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We consider weighted double Hurwitz numbers, with the weight given by arbitrary rational function times an exponent of the completed cycles. Both special singularities are arbitrary, with the lengths of cycles controlled by formal parameters (up to some maximal length on both sides), and on one side there are also distinguished cycles controlled by degrees of formal variables. In these variables the weighted double Hurwitz numbers are presented as coefficients of expansions of some differentials that we prove to satisfy topological recursion. Our results partly resolve a conjecture that we made in [Comm. Math. Phys. 402 (2023), pp. 665–694] and are based on a system of new explicit functional relations for the more general (m,n)-correlation functions, which correspond to the case when there are distinguished cycles controlled by formal variables in both special singular fibers. These (m,n)-correlation functions are the main theme of this paper and the latter explicit functional relations are of independent interest for combinatorics of weighted double Hurwitz numbers. We also put our results in the context of what we call the "symplectic duality", which is a generalization of the x-y duality, a phenomenon known in the theory of topological recursion. [ABSTRACT FROM AUTHOR]

A. Tarski uses in his system for elementary Euclidean geometry only the primitive concept of point , and the two primitive relations betweenness and equidistance. Another approach is for the relations to be on lines instead of points. W. Schwabhäuser and L. Szczerba [Fund. Math. 82 (1974/75), pp. 347–355] showed that perpendicularity together with the ternary relation of co-punctuality are sufficient for dimension two, i.e. they may be used as a system of primitive relations for elementary plane Euclidean geometry. In this paper we give a complete axiomatization for the fragment of elementary plane Euclidean geometry based on perpendicularity alone. We show that this theory is not finitely axiomatizable, that it is decidable and that the complexity is PSPACE-complete. In contrast the complexity of elementary plane Euclidean geometry is exponential. [ABSTRACT FROM AUTHOR]

Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s [Israel J. Math. 44 (1983), pp. 345–360]. A set S\subset \mathbb {N} is an interpolation set for a class of topological dynamical systems \mathcal {C} if any bounded sequence on S can be extended to a sequence that arises from a system in \mathcal {C}. In this paper, we provide combinatorial characterizations of interpolation sets for: [ABSTRACT FROM AUTHOR]

The Gan-Gross-Prasad problem is to describe the restriction of representations of a classical group G to smaller groups H of the same kind. In previous work by Liu and Wang [Trans. Amer. Math. Soc. 373 (2020), pp. 4223–4253; Manuscripta Math. 165 (2021), pp. 159–189; Math. Z. 297 (2021), pp. 997–1021] and Wang [Adv. Math. 393 (2021), p. 72], we study the Gan-Gross-Prasad problem for unipotent representations of finite classical groups. In this paper, we solved the Gan-Gross-Prasad problem over finite fields completely. The main tools used are the Lusztig correspondence and a formula of Reeder [J. Amer. Math. Soc. 20 (2007), pp. 573–602] for the pairings of Deligne-Lusztig characters. We give a reduction decomposition of Reeder's formula and reduce the Gan-Gross-Prasad problem for arbitrary representations to the unipotent representations by Lusztig correspondence. [ABSTRACT FROM AUTHOR]

In a series of work by Wormleighton [Selecta Math. (N.S.) 28 (2022), p 9], Wormleighton [J. Symplectic Geom. 19 (2021), pp. 475–506], and Chaidez and Wormleighton [ECH embedding obstructions for rational surfaces, arXiv: 2008.10125 , 2020], algebraic capacities were introduced in an algebraic manner for polarized algebraic surfaces and applied to the symplectic embedding problems. In this paper, we give a reformulation of algebraic capacities in terms of only a tamed pair of symplectic form and almost complex structure. We show that they actually only depend on the cohomology class of the symplectic form for a rational manifold. Since it is not known that any symplectic form on a rational manifold is Kähler, this novel formulation potentially is more general on a rational manifold. Additionally, for manifolds with b_2^+=1, we derive asymptotic results that are parallel to the context of ECH (Embedded Contact Homology) and algebraic settings. When assuming c_1\cdot [\omega ]>0 on rational manifolds, we further introduce a sequence of tropical polynomials which will succinctly describe those capacities viewed as functions over the domain parametrizing such symplectic forms. As an application, we give a purely symplectic proof of the correspondence between algebraic capacities and ECH capacities for smooth toric surfaces. [ABSTRACT FROM AUTHOR]

It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds (M^{n},g) with boundary and with dimension n<8 that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static (M^{n},g) which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension n < 8 under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22]. [ABSTRACT FROM AUTHOR]

We show that for any polynomial f:\mathbb {Z}\to \mathbb {Z} with positive leading coefficient and irreducible over \mathbb {Q}, if x is large enough then there is a string of (\log x)(\log \log x)^{1/835} consecutive integers n\in [1,x] for which f(n) is composite. This improves the result by Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and Terence Tao [J. Eur. Math. Soc. (JEMS) 23 (2023), pp. 667–700], which has the exponent of \log \log x being a constant depending on f which can be exponentially small in the degree of f. [ABSTRACT FROM AUTHOR]

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller [Proc. Amer. Math. Soc. 150 (2022), pp. 4159–4172]. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that, for any number of summands, a connected sum of doublings is the doubling of a fiber product ring. [ABSTRACT FROM AUTHOR]

We adapt an induction-on-scales argument of Bennett, Bez, Buschenhenke, Cowling, and Flock [Duke Math J. 169 (2020), pp. 3291–3338] to establish a global near-monotonicity statement for the nonlinear Brascamp–Lieb functional under a certain heat-flow, from which follows a global stability result for nonlinear Brascamp–Lieb inequalities under bounded perturbations. [ABSTRACT FROM AUTHOR]

Let W/K be a nonempty scheme over the field of fractions of a Henselian local ring R. A result of Gabber, Liu and Lorenzini [Invent. Math. 192 (2013), pp. 567–626] shows that the GCD of the set of degrees of closed points on W (which is called the index of W/K) can be computed from data pertaining only to the special fiber of a proper regular model of W over R. We show that the entire set of degrees of closed points on W can be computed from data pertaining only to the special fiber, provided the special fiber is a strict normal crossings divisor. As a consequence we obtain an algorithm to compute the degree set of any smooth curve over a Henselian field with finite or algebraically closed residue field. Using this we show that degree sets of curves over such fields can be dramatically different than degree sets of curves over finitely generated fields. For example, while the degree set of a curve over a finitely generated field contains all sufficiently large multiples of the index, there are curves over p-adic fields with index 1 whose degree set excludes all integers that are coprime to 6. [ABSTRACT FROM AUTHOR]

We provide a probabilistic proof of the fundamental gap estimate for Schrödinger operators in convex domains on the sphere, which extends the probabilistic proof of F. Gong, H. Li, and D. Luo [Potential Anal. 44 (2016), pp. 423–442] for the Euclidean case. Our results further generalize the results achieved for the Laplacian by S. Seto, L. Wang, and G. Wei [J. Differential Geom. 112 (2019), pp. 347–389], as well as by C. He, G. Wei, and Qi S. Zhang [Amer. J. Math. 142 (2020), pp. 1161–1191]. The essential ingredient in our analysis is the reflection coupling method on Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We determine the distribution of the sandpile group (also known as the Jacobian) of the Erdős–Rényi random graph G(n,q) along with its canonical duality pairing as n tends to infinity, fully resolving a conjecture due to Clancy, Leake, and Payne [Exp. Math. 24 (2015), pp. 1–7] and generalizing the result by Wood [J. Amer. Math. Soc. 30 (2017), pp. 915–958] on the groups. In particular, we show that a finite abelian p-group G equipped with a perfect symmetric pairing \delta appears as the Sylow p-part of the sandpile group and its pairing with frequency inversely proportional to |G| |\operatorname {Aut}(G,\delta)|, where \operatorname {Aut}(G,\delta) is the set of automorphisms of G preserving the pairing \delta. While this distribution is related to the Cohen–Lenstra distribution, the two distributions are not the same on account of the additional algebraic data of the pairing. The proof utilizes the moment method: we first compute a complete set of moments for our random variable (the average number of epimorphisms from our random object to a fixed object in the category of interest) and then show the moments determine the distribution. To obtain the moments, we prove a universality result for the moments of cokernels of random symmetric integral matrices whose dual groups are equipped with symmetric pairings that is strong enough to handle both the dependence in the diagonal entries and the additional data of the pairing. We then apply results due to Sawin and Wood [ The moment problem for random objects in a category , preprint] to show that these moments determine a unique distribution. [ABSTRACT FROM AUTHOR]

We consider partially hyperbolic diffeomorphisms f with a one-dimensional central direction such that the unstable entropy is different from the stable entropy. Our main result proves that such maps have a finite number of ergodic measures of maximal entropy. Moreover, any C^{1+} diffeomorphism near f in the C^1 topology possesses at most the same number of ergodic measures of maximal entropy. These results extend the findings in Buzzi, Crovisier, and Sarig [Ann. of Math. (2) 195 (2022), pp. 421–508] to arbitrary dimensions and provides an open class of non-Axiom A systems of diffeomorphisms exhibiting a finite number of ergodic measures of maximal entropy. We believe our technique, essentially distinct from the one in Buzzi et al., is robust and may find applications in further contexts. [ABSTRACT FROM AUTHOR]

We characterize the permutative automorphisms of the Cuntz algebra \mathcal {O}_n (namely, stable permutations) in terms of two sequences of graphs that we associate to any permutation of a discrete hypercube [n]^t. As applications we show that in the limit of large t (resp. n) almost all permutations are not stable, thus proving Conj. 12.5 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], characterize (and enumerate) stable quadratic 4 and 5-cycles, as well as a notable class of stable quadratic r-cycles, i.e. those admitting a compatible cyclic factorization by stable transpositions. Some of our results use new combinatorial concepts that may be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]

Let k \ge 1 be an integer and f a holomorphic endomorphism of \mathbb {P}^k (\mathbb {C}) of algebraic degree d\geq 2. We introduce a dynamical volume dimension for ergodic f-invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than (k-1)\log d, a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k=1, but depends on the dynamics of f to incorporate the absence of an analogue of Koebe's theorem and the non-conformality of holomorphic endomorphisms for k\geq 2. If \nu is an ergodic f-invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of \nu. As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k\geq 1. Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents. [ABSTRACT FROM AUTHOR]

We prove that the curl operator on closed oriented 3-manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has 1-dimensional eigenspaces, even along 1-parameter families of \mathcal {C}^k Riemannian metrics, where k\geq 2. We show further that the Hodge Laplacian in dimension 3 has two possible sources for nonsimple eigenspaces along generic 1-parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel [Comm. Pure Appl. Math. 52 (1999), pp. 917–934], allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension 3 is a meagre codimension 1 property with respect to the \mathcal {C}^k topology as proven by Enciso and Peralta-Salas in [Trans. Amer. Math. Soc. 364 (2012), pp. 4207–4224], it is not a meagre codimension 2 property. [ABSTRACT FROM AUTHOR]

In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg's representation of the group G=\text {PSO}(2n+2) and a Hitchin fibration. Consistent with the result over \mathbb {Q} of Arul Shankar and Xiaoheng Wang [Compos. Math. 154 (2018), pp. 188–222], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperelliptic curves, we obtain precisely average number 6. [ABSTRACT FROM AUTHOR]

Alfred Tarski [J. Symbolic Logic 13 (1948), pp. 107–111] defined \mathbf {D}_{pm} to be the set of all sets of type p, type-theoretically definable by parameterfree formulas of type {\le m}, and asked whether it is true that \mathbf {D}_{1m}\in \mathbf {D}_{2m} for m\ge 1. Tarski noted that the negative solution is consistent because the axiom of constructibility \mathbf {V}=\mathbf {L} implies \mathbf {D}_{1m}\notin \mathbf {D}_{2m} for all m\ge 1, and he left the consistency of the positive solution as a major open problem. This was solved in our recent paper [Mathematics 8 (2020), pp. 1–36], where it is established that for any m\ge 1 there is a generic extension of \mathbf {L}, the constructible universe, in which it is true that \mathbf {D}_{1m}\in \mathbf {D}_{2m}. In continuation of this research, we prove here that Tarski's sentences \mathbf {D}_{1m}\in \mathbf {D}_{2m} are not only consistent, but also independent of each other, in the sense that for any set Y\subseteq \omega \smallsetminus \{0\} in \mathbf {L} there is a generic extension of \mathbf {L} in which it is true that \mathbf {D}_{1m}\in \mathbf {D}_{2m} holds for all m\in Y but fails for all m\ge 1, m\notin Y. This gives a full and conclusive solution of the Tarski problem. The other main result of this paper is the consistency of \mathbf {D}_{1}\in \mathbf {D}_{2} via another generic extension of \mathbf {L}, where \mathbf {D}_{p}=\bigcup _m\mathbf {D}_{pm}, the set of all sets of type p, type-theoretically definable by formulas of any type. Our methods are based on almost-disjoint forcing of Jensen and Solovay [Some applications of almost disjoint sets, North-Holland, Amsterdam, 1970, pp. 84–104]. [ABSTRACT FROM AUTHOR]

In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself. [ABSTRACT FROM AUTHOR]

Let K/\mathbb {Q}_p be a finite unramified extension, \overline {\rho }:\mathrm {Gal}(\overline {\mathbb {Q}}_p/K)\rightarrow \mathrm {GL}_n(\overline {\mathbb {F}}_p) a continuous representation, and \tau a tame inertial type of dimension n. We explicitly determine, under mild regularity conditions on \tau, the potentially crystalline deformation ring R^{\eta,\tau }_{\overline {\rho }} in parallel Hodge–Tate weights \eta =(n-1,\cdots,1,0) and inertial type \tau when the shape of \overline {\rho } with respect to \tau has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre's conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150]. [ABSTRACT FROM AUTHOR]

We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting J_{k-1}(s)=\int ^s_0 t^{k-1} e^{-\frac {t^2}{2}}dt and c_{k-1}=J_{k-1}(+\infty), we conjecture that the function F:[0,1]\rightarrow \mathbb {R}, given by \begin{equation*} F(a)= \sum _{k=1}^n 1_{a\in E_k}\cdot (\beta _k J_{k-1}^{-1}(c_{k-1} a)+\alpha _k) \end{equation*} (with an appropriate choice of a decomposition [0,1]=\cup _{i} E_i and coefficients \alpha _i, \beta _i) satisfies, for all symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} F\left (\gamma (\lambda K+(1-\lambda)L)\right)\geq \lambda F\left (\gamma (K)\right)+(1-\lambda) F\left (\gamma (L)\right). \end{equation*} We explain that this conjecture is "the most optimistic possible", and is equivalent to the fact that for any symmetric convex set K, its Gaussian concavity power p_s(K,\gamma) is greater than or equal to p_s(RB^k_2\times \mathbb {R}^{n-k},\gamma), for some k\in \{1,\dots,n\}. We call the sets RB^k_2\times \mathbb {R}^{n-k} round k-cylinders ; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman [Amer. J. Math. 143 (2021), pp. 53–94]. In this manuscript, we make progress towards this question, and show that for any symmetric convex set K in \mathbb {R}^n, \begin{equation*} p_s(K,\gamma)\geq \sup _{F\in L^2(K,\gamma)\cap Lip(K):\,\int F=1} \left (2T_{\gamma }^F(K)-Var(F)\right)+\frac {1}{n-\mathbb {E}X^2}, \end{equation*} where T_{\gamma }^F(K) is the F-torsional rigidity of K with respect to the Gaussian measure. Moreover, the equality holds if and only if K=RB^k_2\times \mathbb {R}^{n-k} for some R>0 and k=1,\dots,n. As a consequence, we get \begin{equation*} p_s(K,\gamma)\geq Q(\mathbb {E}|X|^2, \mathbb {E}\|X\|_K^4, \mathbb {E}\|X\|^2_K, r(K)), \end{equation*} where Q is a certain rational function of degree 2, the expectation is taken with respect to the restriction of the Gaussian measure onto K, \|\cdot \|_K is the Minkowski functional of K, and r(K) is the in-radius of K. The result follows via a combination of some novel estimates, the L2 method (previously studied by several authors, notably Kolesnikov and Milman [J. Geom. Anal. 27 (2017), pp. 1680–1702; Amer. J. Math. 140 (2018), pp. 1147–1185; Geometric aspects of functional analysis , Springer, Cham, 2017; Mem. Amer. Math. Soc. 277 (2022), v+78 pp.], Kolesnikov and the author [Adv. Math. 384 (2021), 23 pp.], Hosle, Kolesnikov, and the author [J. Geom. Anal. 31 (2021), pp. 5799–5836], Colesanti [Commun. Contemp. Math. 10 (2008), pp. 765–772], Colesanti, the author, and Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139], Eskenazis and Moschidis [J. Funct. Anal. 280 (2021), 19 pp.]), and the analysis of the Gaussian torsional rigidity. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the "convex set version" of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest. All the equality case characterizations rely on the careful analysis of the smooth case, the stability versions via trace theory, and local approximation arguments. In addition, we provide a non-sharp estimate for a function F whose composition with \gamma (K) is concave in the Minkowski sense for all symmetric convex sets. [ABSTRACT FROM AUTHOR]

We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai's factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [ Finitary isomorphism of m-dependent processes , Amer. Math. Soc., Birkhauser, Providence, RI, 1992, pp. 373–376], which was first suggested by D. Rudolph [ A characterization of those processes finitarily isomorphic to a Bernoulli shift , Birkhäuser, Boston, Mass., 1981, pp. 1–64]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai's factor theorem for ergodic processes. Another consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [Math. Inst. Hautes Études Sci. 128 (2018), 1–119]. [ABSTRACT FROM AUTHOR]

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in \mathbb {R}^4, which can be finitely presented algebraically as a circuit algebra , or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups \mathsf {KV} and \mathsf {KRV}—the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group \mathsf {GRT}_1 as automorphisms of arrow diagrams. [ABSTRACT FROM AUTHOR]

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps \pi : (X, T) \to (Y, S) between dynamical systems and assume that the mean dimension of the domain (X, T) is larger than the mean dimension of the target (Y, S). We exhibit several situations for which the maps \pi necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss [Israel J. Math. 115 (2000), pp. 1–24] about minimal dynamical systems non-embeddable in [0,1]^{\mathbb {Z}}. [ABSTRACT FROM AUTHOR]

The theory of Moufang sets essentially deals with groups having a split BN-pair of rank one. Every quadratic Jordan division algebra gives rise to a Moufang set such that its root groups are abelian and a certain condition called special is satisfied. It is a major open question if also the converse is true, i.e. if every special Moufang set with abelian root groups comes from a quadratic Jordan division algebra. De Medts and Segev [Amer. Math. Soc. 360 (2008), pp. 5831–5852] proved in Theorem 5.11 that this is the case for special Moufang set satisfying two conditions. In this paper we prove that these conditions are in fact equivalent and hence either of them suffices. Even more, we can replace them by weaker conditions. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring R of prime characteristic. The methods used employ Lyubeznik's F-modules over R, particularly his F-finite F-modules, and also the modules of generalized fractions of Sharp and Zakeri [Mathematika 29 (1982), pp. 32–41]. It is shown that many modules of generalized fractions over R have natural structures as F-modules; these lead to F-module structures on certain local cohomology modules over R, which are exploited, in conjunction with F-module structures on injective R-modules that result from work of Huneke and Sharp [Trans. Amer. Math. Soc. 339 (1993), pp. 765–779], to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2. [ABSTRACT FROM AUTHOR]

Using the Fredholm setup of Farajzadeh-Tehrani [Peking Math. J. (2023), https://doi.org/10.1007/s42543-023-00069-1], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of Gross [Duke Math. J. 153 (2010), pp. 297–362, Cnj. 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in Gross [Prp. 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstruction spaces for maps that have components in a destabilizing (or rubber) component of the target. Finally, we study a higher genus version of these invariants and explain a decomposition of genus one invariants into different contributions. [ABSTRACT FROM AUTHOR]

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case. [ABSTRACT FROM AUTHOR]

We correct a mistake that appeared in the first section of the original article, which appeared in Tran. Amer. Math. Soc. 370 (2018), 99-130. Namely, Corollary 1.16 was false as stated and was subsequently used in later proofs in the paper. In this note it is argued that all the relevant statements after Corollary 1.16 can be saved with at most minor modifications. In particular, all the main results of the original paper remain valid as stated, but some intermediate claims are slightly modified or proved more directly without Corollary 1.16. [ABSTRACT FROM AUTHOR]

The category of framed correspondences Fr*(k) and framed sheaves were invented by Voevodsky in his unpublished notes [ Notes on framed correspondences , 2001, https://www.math.ias.edu/vladimir/publications]. Based on the theory, framed motives are introduced and studied in Garkusha and Panin [J. Amer. Math. Soc. 34 (2021), pp. 261–313]. These are Nisnivich sheaves of S1-spectra and the major computational tool of Garkusha and Panin. The aim of this paper is to show the following result which is essential in proving the main theorem of Garkusha and Panin: given an infinite perfect base field k, any k-smooth scheme X and any n ≥ 1, the map of simplicial pointed Nisnevich sheaves (−,A1//Gm)∧n+ → Tn induces a Nisnevich local level weak equivalence of S1-spectra Mfr(X × (A1//Gm)∧n) → Mfr(X × Tn). Moreover, it is proven that the sequence of S1-spectra Mfr(X ×Tn × Gm) → Mfr(X × Tn × A1) → Mfr(X × Tn+1) is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of Garkusha and Panin. This computation is crucial for the whole machinery of framed motives. [ABSTRACT FROM AUTHOR]

Let p be a prime and G a finite group. We propose a strong bound for the number of p'-degree irreducible characters of G in terms of the commutator factor group of a Sylow p-subgroup of G. The bound arises from a recent conjecture of Navarro and Tiep [Forum Math. Pi 9 (2021), pp. 1–28] on fields of character values and a phenomenon called the continuity of p-rationality level of p'-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture (see G. Navarro [Ann. of Math. (2) 160 (2004), pp. 1129–1140]). We achieve both the bound and the continuity property for p=2. [ABSTRACT FROM AUTHOR]

We show that given a closed n-manifold M, for a Baire-generic set of Riemannian metrics g on M there exists a sequence of closed geodesics that are equidistributed in M if n=2; and an equidistributed sequence of embedded stationary geodesic nets if n=3. One of the main tools that we use is the Weyl law for the volume spectrum for 1-cycles, proved by Liokumovich, Marques, and Neves [Ann. of Math. (2) 187 (2018), pp. 933–961] for n=2 and by Guth and Liokumovich [Preprint, arXiv:2202.11805, 2022] for n=3. We show that our proof of the equidistribution of stationary geodesic nets can be generalized for any dimension n\geq 2 provided the Weyl Law for 1-cycles in n-manifolds holds. [ABSTRACT FROM AUTHOR]

We calculate the E-polynomial for a class of (complex) character varieties \mathcal {M}_n^{\tau } associated to a genus g Riemann surface \Sigma equipped with an orientation reversing involution \tau. Our formula expresses the generating function \sum _{n=1}^{\infty } E(\mathcal {M}_n^{\tau }) T^n as the plethystic logarithm of a product of sums indexed by Young diagrams. The proof uses point counting over finite fields, emulating Hausel and Rodriguez-Villegas [Invent. Math. 174 (2008), pp. 555–624]. [ABSTRACT FROM AUTHOR]

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler–Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene. [ABSTRACT FROM AUTHOR]

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R+ or R with values in a general Riemannian manifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper of Röckner and the second, third, and fourth authors [SIAM J. Math. Anal. 52 (2020), pp. 2237-2274] on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution of the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative. [ABSTRACT FROM AUTHOR]

In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [Compos. Math. 154 (2018), pp. 2403-2425]. [ABSTRACT FROM AUTHOR]

We show that for any even log-concave probability measure \mu on \mathbb {R}^n, any pair of symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} \mu ((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*} where c_n\geq n^{-4-o(1)}. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. [ABSTRACT FROM AUTHOR]

In this short note we correct the statement of the main result of [Trans. Amer. Math. Soc. 373 (2020), no. 3, 1909-1940]. That paper presented the rational cohomology ring of a toric arrangement by generators and relations. One of the series of relations given in the paper is indexed over the set circuits in the arrangement's arithmetic matroid. That series of relations should however be indexed over all sets X with |X| = rk(X)+1. Below we give the complete and correct presentation of the rational cohomology ring. [ABSTRACT FROM AUTHOR]

In this paper, we develop an elementary and unified treatment, in the spirit of Rivière and Struwe (Comm. Pure. Appl. Math. 2008), to explore regularity of weak solutions of higher order geometric elliptic systems in critical dimensions without using conservation law. As a result, we obtain an interior Hölder continuity for solutions of the higher order elliptic system of de Longueville and Gastel in critical dimensions Δku = ∑i=0k−1Δi⟨Vi,du⟩ + ∑i=0 k−2Δiδ (widu) quad in B2k, under critical regularity assumptions on the coefficient functions. This verifies an expectation of Rivière, and provides an affirmative answer to an open question of Struwe in dimension four when k = 2. The Hölder continuity is also an improvement of the continuity result of Lamm and Rivière and de Longueville and Gastel. [ABSTRACT FROM AUTHOR]

For 2\leq p<4, we study the L^p norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact 2-dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov [Duke Math. J. 138 (2007), pp. 445–486], and Hu [Forum Math. 21 (2009), pp. 1021–1052] found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For p=4, we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [Comm. Math. Phys. 350 (2017), pp. 1299–1325], Sogge [Math. Res. Lett. 24 (2017), pp. 549–570], and Bourgain [Geom. Funct. Anal. 1 (1991), pp. 147–187]. [ABSTRACT FROM AUTHOR]

In this article, we prove a new general identity involving the Theta operators introduced by the first author, Iraci, and Vanden Wyngaerd [Adv. Math. 376 (2021), p.59]. From this result, we can easily deduce several new identities that have combinatorial consequences in the study of Macdonald polynomials and diagonal coinvariants. In particular, we provide a unifying framework from which we recover many identities scattered in the literature, often resulting in drastically shorter proofs. [ABSTRACT FROM AUTHOR]

We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has a continuous or homeomorphic extension to the closures of the domains. Our conditions involve the notions of cofat domains and CNED sets, i.e., countably negligible for extremal distances, recently introduced by the author. We use this result towards establishing conformal rigidity of a class of circle domains. A circle domain is conformally rigid if every conformal map onto another circle domain is the restriction of a Möbius transformation. We show that circle domains whose point boundary components are CNED are conformally rigid. This result is the strongest among all earlier works and provides substantial evidence towards the rigidity conjecture of He–Schramm [Invent. Math. 115 (1994), no. 2, 297–310], relating the problems of conformal rigidity and removability. [ABSTRACT FROM AUTHOR]

In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least d contains H. In particular, they raise the questions of determining the uniform Turán densities of K_4^{(3)-} and K_4^{(3)}. The former question was solved only recently by Glebov, Král', and Volec [Israel J. Math. 211 (2016), pp. 349–366] and Reiher, Rödl, and Schacht [J. Eur. Math. Soc. 20 (2018), pp. 1139–1159], while the latter still remains open for almost 40 years. In addition to K_4^{(3)-}, the only 3-uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), pp. 77–97] and a specific family with uniform Turán density equal to 1/27. We develop new tools for embedding hypergraphs in host hypergraphs with positive uniform density and apply them to completely determine the uniform Turán density of a fundamental family of 3-uniform hypergraphs, namely tight cycles C_\ell ^{(3)}. The uniform Turán density of C_\ell ^{(3)}, \ell \ge 5, is equal to 4/27 if \ell is not divisible by three, and is equal to zero otherwise. The case \ell =5 resolves a problem suggested by Reiher. [ABSTRACT FROM AUTHOR]

Given two polynomials p(x), q(x) of degree d, we give a combinatorial formula for the finite free cumulants of p(x)\boxtimes _d q(x). We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that \boxtimes _d converges to \boxtimes as d goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers. [ABSTRACT FROM AUTHOR]

We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana [Ann. Inst. Fourier (Grenoble) 54 (2004), pp. 499–630] and Corvaja–Zannier [Math. Z. 286 (2017), pp. 579–602] holds for Enriques surfaces, as well as K3 surfaces of Picard rank \geq 6 apart from a finite list of geometric Picard lattices. Concretely, we prove that such surfaces over finitely generated fields of characteristic 0 satisfy the weak Hilbert Property after a finite field extension of the base field. The degree of the field extension can be uniformly bounded. [ABSTRACT FROM AUTHOR]

We consider a nonlinear heat equation with a double source: |u|p-1u and |∇u|q. This equation has a double interest: in ecology, it was used by Souplet (1996) as a population dynamics model; in mathematics, it was introduced by Chipot and Weissler (1989) as an intermediate equation between the semilinear heat equation and the Hamilton-Jacobi equation. Further interest in this equation comes from its lack of variational structure. In this paper, we intend to see whether the standard blow-up dynamics known for the standard semilinear heat equation (with |u|p-1u as the only source) can be modified by the addition of the second source (|∇u|q). Here arises a nice critical phenomenon at blow-up: - when q < 2p/(p+1), the second source is subcritical in size with respect to the first, and we recover the classicial blow-up profile known for the standard semilinear case; - when q = 2p/(p + 1), both terms have the same size, and only partial blow-up descriptions are available. In this paper, we focus on this case, and start from scratch to: - first, formally justify the occurrence of a new blow-up profile, which is different from the standard semilinear case; - second, to rigorously justify the existence of a solution obeying that profile, thanks to the constructive method introduced by Bricmont and Kupiainen together with Merle and Zaag. Note that our method yields the stability of the constructed solution. Moreover, our method is far from being a straightforward adaptation of earlier literature and should be considered as a source of novel ideas whose application goes beyond the particular equation we are considering, as we explain in the introduction. [ABSTRACT FROM AUTHOR]

Let R be a standard graded Gorenstein algebra over a field presented by quadrics. In [Compositio Math. 129 (2001), no. 1, 95-121], Conca-Rossi-Valla show that such a ring is Koszul if reg R ≤ 2 or if reg R = 3 and c = codim R ≤ 4, and they ask whether this is true for reg R = 3 in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring R that guarantee the Nagata idealization ~ R = R × ωR(−a−1) is a non-Koszul quadratic Gorenstein ring. We prove there exist rings of regularity 3 satisfying our conditions for all c ≥ 9; this yields a negative answer to the question from the above-mentioned paper. [ABSTRACT FROM AUTHOR]

Let g be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ be a diagram automorphism of g, and let L(g,μ) be the loop algebra of g associated to μ. In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g[μ] of L(g,μ). The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212-216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283-307] as special examples. As an application, when g is of simply-laced-type, we prove that the classical limit of the μ-twisted quantum affinization of the quantum Kac-Moody algebra associated to g introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g[μ]. [ABSTRACT FROM AUTHOR]