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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]

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 this short note we show that Doyle-Grigor'yan criterion for non-parabolicity is not necessary in dimension greater than or equal to four. This gives an negative answer to Problem # 1 of Grigor'yan [Bull. Amer. Math. Soc. (N.S) 36 (1999). pp. 135–249] in this dimensional range. [ABSTRACT FROM AUTHOR]

Let X_1,X_2,\ldots be independent and identically distributed random variables in {\mathbb {C}} chosen from a probability measure \mu and define the random polynomial \begin{align*} P_n(z)=(z-X_1)\ldots (z-X_n)\,. \end{align*} We show that for any sequence k = k(n) satisfying k \leq \log n / (5 \log \log n), the zeros of the kth derivative of P_n are asymptotically distributed according to the same measure \mu. This extends work of Kabluchko, which proved the k = 1 case, as well as Byun, Lee and Reddy [Trans. Amer. Math. Soc. 375, pp. 6311–6335] who proved the fixed k case. [ABSTRACT FROM AUTHOR]

Suppose that B is a separable stable C^*-algebra with real rank zero, stable rank one and (\mathrm {K}_0(B), \mathrm {K}_0^+(B)) is weakly unperforated in the sense of Elliott [Internat. J. Math. 1 (1990), no. 4, pp. 361–380]. Let A be a unital simple separable nuclear \mathrm {C}^*-algebra. We show that B has the corona factorization property and any unital extension of A by B is absorbing. [ABSTRACT FROM AUTHOR]

We prove two results concerning extendible cardinals. First, we show that the first C^{(n)}-extendible cardinal is strictly greater than the first C^{(n)}-supercompact. This answers a question posed by Bagaria in [Arch. Math. Logic 51 (2012), pp. 213–240]. Second, assuming the existence of strong enough large cardinals, we prove the consistency of the following: there are cardinals \kappa <\lambda such that \lambda is singular of countable cofinality, \kappa is {<}\lambda-extendible (but not \lambda-extendible) and (\lambda ^+)^{\mathrm {HOD}}<\lambda ^+. [ABSTRACT FROM AUTHOR]

In this paper, using the method of blow-up analysis, we establish a trace Trudinger-Moser inequality involving L^p-norm and obtain the corresponding extremal on a compact Riemann surface with a smooth boundary. The result generalizes those of Li-Liu [Math. Z. 250 (2005), pp. 363–686] and Zhang [Commun. Pure Appl. Anal. 20 (2021), pp. 1721–1735]. [ABSTRACT FROM AUTHOR]

The chord log-Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-chord measure of a convex body. The chord log-Minkowski problem has been extensively studied by Guo, Xi, and Zhao [Math. Ann. (2023), DOI 10.1007/s00208-023-02721-8]; Lutwak, Xi, Yang, and Zhang [Commun. Pure Appl. Math. (2023), DOI 10.1002/cpa.22190]; Qin [Adv. Math. 427 (2023), Paper No. 109132]. In this paper, we solve the chord log-Minkowski problem when q\in (0,1), without symmetry assumptions. [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]

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]

I argue that since technology is already changing the way we do mathematics, we can use this technology to allow mathematics to be more congressive without human mathematicians becoming redundant in the face of technological advances. In thinking about what it means to "do mathematics," I examine the following aspects of technology in mathematics: teaching and learning, asking questions, collaboration, dissemination, and the act of doing research. This is not intended to be a rigorous analysis but an informed reflection based on my experience as a mathematician. [ABSTRACT FROM AUTHOR]

In this article we discuss how abstraction boundaries can help tame complexity in mathematical research with the help of an interactive theorem prover. While many of the ideas we present here have been used implicitly by mathematicians for some time, we argue that the use of an interactive theorem prover introduces additional qualitative benefits in the implementation of these ideas. [ABSTRACT FROM AUTHOR]

The author takes colleagues to task for failing to "live deliberately," and specifically for failing to pay attention to debates over technology and artificial intelligence in the wider society, when contemplating a mechanical future for mathematics. [ABSTRACT FROM AUTHOR]

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

Let \mathbb {F}_q denote the finite field of characteristic p and order q. Let \mathbb {Z}_q denote the unramified extension of the p-adic rational integers \mathbb {Z}_p with residue field \mathbb {F}_q. Given two positive integers m,n, define a box \mathcal B_m to be a subset of \mathbb {Z}_q^n with q^{nm} elements such that \mathcal B_m modulo p^m is equal to (\mathbb {Z}_q/p^m \mathbb {Z}_q)^n. For a collection of nonconstant polynomials f_1,\dots,f_s\in \mathbb {Z}_q[x_1,\ldots,x_n] and positive integers m_1,\dots,m_s, define the set of common zeros inside the box \mathcal B_m to be \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} It is an interesting problem to give the sharp estimates for the p-divisibility of |V|. This problem has been partially solved for the three cases: (i) m=m_1=\cdots =m_s=1, which is just the Ax-Katz theorem, (ii) m=m_1=\cdots =m_s>1, which was solved by Katz [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m=1, and m_1,\dots,m_s\geq 1, which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. Based on the multi-fold addition and multiplication of the finite Witt rings over \mathbb {F}_q, we investigate the remaining unconsidered case of m>1 and m\neq m_j for some 1\leq j\leq s, and finally provide a complete answer to this problem. [ABSTRACT FROM AUTHOR]

The set of degrees of maps D(M,N), where M,N are closed oriented n-manifolds, always contains 0 and the set of degrees of self-maps D(M) always contains 0 and 1. Also, if a,b\in D(M), then ab\in D(M); a set A\subseteq \mathbb {Z} so that ab\in A for each a,b\in A is called multiplicative. On the one hand, not every infinite set of integers (containing 0) is a mapping degree set (Neofytidis, Wang, and Wang [Bull. Lond. Math. Soc. 55 (2023), pp. 1700–1717]) and, on the other hand, every finite set of integers (containing 0) is the mapping degree set of some 3-manifolds (Costoya, Muñoz and Viruel [ Finite sets containing zero are mapping degree sets , arXiv: 2301.13719 ]). We show the following: Not every multiplicative set A containing 0,1 is a self-mapping degree set. For each n\in \mathbb {N} and k\geq 3, every D(M,N) for n-manifolds M and N is D(P,Q) for some (n+k)-manifolds P and Q. As a consequence of (ii) and Costoya, Muñoz and Viruel, every finite set of integers (containing 0) is the mapping degree set of some n-manifolds for all n\neq 1,2,4,5. [ABSTRACT FROM AUTHOR]

In this article we establish a new analytic criterion for univalence of typically-real functions. Moreover, we find geometric properties for functions fulfilling the criterion. The new subclass of univalent functions is defined via these geometric properties. This class can be useful for verifying the univalence of potentially extremal polynomials associated to the work of Dmitrishin, Dyakonov and Stokolos [Anal. Math. Phys. 9 (2019), pp. 991–1004] on univalent polynomials and Koebe's One-Quarter Theorem. [ABSTRACT FROM AUTHOR]

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui–Hà–Jayanthan–Thomas [Collect. Math. 72 (2021), pp. 605–614]. Complete solutions are delivered for two conjectures proposed by these authors. For given real numbers a and b, we consider the set Res(a,b) of possible values of the resurgence of I+J where I and J are ideals in disjoint sets of variables having resurgence a and b, respectively. Some questions and partial results about Res(a,b) are discussed. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

We prove that if a symplectic 4-manifold X becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold X is originally rational. That is, a symplectic rational blow-up of a rational symplectic 4-manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition. [ABSTRACT FROM AUTHOR]

A paper of Reeder–Yu [J. Amer. Math. Soc. 27 (2014), pp. 437–477] gives a construction of epipelagic supercuspidal representations of p-adic groups. The input for this construction is a pair (\lambda, \chi) where \lambda is a stable vector in a certain representation coming from a Moy–Prasad filtration, and \chi is a character of the additive group of the residue field. We say two such pairs are equivalent if the resulting supercuspidal representations are isomorphic. In this paper we describe the equivalence classes of such pairs. As an application, we give a classification of the simple supercuspidal representations for split adjoint groups. Finally, under an assumption about unramified base change, we describe properties of the Langlands parameters associated to these simple supercuspidals, showing that they have trivial L-functions and minimal Swan conductors, and showing that each of these simple supercuspidals lies in a singleton L-packet. [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]

We give a simple proof that the orbit space of the p-subgroup complex of a finite group is contractible using Brown-Forman discrete Morse theory. This result was originally conjectured by Webb [Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 349–365] and proved by Symonds [Comment. Math. Helv. 73 (1998), pp. 400–405]. [ABSTRACT FROM AUTHOR]

A study of the class \Delta consisting of topological \Delta-spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes \Delta _2 \subset \Delta _1 properly containing \Delta. We observe that for every first-countable X the following equivalences hold: X\in \Delta _1 iff X\in \Delta _2 iff each countable subset of X is G_{\delta }. Thus, new proposed concepts provide a natural extension of the family of all \lambda-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class \Delta _1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class \Delta _2. We investigate whether the classes \Delta _1 and \Delta _2 are invariant under the basic topological operations. Similarly to \Delta, both classes \Delta _1 and \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to \Delta, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. C_p(X) admits a continuous linear map onto C_p(Y). We show that Y \in \Delta _1 whenever X \in \Delta _1. Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone–Čech compactification \beta Y is scattered. [ABSTRACT FROM AUTHOR]

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case. [ABSTRACT FROM AUTHOR]

Karn and Sinha [Glasg. Math. J. 56 (2014), pp. 427–437] introduced the p- limited (1 \leq p < \infty) sets (see the definition below). We show that p-limited sets are preserved by continuous polynomials as well as by the projective tensor product and that scalar-valued polynomials are p-summable on p-limited sets. Considering the notion of p-limited set from the \ell _p-valued operators point of view, we introduce in Section \ref{sec:3} two weaker types of p-limitedness in the setting of Banach lattices and study their basic properties. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

We show that R is regular if Tor_{i}^{R}(R^{+},k) = 0 for some i\geq 1 assuming further that R is a \mathbb {N}-graded ring of dimension 2 finitely generated over an algebraically closed equicharacteristic zero field k. This answers a question of Bhatt, Iyengar, and Ma [Comm. Algebra 47 (2019), pp. 2367–2383]. We use almost mathematics over R^{+} to deduce properties of the noetherian ring R and rational surface singularities. Moreover we observe that R^{+} in equicharacteristic zero has a rich module-theoretic structure; it is m-adically ideal(wise) separated, (weakly) intersection flat, and Ohm-Rush. As an application we show that the hypothesis can be astonishingly vacuous for i \ll dim(R). We show that a positive answer to an old question of Aberbach and Hochster [J. Pure Appl. Algebra 122 (1997), pp. 171–184] also answers this question and we use our techniques to study a question of André and Fiorot [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), pp. 81–114] regarding 'fpqc analogues' of splinters. [ABSTRACT FROM AUTHOR]

Langlands [ On the classification of irreducible representations of real algebraic groups , Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170] defined L-packets for real reductive groups. In order to refine the local Langlands correspondence, Adams-Barbasch-Vogan [ The Langlands classification and irreducible characters for real reductive groups , Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992] combined L-packets over all real forms belonging to an inner class. In the tempered setting, using different methods, Kaletha [Ann. of Math. (2) 184 (2016), pp. 559–632] also defines such combined L-packets with a refinement to the local Langlands correspondence. We prove that the tempered L-packets of Adams-Barbasch-Vogan and Kaletha are the same and are parameterized identically. [ABSTRACT FROM AUTHOR]

Matroids are combinatorial abstractions of independence, a ubiquitous notion that pervades many branches of mathematics. June Huh and his collaborators recently made spectacular breakthroughs by developing a Hodge theory of matroids that resolved several long-standing conjectures in matroid theory. We survey the main results in this development and ideas behind them. [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 propose and study a fully discrete finite volume scheme for the linear Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted L^2 relative entropy. Then, we adapt the arguments developed in Dolbeault, Mouhot, and Schmeiser [Trans. Amer. Math. Soc. 367 (2015), pp. 3807–3828] based on hypocoercivity methods to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play. [ABSTRACT FROM AUTHOR]

Kleban and Özlük [Comm. Math. Phys. 203 (1999), pp. 635–647] introduced a 'Farey fraction spin chain' and made a conjecture regarding its asymptotic number of states with given energy, the latter being given (up to some normalisation) by the number \Phi (N) of 2{\times }2 matrices arising as products of \bigl (\begin {smallmatrix} 1 & 0 \\ 1 & 1 \end {smallmatrix}\bigr) and \bigl (\begin {smallmatrix} 1 & 1 \\ 0 & 1 \end {smallmatrix}\bigr) whose trace equals N. Although their conjecture was disproved by Peter [J. Number Theory 90 (2001), pp. 265–280], quite precise results are known on average by works of Kallies–Özlük–Peter–Snyder [Comm. Math. Phys. 203 (1999), pp. 635–647], Boca [J. Reine Angew. Math. 606 (2007), pp. 149–165] and Ustinov [Mat. Sb. 204 (2013), pp. 143–160]. We show that the problem of estimating \Phi (N) can be reduced to a problem on divisors of quadratic polynomials which was already solved by Hooley [Math. Z. 69 (1958), pp. 211–227] in a special case and, quite recently, in full generality by Bykovskiĭ and Ustinov [Dokl. Math. 99 (2019), pp. 195–200]. This produces an unconditional estimate for \Phi (N), which hitherto was only (implicitly) known, conditionally on the availability on wide zero-free regions for certain Dirichlet L-functions, by the work of Kallies–Özlük–Peter–Snyder. [ABSTRACT FROM AUTHOR]

On negatively curved compact manifolds, it is possible to associate to every closed form a bounded cocycle – hence a bounded cohomology class – via integration over straight simplices. The kernel of this map is contained in the space of exact forms. We show that in degree 2 this kernel is trivial, in contrast with higher degree. In other words, exact non-zero 2-forms define non-trivial bounded cohomology classes. This result is the higher dimensional version of a classical theorem by Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] for surfaces. As a consequence, one gets that the second bounded cohomology of negatively curved manifolds contains an infinite dimensional space, whose classes are explicitly described by integration of forms. This also showcases that some recent results by Marasco [Proc. Amer. Math. Soc. 151 (2023), pp. 2707–2715] can be applied in higher dimension to obtain new non-trivial results on the vanishing of certain cup products and Massey products. Some other applications are discussed. [ABSTRACT FROM AUTHOR]

We investigate Fourier decay for Frostman measures supported on curves with nonzero curvature. We combine decoupling with known lower bounds for Furstenberg sets to extend the main result of Orponen [Ann. Fenn. Math. 48 (2023), pp. 113–139]. [ABSTRACT FROM AUTHOR]

Let R be a finitely generated \mathbb N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P\in Ass(R/I) one defines the v-invariant v_P(I) of I at P as the least c\in \mathbb N such that P=I:f for some f\in R_c. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/I^n) is constant for large n. So it makes sense to consider a prime ideal P\in Ass(R/I^n) for all the large n and investigate how v_P(I^n) depends on n. We prove that v_P(I^n) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [ Asymptotic behaviour of the \text {v}-number of homogeneous ideals , https://arxiv.org/abs/2306.14243, 2023]. [ABSTRACT FROM AUTHOR]

This corrigendum makes a correction to K. Salame, "On Lau's conjecture", Proc. Amer. Math. Soc. 148 (2020), no. 1, pp. 343–350. [ABSTRACT FROM AUTHOR]

In Theorem 0.1 of the paper "A Severi type theorem for surfaces in \mathbb {P}^6" [Proc. Amer. Math. Soc. 149 (2021), pp. 591–605], we claimed to have given a complete classification of smooth surfaces in \mathbb {P}^6 with one 4-secant plane through the general point of \mathbb {P}^6, but the classification is still incomplete. [ABSTRACT FROM AUTHOR]