-
Polypositroids Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-18 Thomas Lam, Alexander Postnikov
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
-
Undecidability of polynomial inequalities in weighted graph homomorphism densities Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-18 Grigoriy Blekherman, Annie Raymond, Fan Wei
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 $[0,1]^2 \to \mathbb {R}$. In 2011, Hatami
-
Finite skew braces of square-free order and supersolubility Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-18 A. Ballester-Bolinches, R. Esteban-Romero, M. Ferrara, V. Pérez-Calabuig, M. Trombetti
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
-
On the Turán number of the hypercube Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-15 Oliver Janzer, Benny Sudakov
In 1964, Erdős proposed the problem of estimating the Turán number of the d-dimensional hypercube $Q_d$ . Since $Q_d$ is a bipartite graph with maximum degree d, it follows from results of Füredi and Alon, Krivelevich, Sudakov that $\mathrm {ex}(n,Q_d)=O_d(n^{2-1/d})$ . A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm {ex}(n,Q_d)=o(n^{2-1/d})$ . We obtain the
-
Asymptotic expansions relating to the distribution of the length of longest increasing subsequences Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-15 Folkmar Bornemann
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 $n^{-1/3}$ . 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
-
Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-14 Ruiyuan Chen
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
-
Equivariant Hodge polynomials of heavy/light moduli spaces Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-14 Siddarth Kannan, Stefano Serpente, Claudia He Yun
Let $\overline {\mathcal {M}}_{g, m|n}$ denote Hassett’s moduli space of weighted pointed stable curves of genus g for the heavy/light weight data $$\begin{align*}\left(1^{(m)}, 1/n^{(n)}\right),\end{align*}$$ and let $\mathcal {M}_{g, m|n} \subset \overline {\mathcal {M}}_{g, m|n}$ be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which
-
Topology of moduli spaces of curves and anabelian geometry in positive characteristic Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-14 Zhi Hu, Yu Yang, Runhong Zong
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
-
Hyperfiniteness of boundary actions of acylindrically hyperbolic groups Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-11 Koichi Oyakawa
We prove that, for any countable acylindrically hyperbolic group G, there exists a generating set S of G such that the corresponding Cayley graph $\Gamma (G,S)$ is hyperbolic, $|\partial \Gamma (G,S)|>2$ , the natural action of G on $\Gamma (G,S)$ is acylindrical and the natural action of G on the Gromov boundary $\partial \Gamma (G,S)$ is hyperfinite. This result broadens the class of groups that
-
Components of moduli stacks of two-dimensional Galois representations Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-11 Ana Caraiani, Matthew Emerton, Toby Gee, David Savitt
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
-
Delta and Theta Operator Expansions Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-07 Alessandro Iraci, Marino Romero
We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking functions. Here, $\Delta _F$ and $\Pi $ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$ . Our main results
-
Relative rank and regularization Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-06 Amichai Lampert, Tamar Ziegler
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
-
Nef cones of fiber products and an application to the cone conjecture Forum Math. Sigma (IF 1.389) Pub Date : 2024-03-05 Cécile Gachet, Hsueh-Yung Lin, Long Wang
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
-
Dirac geometry II: coherent cohomology Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-27 Lars Hesselholt, Piotr Pstrągowski
Dirac rings are commutative algebras in the symmetric monoidal category of $\mathbb {Z}$ -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 $\infty $ -category of Dirac stacks, which also contains formal Dirac schemes
-
On the lack of compactness in the axisymmetric neo-Hookean model Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-26 Marco Barchiesi, Duvan Henao, Carlos Mora-Corral, Rémy Rodiac
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
-
On determining and breaking the gauge class in inverse problems for reaction-diffusion equations Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-26 Yavar Kian, Tony Liimatainen, Yi-Hsuan Lin
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
-
Two-Point Concentration of the Independence Number of the Random Graph Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-23 Tom Bohman, Jakob Hofstad
We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ \epsilon } < p \le 1$ . 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 $ p = o ( (\log (n)/n)^{2/3} )$ . The extent of concentration of the independence number of $ G_{n,p}$ for $ \omega (1/n) < p
-
Minimal log discrepancies of hypersurface mirrors Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-19 Louis Esser
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
-
Combinatorial formulas for shifted dual stable Grothendieck polynomials Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-13 Joel Lewis, Eric Marberg
The K-theoretic Schur P- and Q-functions $G\hspace {-0.2mm}P_\lambda $ and $G\hspace {-0.2mm}Q_\lambda $ 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
-
Local-global compatibility for regular algebraic cuspidal automorphic representations when Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-12 Ila Varma
We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation
-
Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts Forum Math. Sigma (IF 1.389) Pub Date : 2024-02-01 Jarod Alper, Jack Hall, Daniel Halpern-Leistner, David Rydh
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
-
Δ–Springer varieties and Hall–Littlewood polynomials Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-31 Sean T. Griffin
The $\Delta $ -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 $\Delta $ -Springer variety. We do this by interpreting the Frobenius characteristic in
-
Lie algebra actions on module categories for truncated shifted yangians Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-31 Joel Kamnitzer, Ben Webster, Alex Weekes, Oded Yacobi
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 $\Gamma
-
A Pipe Dream Perspective on Totally Symmetric Self-Complementary Plane Partitions Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-29 Daoji Huang, Jessica Striker
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
-
Almost Everywhere Behavior of Functions According to Partition Measures Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-29 William Chan, Stephen Jackson, Nam Trang
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
-
Functorial Fast-Growing Hierarchies Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-26 J. P. Aguilera, F. Pakhomov, A. Weiermann
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 $\mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc
-
PL-Genus of surfaces in homology balls Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-25 Jennifer Hom, Matthew Stoffregen, Hugo Zhou
We consider manifold-knot pairs $(Y,K)$ , where Y is a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface $\Sigma $ in a homology ball X, such that $\partial (X, \Sigma ) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof
-
Asymptotic expansion of matrix models in the multi-cut regime Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-24 Gaëtan Borot, Alice Guionnet
We establish the asymptotic expansion in $\beta $ 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 $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to
-
The exact consistency strength of the generic absoluteness for the universally Baire sets Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-18 Grigor Sargsyan, Nam Trang
A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. $\mathsf {Sealing}$ 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 $\mathsf {Largest\ Suslin\ Axiom}$ ( $\mathsf {LSA}$ ) is a determinacy axiom isolated by
-
The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-18 Valentin Blomer, Jörg Brüdern, Ulrich Derenthal, Giuliano Gagliardi
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.
-
Cogroupoid structures on the circle and the Hodge degeneration Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-15 Tasos Moulinos
We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$ -fold delooping of the filtered loop space $E_2$ -groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$ -cogroupoid object in the
-
Positivity of Schur forms for strongly decomposably positive vector bundles Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-08 Xueyuan Wan
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,
-
Exact solutions to the Erdős-Rothschild problem Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-08 Oleg Pikhurko, Katherine Staden
Let $\boldsymbol {k} := (k_1,\ldots ,k_s)$ be a sequence of natural numbers. For a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ such that, for every $c \in \{1,\dots ,s\}$ , the edges of colour c contain no clique of order $k_c$ . Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices
-
Hypercontractivity on the symmetric group Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-08 Yuval Filmus, Guy Kindler, Noam Lifshitz, Dor Minzer
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, $S_n$ , one of the most
-
Chern classes in equivariant bordism Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-05 Stefan Schwede
We introduce Chern classes in $U(m)$ -equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$ -cohomology of $B U(m)$ . 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
-
Tropical Fock–Goncharov coordinates for -webs on surfaces I: construction Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-05 Daniel C. Douglas, Zhe Sun
For a finite-type surface $\mathfrak {S}$ , we study a preferred basis for the commutative algebra $\mathbb {C}[\mathscr {R}_{\mathrm {SL}_3(\mathbb {C})}(\mathfrak {S})]$ of regular functions on the $\mathrm {SL}_3(\mathbb {C})$ -character variety, introduced by Sikora–Westbury. These basis elements come from the trace functions associated to certain trivalent graphs embedded in the surface $\mathfrak
-
Every complex Hénon map is exponentially mixing of all orders and satisfies the CLT Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-05 Fabrizio Bianchi, Tien-Cuong Dinh
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.
-
Modularity of trianguline Galois representations Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-05 Rebecca Bellovin
We use the theory of trianguline $(\varphi ,\Gamma )$ -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
-
The stable cohomology of self-equivalences of connected sums of products of spheres Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-05 Robin Stoll
We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of ${\mathrm {S}^{k}} \times {\mathrm {S}^{l}}$ , where $3 \le k < l \le 2k - 2$ . The result is expressed in terms of Lie graph complex homology.
-
Base sizes of primitive groups of diagonal type Forum Math. Sigma (IF 1.389) Pub Date : 2024-01-04 Hong Yi Huang
Let G be a permutation group on a finite set $\Omega $ . The base size of G is the minimal size of a subset of $\Omega $ 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
-
Lim Ulrich sequences and Boij-Söderberg cones Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-18 Srikanth B. Iyengar, Linquan Ma, Mark E. Walker
This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings.
-
Persistence and the Sheaf-Function Correspondence Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-18 Nicolas Berkouk
The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of
-
E-Polynomials of Generic -Character Varieties: Branched Case Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-18 Cheng Shu
For any branched double covering of compact Riemann surfaces, we consider the associated character varieties that are unitary in the global sense, which we call $\operatorname {\mathrm {GL}}_n\rtimes \!<\!\sigma {>}$ -character varieties. We restrict the monodromies around the branch points to generic semi-simple conjugacy classes contained in $\operatorname {\mathrm {GL}}_n\sigma $ and compute the
-
Counting geodesics of given commutator length Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-15 Viveka Erlandsson, Juan Souto
Let $\Sigma $ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in $\Sigma $ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $\Sigma $ . In the appendix, we use the same strategy to give a proof
-
Two-variable fibrations, factorisation systems and -categories of spans Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-07 Rune Haugseng, Fabian Hebestreit, Sil Linskens, Joost Nuiten
We prove a universal property for $\infty $ -categories of spans in the generality of Barwick’s adequate triples, explicitly describe the cocartesian fibration corresponding to the span functor, and show that the latter restricts to a self-equivalence on the class of orthogonal adequate triples, which we introduce for this purpose. As applications of the machinery we develop, we give a quick proof
-
The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-06 Chokri Manai, Simone Warzel
A semiclassical analysis based on spin-coherent states is used to establish a classification and novel simple formulae for the spectral gap of mean-field spin Hamiltonians. For gapped systems, we provide a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian, hence justifying fluctuation theory at zero temperature for this case. We also point
-
A question of Frohardt on -groups, skew translation quadrangles of even order and cyclic STGQs Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-06 Koen Thas
We solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite $2$ -groups with Kantor familes, by showing that finite groups K with a Kantor family $(\mathcal {F},\mathcal {F}^*)$ having distinct members $A, B \in \mathcal {F}$ such that $A^* \cap B^*$ is a central subgroup of K and the quotient $K/(A^* \cap B^*)$ is abelian cannot exist if the center of K has exponent $4$ and the members
-
Gluing approximable triangulated categories Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-05 Jesse Burke, Amnon Neeman, Bregje Pauwels
Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is approximable if this modified procedure is possible. Not surprisingly this has
-
Categorical and K-theoretic Donaldson–Thomas theory of (part II) Forum Math. Sigma (IF 1.389) Pub Date : 2023-12-04 Tudor Pădurariu, Yukinobu Toda
Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory. We first prove a categorical analogue of Davison’s
-
Thermalization in Kitaev’s quantum double models via tensor network techniques Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-28 Angelo Lucia, David Pérez-García, Antonio Pérez-Hernández
We show that every ergodic Davies generator associated to any 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians
-
On the nilpotent orbit theorem of complex variations of Hodge structure Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-24 Ya Deng
We prove some results on the nilpotent orbit theorem for complex variations of Hodge structure.
-
t-Design Curves and Mobile Sampling on the Sphere Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-23 Martin Ehler, Karlheinz Gröchenig
In analogy to classical spherical t-design points, we introduce the concept of t-design curves on the sphere. This means that the line integral along a t-design curve integrates polynomials of degree t exactly. For low degrees, we construct explicit examples. We also derive lower asymptotic bounds on the lengths of t-design curves. Our main results prove the existence of asymptotically optimal t-design
-
Stability conditions for polarised varieties Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-20 Ruadhaí Dervan
We introduce an analogue of Bridgeland’s stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical Kähler metric, modelled on the constant scalar curvature
-
-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-17 Philipp Lücke, Sandra Müller
Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets
-
Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-17 Gebhard Böckle, Ann-Kristin Juschka
Let K be a finite extension of the p-adic field ${\mathbb {Q}}_p$ of degree d, let ${{\mathbb {F}}\,\!{}}$ be a finite field of characteristic p and let ${\overline {{D}}}$ be an n-dimensional pseudocharacter in the sense of Chenevier of the absolute Galois group of K over the field ${{\mathbb {F}}\,\!{}}$ . For the universal mod p pseudodeformation ring ${\overline {R}{{\phantom {\overline {\overline
-
Length functions in Teichmüller and anti-de Sitter geometry Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-17 Filippo Mazzoli, Gabriele Viaggi
We establish a link between the behavior of length functions on Teichmüller space and the geometry of certain anti-de Sitter $3$ -manifolds. As an application, we give new purely anti-de Sitter proofs of results of Teichmüller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend
-
Double Schubert polynomials do have saturated Newton polytopes Forum Math. Sigma (IF 1.389) Pub Date : 2023-11-03 Federico Castillo, Yairon Cid-Ruiz, Fatemeh Mohammadi, Jonathan Montaño
We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal
-
From topological recursion to wave functions and PDEs quantizing hyperelliptic curves Forum Math. Sigma (IF 1.389) Pub Date : 2023-10-31 Bertrand Eynard, Elba Garcia-Failde
Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second
-
Prefix monoids of groups and right units of special inverse monoids Forum Math. Sigma (IF 1.389) Pub Date : 2023-10-30 Igor Dolinka, Robert D. Gray
A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this
-
Post-Lie algebras in Regularity Structures Forum Math. Sigma (IF 1.389) Pub Date : 2023-10-27 Yvain Bruned, Foivos Katsetsiadis
In this work, we construct the deformed Butcher-Connes-Kreimer Hopf algebra coming from the theory of Regularity Structures as the universal envelope of a post-Lie algebra. We show that this can be done using either of the two combinatorial structures that have been proposed in the context of singular SPDEs: decorated trees and multi-indices. Our construction is inspired from multi-indices where the