DOI: 10.1017/fms.2024.5

Authors:Lam; Thomas, Postnikov, Alexander First page: 42 Abstract: 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. PubDate: 2024-03-18 DOI: 10.1017/fms.2024.11

Authors:Alpern; Itai, Kupferman, Raz, Maor, Cy First page: 43 Abstract: We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to -perturbations of their fundamental forms: For a manifold endowed with a reference metric and a reference shape operator, we show that a sequence of immersions , whose pullback metrics and shape operators are arbitrary close in to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold , removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds. PubDate: 2024-04-02 DOI: 10.1017/fms.2024.30

Authors:Kim; Minki, Lew, Alan First page: 44 Abstract: We present extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ‘very colorful’ Helly theorem introduced by Arocha, Bárány, Bracho, Fabila and Montejano and the ‘semi-intersecting’ colorful Helly theorem proved by Montejano and Karasev.As an application, we obtain the following extension of Tverberg’s theorem: Let A be a finite set of points in with . Then, there exist a partition of A and a subset of size such that for all . That is, we obtain a partition of A into r parts that remains a Tverberg partition even after removing all but one arbitrary point from . PubDate: 2024-04-02 DOI: 10.1017/fms.2024.23

Authors:Habegger; P., Schmidt, H. First page: 45 Abstract: In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form , where p is a prime number and where the orbit of is finite. For example, if and is periodic under with , we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these f, our method has application to the irreducibility of polynomials. Indeed, say y is preperiodic under f but not periodic. Then any iteration of f minus y is irreducible in . PubDate: 2024-04-02 DOI: 10.1017/fms.2023.112

Authors:Charlton; Steven, Keilthy, Adam First page: 46 Abstract: In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration [9]. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for . In contrast, a simple combinatorial filtration, the block filtration [13, 28] is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple t values [22] in terms of classical double zeta values. PubDate: 2024-04-02 DOI: 10.1017/fms.2024.16

Authors:Lim; Woonam, Moreira, Miguel, Pi, Weite First page: 47 Abstract: We prove that the cohomology rings of the moduli space of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the -independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties. PubDate: 2024-04-01 DOI: 10.1017/fms.2024.31

Authors:He; Tongmu First page: 48 Abstract: Faltings ringed topos, the keystone of Faltings’ approach to p-adic Hodge theory for a smooth variety over a local field, relies on the choice of an integral model, and its good properties depend on the (logarithmic) smoothness of this model. Inspired by Deligne’s approach to classical Hodge theory for singular varieties, we establish a cohomological descent result for the structural sheaf of Faltings topos, which makes it possible to extend Faltings’ approach to any integral model, that is, without any smoothness assumption. An essential ingredient of our proof is a variation of Bhatt–Scholze’s arc-descent of perfectoid rings. PubDate: 2024-04-02 DOI: 10.1017/fms.2024.26

Authors:Kaplan; Itay, Ramsey, Nicholas, Simon, Pierre First page: 49 Abstract: We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP theory is simple. PubDate: 2024-04-08 DOI: 10.1017/fms.2024.35

Authors:Oh; Tadahiro, Seong, Kihoon, Tolomeo, Leonardo First page: 50 Abstract: We study Gibbs measures with log-correlated base Gaussian fields on the d-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson’s argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove nonnormalizability of the Gibbs measure. When , our argument provides an alternative proof of the nonnormalizability result for the focusing -measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein’s inequality on . We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) nonnormalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction. PubDate: 2024-04-08 DOI: 10.1017/fms.2024.28

Authors:Coskun; Izzet, Larson, Eric, Vogt, Isabel First page: 51 Abstract: Let be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under is semistable if the genus of Y is at least and stable if the genus of Y is at least . We prove this conjecture if the map is general in any component of the Hurwitz space of covers of an arbitrary smooth curve Y. PubDate: 2024-04-08 DOI: 10.1017/fms.2024.21

Authors:Matsuda; Toyomu, Perkowski, Nicolas First page: 52 Abstract: We give an extension of Lê’s stochastic sewing lemma. The stochastic sewing lemma proves convergence in of Riemann type sums for an adapted two-parameter stochastic process A, under certain conditions on the moments of and of conditional expectations of given . Our extension replaces the conditional expectation given by that given for , and it allows to make use of asymptotic decorrelation properties between and by including a singularity in . We provide three applications for which Lê’s stochastic sewing lemma seems to be insufficient. The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence. PubDate: 2024-04-11 DOI: 10.1017/fms.2024.32

Authors:Schremmer; Felix First page: 53 Abstract: We give new descriptions of the Bruhat order and Demazure products of affine Weyl groups in terms of the weight function of the quantum Bruhat graph. These results can be understood to describe certain closure relations concerning the Iwahori–Bruhat decomposition of an algebraic group. As an application towards affine Deligne–Lusztig varieties, we present a new formula for generic Newton points. PubDate: 2024-04-15 DOI: 10.1017/fms.2024.33

Authors:Corey; Daniel, Li, Wanlin First page: 54 Abstract: We define a new algebraic invariant of a graph G called the Ceresa–Zharkov class and show that it is trivial if and only if G is of hyperelliptic type, equivalently, G does not have as a minor the complete graph on four vertices or the loop of three loops. After choosing edge lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph G that is closely related to the Ceresa cycle for an algebraic curve defined over . PubDate: 2024-04-25 DOI: 10.1017/fms.2024.36

Authors:Mucciconi; Matteo, Sasada, Makiko, Sasamoto, Tomohiro, Suda, Hayate First page: 55 Abstract: The box-ball system (BBS), which was introduced by Takahashi and Satsuma in 1990, is a soliton cellular automaton. Its dynamics can be linearized by a few methods, among which the best known is the Kerov–Kirillov–Reschetikhin (KKR) bijection using rigged partitions. Recently, a new linearization method in terms of ‘slot configurations’ was introduced by Ferrari–Nguyen–Rolla–Wang, but its relations to existing ones have not been clarified. In this paper, we investigate this issue and clarify the relation between the two linearizations. For this, we introduce a novel way of describing the BBS dynamics using a carrier with seat numbers. We show that the seat number configuration also linearizes the BBS and reveals explicit relations between the KKR bijection and the slot configuration. In addition, by using these explicit relations, we also show that even in case of finite carrier capacity the BBS can be linearized via the slot configuration. PubDate: 2024-05-10 DOI: 10.1017/fms.2024.39

Authors:Brown; Michael K., Erman, Daniel First page: 56 Abstract: We give a short new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties. As in their work, this leads to a proof of a conjecture of Berkesch-Erman-Smith on virtual resolutions and to a resolution of the diagonal in the simplicial case. PubDate: 2024-05-10 DOI: 10.1017/fms.2024.40

Authors:Mutanguha; Jean Pierre First page: 57 Abstract: To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the real pretree that represents the free group automorphism. Finally and crucially, the loxodromic elements are exactly those whose (conjugacy class) length grows exponentially under iteration of the automorphism; thus, the action on the real pretree is able to detect the growth type of an element.This construction extends the theory of metric trees that has been used to study free group automorphisms. The new idea is that one can equivariantly blow up an isometric action on a real tree with respect to other real trees and get a rigid action on a treelike structure known as a real pretree. Topology plays no role in this construction as all the work is done in the language of pretrees (intervals). PubDate: 2024-05-10 DOI: 10.1017/fms.2024.38

Authors:Frisch; Joshua, Seward, Brandon, Zucker, Andy First page: 58 Abstract: Given a countable group G and a G-flow X, a probability measure on X is called characteristic if it is -invariant. Frisch and Tamuz asked about the existence of a minimal G-flow, for any group G, which does not admit a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection of infinite subgroups , when is there a faithful G-flow for which every acts minimally' PubDate: 2024-05-10 DOI: 10.1017/fms.2024.41

Authors:Friedman; Robert, Laza, Radu First page: 59 Abstract: The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least . For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with -liminal singularities (which are exactly the ordinary double points in dimension but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneous k-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions. PubDate: 2024-05-10 DOI: 10.1017/fms.2024.44

Authors:Felisetti; Camilla, Giovenzana, Franco, Grossi, Annalisa First page: 60 Abstract: We give a lattice-theoretic characterization for a manifold of type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li–Pertusi–Zhao variety of type associated to any smooth cubic fourfold. Moreover, we determine when a birational transformation is induced by an automorphism of the K3 surface, and we use this to classify all induced birational symplectic involutions. PubDate: 2024-05-10 DOI: 10.1017/fms.2024.46

Authors:Nesterov; Denis First page: 61 Abstract: In this article, we study quasimaps to moduli spaces of sheaves on a surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of -stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of , where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on , if ; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on . PubDate: 2024-05-17 DOI: 10.1017/fms.2024.48

Authors:Medina-Mardones; Anibal M., Rivera, Manuel First page: 62 Abstract: We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal -coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures. PubDate: 2024-05-15 DOI: 10.1017/fms.2024.50

Authors:Benedetti; Vladimiro, Bolognesi, Michele, Faenzi, Daniele, Manivel, Laurent First page: 63 Abstract: Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover , the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of . In fact, each point defines a natural embedding of in . We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of and thus deserves to be coined the Coble quadric of the pointed curve . PubDate: 2024-05-17 DOI: 10.1017/fms.2024.52

Authors:Naranjo; J.C., Ortega, A., Spelta, I. First page: 64 Abstract: We consider cyclic unramified coverings of degree d of irreducible complex smooth genus curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for prime such that is also prime. We use results of arithmetic nature on -type abelian varieties combined with theta-duality techniques. PubDate: 2024-05-21 DOI: 10.1017/fms.2024.42

Authors:Dervan; Ruadhaí, Papazachariou, Theodoros Stylianos First page: 65 Abstract: Divisorial stability of a polarised variety is a stronger – but conjecturally equivalent – variant of uniform K-stability introduced by Boucksom–Jonsson. Whereas uniform K-stability is defined in terms of test configurations, divisorial stability is defined in terms of convex combinations of divisorial valuations on the variety.We consider the behaviour of divisorial stability under finite group actions and prove that equivariant divisorial stability of a polarised variety is equivalent to log divisorial stability of its quotient. We use this and an interpolation technique to give a general construction of equivariantly divisorially stable polarised varieties. PubDate: 2024-05-22 DOI: 10.1017/fms.2024.47

Authors:Dudas; Olivier, Ivanov, Alexander B. First page: 66 Abstract: In this paper, we prove some orthogonality relations for representations arising from deep level Deligne–Lusztig schemes of Coxeter type. This generalizes previous results of Lusztig [Lus04], and of Chan and the second author [CI21b]. Applications include the study of smooth representations of p-adic groups in the cohomology of p-adic Deligne–Lusztig spaces and their relation to the local Langlands correspondences. Also, the geometry of deep level Deligne–Lusztig schemes gets accessible, in the spirit of Lusztig’s work [Lus76]. PubDate: 2024-05-22 DOI: 10.1017/fms.2024.55

Authors:Brandt; Madeline, Chan, Melody, Kannan, Siddarth First page: 67 Abstract: For and , let denote the complex moduli stack of n-marked smooth hyperelliptic curves of genus g. A normal crossings compactification of this space is provided by the theory of pointed admissible -covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of . Using this graph complex, we give a sum-over-graphs formula for the -equivariant weight zero compactly supported Euler characteristic of . This formula allows for the computer-aided calculation, for each , of the generating function for these equivariant Euler characteristics for all n. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible G-covers of genus zero curves, when G is abelian, as a symmetric -complex. We use these complexes to generalize our formula for to moduli spaces of n-pointed smooth abelian covers of genus zero curves. PubDate: 2024-05-27 DOI: 10.1017/fms.2024.53

Authors:Guedj; Vincent, Guenancia, Henri, Zeriahi, Ahmed First page: 68 Abstract: Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on , and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form. PubDate: 2024-05-27 DOI: 10.1017/fms.2024.54

Authors:Wang; Jianping, Wen, Xueqing First page: 69 Abstract: We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F-regular type. As a consequence, all higher cohomologies of the theta line bundle vanish. During the proof, we develop a method to estimate codimension. PubDate: 2024-05-27 DOI: 10.1017/fms.2024.57

Authors:Blekherman; Grigoriy, Chen, Justin, Jung, Jaewoo First page: 70 Abstract: The Hankel index of a real variety X is an invariant that quantifies the difference between nonnegative quadrics and sums of squares on X. In [5], the authors proved an intriguing bound on the Hankel index in terms of the Green–Lazarsfeld index, which measures the ‘linearity’ of the minimal free resolution of the ideal of X. In all previously known cases, this bound was tight. We provide the first class of examples where the bound is not tight; in fact, the difference between Hankel index and Green–Lazarsfeld index can be arbitrarily large. Our examples are outer projections of rational normal curves, where we identify the center of projection with a binary form F. The Green–Lazarsfeld index of the projected curve is given by the complex Waring border rank of F [16]. We show that the Hankel index is given by the almost real rank of F, which is a new notion that comes from decomposing F as a sum of powers of almost real forms. Finally, we determine the range of possible and typical almost real ranks for binary forms. PubDate: 2024-05-30 DOI: 10.1017/fms.2024.45

Authors:Gehrmann; Lennart, Pati, Maria Rosaria First page: 71 Abstract: Let be a cuspidal, cohomological automorphic representation of an inner form G of over a number field F of arbitrary signature. Further, let be a prime of F such that G is split at and the local component of at is the Steinberg representation. Assuming that the representation is noncritical at , we construct automorphic -invariants for the representation . If the number field F is totally real, we show that these automorphic -invariants agree with the Fontaine–Mazur -invariant of the associated p-adic Galois representation. This generalizes a recent result of Spieß respectively Rosso and the first named author from the case of parallel weight PubDate: 2024-05-30 DOI: 10.1017/fms.2024.51