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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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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 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 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]

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]

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]

Let X be a quotient of the modular curve X_0(N) whose Jacobian J_X is a simple factor of J_0(N)^{new} over \mathbf {Q}. Let f be the newform of level N and weight 2 associated with J_X; assume f has analytic rank 1. We give analytic methods for determining the rational points of X using quadratic Chabauty by computing two p-adic Gross–Zagier formulas for f. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic p-adic L-function of f constructed by Bertolini, Darmon, and Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148], which lies outside of the range of interpolation. [ABSTRACT FROM AUTHOR]

We construct pointed Prym–Brill–Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym–Petri map and prove a pointed version of the Prym–Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters [Ann. Sci. Ëcole Norm. Sup. (4) 18 (1985), pp. 671–683] and De Concini–Pragacz [Math. Ann. 302 (1995), pp. 687–697] on the unpointed case. Finally, we show that Prym varieties are Prym–Tyurin varieties for Prym–Brill–Noether curves of exponent enumerating standard shifted tableaux times a factor of 2, extending to the Prym setting work of Ortega [Math. Ann. 356 (2013), pp. 809–817]. [ABSTRACT FROM AUTHOR]

Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in \mathbb {R}^d. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis [Invent. Math. 138 (1999), pp. 451–494], Khinchin and Gallagher [J. London Math. Soc. 37 (1962), pp. 387–390] can be deduced from our distributional results. [ABSTRACT FROM AUTHOR]

We show that the number of non-isotopic commutative semifields of odd order p^{n} is exponential in n when n = 4t and t is not a power of 2. We introduce a new family of commutative semifields and a method for proving isotopy results on commutative semifields that we use to deduce the aforementioned bound. The previous best bound on the number of non-isotopic commutative semifields of odd order was quadratic in n and given by Zhou and Pott [Adv. Math. 234 (2013), pp. 43–60]. Similar bounds in the case of even order were given in Kantor [J. Algebra 270 (2003), pp. 96–114] and Kantor and Williams [Trans. Amer. Math. Soc. 356 (2004), pp. 895–938]. [ABSTRACT FROM AUTHOR]

We study the category of representations of \mathfrak {sl}_{m+2n} in positive characteristic, with p-character given by a nilpotent with Jordan type (m+n,n). Recent work of Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] implies that this representation category is equivalent to \mathcal {D}_{m,n}^0, the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], we construct functors indexed by affine tangles, between these categories \mathcal {D}_{m,n} (i.e. for different values of n). This allows us to describe the irreducible objects in \mathcal {D}_{m,n}^0 and enumerate them by crossingless (m,m+2n) matchings. We compute the \mathrm {Ext} spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov's arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category \mathcal {O} due to Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al. [ABSTRACT FROM AUTHOR]

We correct some errors in The class number one problem for the normal CM-fields of degree 32 in Trans. Amer. Math. Soc., vol. 359 no. 10 (2007). [ABSTRACT FROM AUTHOR]

We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space \mathbb {R}^3, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in \mathbb {R}^4, to the energy-subcritical equation in \mathbb {R}^3. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the I-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime. [ABSTRACT FROM AUTHOR]

We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman [J. Algebraic Geom. 10 (2001), pp. 623–694], Yoshioka [J. Reine Angew.Math. 515 (1999), pp. 97–123], and Nakajima [ Convolution on homology groups of moduli spaces of sheaves on K3 surfaces , Contemp. Math., vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 75–87]. We show that these sequences can be given the structure of a geometric categorical \mathfrak {sl}_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops. [ABSTRACT FROM AUTHOR]

Using the theory of Macdonald [ Symmetric functions and Hall polynomials , The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q=t. We generalize this to restricted rational Cherednik algebras of wreath product groups C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [ Combinatorics, symmetric functions, and Hilbert schemes , Int. Press, Somerville, MA, 2003, pp. 39–111]. [ABSTRACT FROM AUTHOR]

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100–1126] concerning manifolds with negative k-Ricci curvature and generalize a result of Wu-Yau [Comm. Anal. Geom. 24 (2016), pp. 901–912] and Diverio-Trapani [J. Differential Geom. 111 (2019), pp. 303–314] to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive. [ABSTRACT FROM AUTHOR]

We prove sharp L^{12} estimates for exponential sums associated with nondegenerate curves in \mathbb {R}^4. We place Bourgain's seminal result [J. Amer. Math. Soc. 30 (2017), pp. 205–224] in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves. [ABSTRACT FROM AUTHOR]

Let \mathcal {C}(n,k) be the set of k-dimensional simplicial complexes C over a fixed set of n vertices such that: C has a complete k-1-skeleton; C has precisely {{n-1}\choose {k}} k-faces; the homology group H_{k-1}(C) is finite. Consider the probability measure on \mathcal {C}(n,k) where the probability of a simplicial complex C is proportional to |H_{k-1}(C)|^2. For any fixed k, we determine the local weak limit of these random simplicial complexes as n tends to infinity. This local weak limit turns out to be the same as the local weak limit of the 1-out k-complexes investigated by Linial and Peled [Ann. of Math. (2) 184 (2016), pp. 745-773]. [ABSTRACT FROM AUTHOR]

We introduce and study a notion of relative 1-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative 1-homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1-homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1-homotopy class. Our theorems generalize and strengthen results of Lemaire [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), pp. 91–103], Jost [J. Reine Angew. Math. 359 (1985), pp. 37-54], Schoen–Yau [Ann. of Math. (2) 110 (1979), pp. 127–142], and Sacks–Uhlenbeck [Trans. Amer. Math. Soc. 271 (1982), pp. 639–652]. [ABSTRACT FROM AUTHOR]

Jockusch [J. Combin. Theory Ser. A 72 (1995), pp. 318–321] constructed an infinite family of centrally symmetric (cs, for short) triangulations of 3-spheres that are cs-2-neighborly. Recently, Novik and Zheng [Adv. Math. 370 (2020), 16 pp.] extended Jockusch's construction: for all d and n>d, they constructed a cs triangulation of a d-sphere with 2n vertices, \Delta ^d_n, that is cs-\lceil d/2\rceil-neighborly. Here, several new cs constructions, related to \Delta ^d_n, are provided. It is shown that for all k>2 and a sufficiently large n, there is another cs triangulation of a (2k-1)-sphere with 2n vertices that is cs-k-neighborly, while for k=2 there are \Omega (2^n) such pairwise non-isomorphic triangulations. It is also shown that for all k>2 and a sufficiently large n, there are \Omega (2^n) pairwise non-isomorphic cs triangulations of a (2k-1)-sphere with 2n vertices that are cs-(k-1)-neighborly. The constructions are based on studying facets of \Delta ^d_n, and, in particular, on some necessary and some sufficient conditions similar in spirit to Gale's evenness condition. Along the way, it is proved that Jockusch's spheres \Delta ^3_n are shellable and an affirmative answer to Murai–Nevo's question about 2-stacked shellable balls is given. [ABSTRACT FROM AUTHOR]

Let G be a reductive group, and let X be a smooth quasi-projective complex variety. We prove that any G-irreducible, G-cohomologically rigid local system on X with finite order abelianization and quasi-unipotent local monodromies is integral. This generalizes work of Esnault and Groechenig [Selecta Math. (N. S.) 24 (2018), pp. 4279–4292; Acta Math. 225 (2020), pp. 103–158] when G= \mathrm {GL}_n, and it answers positively a conjecture of Simpson [Inst. Hautes Études Sci. Publ. Math. 75 (1992), pp. 5–95; Inst. Hautes Études Sci. Publ. Math. 80 (1994), pp. 5–79] for G-cohomologically rigid local systems. Along the way we show that the connected component of the Zariski-closure of the monodromy group of any such local system is semisimple; this moreover holds when we relax cohomological rigidity to rigidity. [ABSTRACT FROM AUTHOR]

We show that two-weight L2 bounds for sparse square functions (uniform with respect to sparseness constants, and in both directions) do not imply a two-weight L2 bound for the Hilbert transform. We present an explicit counterexample, making use of the construction due to Reguera–Thiele from [Math. Res. Lett. 19 (2012)]. At the same time, we show that such two-weight bounds for sparse square functions do not imply both separated Orlicz bump conditions on the involved weights for p = 2 (and for Young functions satisfying an appropriate integrability condition). We rely on the domination of L log L bumps by Orlicz bumps observed by Treil–Volberg in [Adv. Math. 301 (2016), pp. 499-548] (for Young functions satisfying an appropriate integrability condition). [ABSTRACT FROM AUTHOR]

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski's conjecture for amenable theories (proved by Krupiński, Newelski, and Simon [J. Math. Log. 19 (2019), no. 2, 1950012, p. 55]) cannot be dropped. [ABSTRACT FROM AUTHOR]

Let E be a CM number field and F its maximal real subfield. We prove a level-raising result for regular algebraic conjugate self-dual automorphic representations of GLn(AE). This generalizes previously known results of Thorne [Forum Math. Sigma 2 (2014)] by removing certain hypotheses occurring in that work. In particular, the level-raising prime p is allowed to be unramified as opposed to inert in F, the field E/F is not assumed to be everywhere unramified, and the field E is allowed to be a general CM field. [ABSTRACT FROM AUTHOR]