Search

We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations of the higher Cantor and Baire spaces.

We study games of length ω2 with moves in ℕ and Borel payoff. These are, e.g., games in which two players alternate turns playing digits to produce a real number in [0, 1] infinitely many times, after which the winner is decided in terms of the sequence belonging to a Borel set in the product space [0,1]ℕ. The main theorem is that Borel games of length ω2 are determined if, and only if, for every countable ordinal α, there is a fine-structural, countably iterable model of Zermelo set theory with α-many iterated powersets above a limit of Woodin cardinals.

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with (measurable) discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibers. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibers have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.

Let $\Gamma$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $\Gamma$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of $G$ is compact in the natural topology. It is shown that there exists a {\it bounded} ergodic $G$-valued cocycle of $\Gamma$.

Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds., Comment: 11 pages

The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X is locally finite, essential, irreducible, non-Euclidean and admits a cocompact action by a group G, Nevo-Sageev show that a subset, B(X), of the Roller boundary is the realization of the Poisson boundary and that the action of G on B(X) is minimal and strongly proximal. Additionally, these authors show B(X) satisfies many other desirable dynamical and topological properties. In this article we give several equivalent characterizations for when B(X) is equal to the entire Roller boundary. As an application we show, under mild hypotheses, that if X is also 2-dimensional then X is G-equivariantly quasi-isometric to a CAT(0) cube complex X' whose Roller boundary is equal to B(X'). Additionally, we use our characterization to show that the usual CAT(0) cube complex for which an infinite right-angled Coxeter/Artin group acts on geometrically has Roller boundary equal to B(X), as long as the corresponding group does not decompose as a direct product., Incorporates referee's comments. This is the final pre-publication version; to appear in Israel Journal of Mathematics

Recently Luo and Zhou showed that for a fixed GL(2) Hecke-Maass cusp form, the natural density of primes at which the Satake parameters fail the Ramanujan Conjecture does not exceed 1/35. In this short note, we investigate the GL(3) case and obtain two similar (conditional) results.

A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics

We compare two different cohomology groups on unitary Shimura curves over totally real number fields; these groups are used in the construction of p-adic families of modular forms in this context. More precisely, we describe the space of finite slope overconvergent modular forms of any weight in terms of a convenient overconvergent cohomology.

We study the lifting of the Schubert stratification of the homogeneous space of complete real flags of $R^{n+1}$ to its universal covering group $Spin_{n+1}$. We call the lifted strata the Bruhat cells of $Spin_{n+1}$, in keeping with the homonymous classical decomposition of reductive algebraic groups. We present explicit parameterizations for these Bruhat cells in terms of minimal-length expressions $\sigma=a_{i_1}... a_{i_k}$ for permutations $\sigma\in S_{n+1}$ in terms of the $n$ generators $a_i=(i,i+1)$. These parameterizations are compatible with the Bruhat orders in the Coxeter-Weyl group $S_{n+1}$. This stratification is an important tool in the study of locally convex curves; we present a few such applications., Comment: 34 pages, 1 figure, minor revisions. There is a strong overlap in content with arXiv:1810.08632

An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $\eta \geq 0$, $\Pr(s_n(M_n) \leq \eta) \lesssim n^{C}\eta + \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a `truly combinatorial' setting., Comment: 28 pages; comments welcome!

Telgarsky’s conjecture states that for each k ∈ ℕ, there is a topological space Xk such that in the Banach-Mazur game on Xk, the player nonempty has a winning (k + 1)-tactic but no winning k-tactic. We prove that this statement is consistently false. More specifically, we prove, assuming GCH+□, that if nonempty has a winning strategy for the Banach-Mazur game on a T3 space X, then she has a winning 2-tactic. The proof uses a coding argument due to Galvin, whereby if X has a π-base with certain nice properties, then nonempty is able to encode, in each consecutive pair of her opponent’s moves, all essential information about the play of the game before the current move. Our proof shows that under GCH + □, every T3 space has a sufficiently nice π-base that enables this coding strategy. Translated into the language of partially ordered sets, what we really show is that GCH + □ implies the following statement, which is equivalent to the existence of the “nice” π-bases mentioned above: ∇: Every separative poset ℙ with the κ-cc contains a dense sub-poset $$\mathbb{D}$$ such that $$\left| {\left\{ {q \in \mathbb{D} :p\;{\rm{extends}}\;q} \right\}} \right| < \kappa $$ for every p ∈ ℙ. We prove that this statement is independent of ZFC: while it holds under GCH + □, it is false even for ccc posets if $$\mathfrak{b} > {\aleph _1}$$ . We also show that if $$\left|\mathbb{P} \right| < {\aleph _\omega }$$ , then ∇-for-ℙ is a consequence of GCH holding below ∣ℙ∣.

A well-known conjecture of Frankl and F\"{u}redi from 1989 states that an initial segment of colex of has the largest Lagrangian of any $r$-uniform hypergraph with $m$ hyperedges. We show that this is true when $r=3$. We also give a new proof of a related conjecture of Nikiforov and a counterexample to an old conjecture of Ahlswede and Katona., Comment: We split our original paper (arXiv:1807.00793v2) into two parts. The first part can be found in arXiv:1807.00793. This is the second part, which consists of 18 pages, including a two-page appendix

This paper concerns the porosity of the class $${\cal D}$$ of diametrically complete sets, as a subset of the metric space of closed, convex bounded sets $${\cal H}$$ endowed with the Hausdorff metric in Banach spaces. We begin by answering in the negative a question posed in [7] regarding the existence of non-trivial diametrically minimal sets in non-reflexive Banach spaces. This result is then used to prove that, if dim X > 1 and Jung’s constant J(X) < 2, $${\cal D}$$ is uniformly very porous. Our lower bound for the extreme porosity of $${\cal D}$$ at each element of $${\cal H}$$ depends uniquely on J(X). When dim X > 1 and the smaller class $${\cal W}$$ of constant width sets is considered, we prove the existence of a universal lower bound for its extreme porosity.

We establish a dimension-free improvement of Talagrand’s Gaussian transport-entropy inequality, under the assumption that the measures satisfy a Poincare inequality. We also study stability of the inequality, in terms of relative entropy, when restricted to measures whose covariance matrix trace is smaller than the ambient dimension. In case the covariance matrix is strictly smaller than the identity, we give dimension-free estimates which depend on its eigenvalues. To complement our results, we show that our conditions cannot be relaxed, and that there exist measures with covariance larger than the identity, for which the inequality is not stable, in relative entropy. To deal with these examples, we show that, without any assumptions, one can always get quantitative stability estimates in terms of relative entropy to Gaussian mixtures. Our approach gives rise to a new point of view which sheds light on the hierarchy between Fisher information, entropy and transportation distance, and may be of independent interest. In particular, it implies that the described results apply verbatim to the log-Sobolev inequality and improve upon some known estimates in the literature.

Assuming Schanuel's Conjecture we prove that for any variety V over the algebraic closure over the rational numbers, of dimension n and with dominant projections, there exists a generic point in V. We obtain in this way many instances of the Strong Exponential Closure introduced by Zilber., Comment: 10 pages

Let p be a prime number and f a positive integer with f < p. In this paper, we determine the structure of simple Breuil modules of type $$ \oplus _{i = 1}^n\omega _f^{{k_i}}$$ corresponding to n-dimensional irreducible representations of $${G_{{{\bf{Q}}_p}}}$$ . We also describe the extensions of those simple Breuil modules if they correspond to mod p reductions of strongly divisible modules that correspond to Galois stable lattices in potentially semistable representations of $${G_{{{\bf{Q}}_p}}}$$ with Hodge-Tate weights {0, 1, …, n − 1} and Galois type $$ \oplus _{i = 1}^n\widetilde\omega _f^{{k_i}}$$ .

For last passage percolation (LPP) on ℤ2 with exponential passage times, let Tn denote the passage time from (1, 1) to (n,n). We investigate the law of iterated logarithm of the sequence {Tn}n≥1; we show that $$\lim \,{\inf _{n \to \infty }}{{{T_n} - 4n} \over {{n^{1/3}}{{\left( {\log \,\log \,n} \right)}^{1/3}}}}$$ almost surely converges to a deterministic negative constant and obtain some estimates on the same. This settles a conjecture of Ledoux (2018) where a related lower bound and similar results for the corresponding upper tail were proved. Our proof relies on a slight shift in perspective from point-to-point passage times to considering point-to-line passage times instead, and exploiting the correspondence of the latter to the largest eigenvalue of the Laguerre Orthogonal Ensemble (LOE). A key technical ingredient, which is of independent interest, is a new lower bound of lower tail deviation probability of the largest eigenvalue of β-Laguerre ensembles, which extends the results proved in the context of the β-Hermite ensembles by Ledoux and Rider (2010).

Given a log smooth morphism f: X → S of toroidal embeddings, we perform a Raynaud-Gruson type operation on f to make it flat and with reduced fibers. We do this by studying the geometry of the associated map of cone complexes C(X) → C(S). As a consequence, we show that the toroidal part of semistable reduction of Abramovich-Karu can be done in a canonical way.

We rectify Theorem 3.4 of [1] which gives a relation between the isomorphism types of a group algebra FG of a finite abelian group G over a field F of characteristic 0, the second cohomology group H2(G, F*) and the isomorphism type of G. We show how this relation depends on the behavior of F under extensions by roots of unity.

We prove that $$\mathfrak{mcf}$$ and $$\mathfrak{mcf}$$ are both consistent relative to ZFC. This answers a question by Banakh, Repovs and Zdomskyy and a question from [MS11].

Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph ${\mathcal M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of $\alpha\chi_1$. We study the diameters of McKay graphs for simple groups $G$. For $G$ a group of Lie type, we show that for any $\alpha$, the diameter is bounded by a quadratic function of the rank, and obtain much stronger bounds for $G={\rm PSL}_n(q)$ or ${\rm PSU}_n(q)$. We also bound the diameter for symmetric and alternating groups.

This paper is devoted to the stochastic properties of dynamical systems preserving an infinite measure. More precisely we prove central limit theorems for Birkhoff sums of observables of ℤ2-extensions of dynamical systems (satisfying some nice spectral properties). The motivation of our paper is the ℤ2-periodic Lorentz process for which we establish a functional central limit theorem for Holder continuous observables (in discrete time as well as in continuous time).

Let (G, Z(G)) be a Camina pair. We prove that ∣G: Z(G)∣ > ∣Z(G)∣2 when G has nilpotence class at least 4. When G has nilpotence class 3, we show that ∣G: Z(G)∣ > ∣Z(G)∣3/2. Under the additional assumption that Z2(G) ∣Z(G)∣3 is obtained.

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. Let us say that a strictly geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed plane curves with this property are ellipses. In the present article we show that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net. We also recall Blaschke’s derivation of the Liouville property from the Ivory property and his proof of Weihnacht’s theorem: the only Liouville nets in the plane are nets of confocal conics and their degenerations. This suggests the following generalization of Birkhoff’s conjecture: If an interior neighborhood of a closed strictly geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.

We develop the theory of smooth principal bundles for a smooth group G, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define D-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling back a D-numerable bundle along smoothly homotopic maps gives isomorphic pullbacks. We then define smooth structures on Milnor’s spaces EG and BG, show that EG → BG is a D-numerable principal bundle, and prove that it classifies all D-numerable principal bundles over any diffeological space. We deduce analogous classification results for D-numerable diffeological bundles and vector bundles.

In the present paper it is proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed *-graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary ℤ2-grading and graded involution.

We prove the existence of a positive solution for the problem $${\rm{\gamma}}{{\rm{\Delta}}^2}u - m\left(u \right){\rm{\Delta}}u = \mu a\left(x \right){u^q} + b\left(x \right){u^p},\,\,{\rm{in}}\,{\rm{\Omega ,}}\,\,\,\,\,u = {\rm{\gamma \Delta}}u = 0,\,\,{\rm{on}}\,\,\partial {\rm{\Omega ,}}$$ where Ω ⊂ ℝN is a bounded smooth domain, γ ∈ {0, 1},0 1 < p, m is weakly continuous in $${H^2}\left({\rm{\Omega}} \right) \cap H_0^1\left({\rm{\Omega}} \right),a \in {L^\infty}\left({\rm{\Omega}} \right)$$ is nonnegative and b is a bounded potential which can change sign. The solution is obtained via a sub-supersolution approach when the parameter µ > 0 is small.

In recent work the authors introduced the notion of near-isomorphism for torsion-free abelian groups of infinite rank. In a natural way this concept extends the classical definition of near-isomorphism in the finite rank case. We prove some further properties of this notion and introduce the new concept of bounded near-isomorphism which is a stronger form of near-isomorphism. As an application we study bounded near-isomorphic bounded completely decomposable abelian groups (bcd-groups). It is shown that bcd-groups allow a theory that is very similar to the theory of almost completely decomposable groups with respect to bounded near-isomorphism.

For a polynomial F(t, A1, …, An) ∈ $$\mathbb{F}$$ p[t, A1, …, An] (p being a prime number) we study the factorization statistics of its specializations $$F\left({t,{a_1}, \ldots ,{a_n}} \right) \in {\mathbb{F}_p}\left[t \right]$$ with (a1, …, an) ∈ S, where $$S \subset \mathbb{F}_p^n$$ is a subset, in the limit p → ∞ and deg F fixed. We show that for a sufficiently large and regular subset $$S \subset \mathbb{F}_p^n$$ , e.g., a product of n intervals of length H1, …, Hn with $$\prod\nolimits_{i = 1}^n {{H_n} > {p^{n - 1/2 +\epsilon}}} $$ , the factorization statistics is the same as for unrestricted specializations (i.e., $$S = \mathbb{F}_p^n$$ ) up to a small error. This is a generalization of the well-known Polya-Vinogradov estimate of the number of quadratic residues modulo p in an interval.

This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a cocompact special group $G$ and a finite collection of subgroups $\mathcal{H}$, then $G$ is hyperbolic relative to $\mathcal{H}$ if and only if (i) each subgroup of $\mathcal{H}$ is convex-cocompact, (ii) $\mathcal{H}$ is an almost malnormal collection, and (iii) every non-virtually cyclic abelian subgroup of $G$ is contained in a conjugate of some group of $\mathcal{H}$. As an application, we show that a virtually cocompact special group is hyperbolic relative to abelian subgroups if and only if it does not contain $\mathbb{F}_2 \times \mathbb{Z}$., Comment: 28 pages, 3 figures. The section dedicated to graph braid groups in the first version is now contained in arXiv:1912.10674. To appear in Israel J. Math

We introduce Turaev bicategories and group braided Turaev bicategories. On one hand, the latter generalize the notions of braided Turaev categories, introduced at the turn of the millennium and originally called “braided crossed group categories” by Turaev himself, and on the other hand, the former generalize the notion of bicategories. For bimonads in 2-categories, which we defined in one of our previous papers, we introduce generalized Yetter-Drinfel’d modules in 2-categories. These generalize to the 2-categorical setting the generalized Yetter-Drinfel’d modules (over a field) of Panaite and Staic, and thus also in particular the anti Yetter-Drinfel’d modules, introduced by Hajac-Khalkhali-Rangipour-Sommerhauser as coefficients for the cyclic cohomology of Hopf algebras, defined by Connes and Moscovici. We construct a Turaev 2-category for bimonads in 2-categories as a Turaev extension of the 2-category of bimonads, generalizing the corresponding Turaev category of Panaite and Staic. We also prove in the 2-categorical setting their results on pairs in involution, which in turn go back to modular pairs in involution of Connes and Moscovici. Moreover, two examples of proper Turaev bicategories are constructed, one of which is group braided.

In this paper, we are concerned with the following nonlinear magnetic Schrodinger equation with critical growth: $$\left\{ {\matrix{ {{{\left( {{\varepsilon \over i}\nabla - A\left( x \right)} \right)}^2}u + V\left( x \right)u = f\left( {{{\left| u \right|}^2}} \right)u + {{\left| u \right|}^{{2^*} - 2}}u\,\,\,\,\,\,{\rm{in}}\,{\mathbb{R}^N},} \hfill \cr {u \in {H^1}\left( {{\mathbb{R}^N},\mathbb{C}} \right),} \hfill \cr } } \right.$$ where ∊ > 0 is a parameter, N ≥ 3 and $${2^*} = {{2N} \over {N - 2}}$$ is the Sobolev critical exponent, V: ℝN → ℝ and A: ℝN → ℝN are continuous potentials, f: ℝ → ℝ is a subcritical nonlinear term. Under a local assumption on the potential V, by the variational methods, the penalization technique and the Ljusternic—Schnirelmann theory, we prove the multiplicity and concentration of nontrivial solutions of the above problem for e small. For the problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

Let $$\mathbb{K}$$ be a field of characteristic zero. We study the graded identities of the special linear Lie algebra with the Pauli and Cartan gradings. Given a prime number p we provide a finite basis for the graded identities of $$s{l_p}\left(\mathbb{K}\right)$$ with the Pauli grading by the group ℤp × ℤp and compute its graded codimensions. We also prove that $${{\mathop{\rm var}} ^{{\mathbb{Z}_p} \times {\mathbb{Z}_p}}}\left( {s{l_p}\left(\mathbb{K} \right)} \right)$$ is a minimal variety and satisfies the Specht property. As a by-product we determine a basis for the identities of certain graded Lie algebras with a grading in which every homogeneous subspace has dimension ≤ 1. For $$s{l_m}\left(\mathbb{K}\right)$$ with the Cartan grading a finite basis for the graded identities is determined, moreover a basis for the subspace of the multilinear polynomials in the relatively free algebra $$L\left\langle {{X_G}} \right\rangle /{T_G}\left( {s{l_m}\left( \right)} \right)$$ , as a vector space, is exhibited. As a consequence we compute the graded codimensions for m = 2 and provide bases for the graded identities and for the subspace of the multilinear polynomials in the relatively free algebra of certain Lie subalgebras of $${M_m}{\left(\mathbb{K}\right)^{\left( - \right)}}$$ with the Cartan grading.

We prove that the symmetrized deleted join SymmDelJoin( $$\mathcal{K}$$ ) of a “balanced family” $$\mathcal{K}$$ = 〈Ki〉 =1 of collectively r-unavoidable subcomplexes of 2[m] is (m−r−1)-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result which is more conceptual and more general than previously known results. Already the case r = 2 of this result seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman’s discrete Morse theory.

Let E be a Hermitian vector bundle over a complete Kahler manifold (X, ω), dimℂX = n, with a d(bounded) Kahler form ω, and let dA be a Hermitian connection on E. The goal of this article is to study the L2-Hodge theory on the vector bundle E. We extend the results of Gromov [18] to the Hermitian vector bundle. Finally, as an application, we prove a gap result for the Yang-Mills connection on the bundle E over X.

In this paper we study a connective commutative differential graded algebra (CDGA) which is piecewise Noetherian. The principal aim is to analyze a dualizing complex of CDGA’s and investigate a Gorenstein CDGA. We also establish many other results which can be expected to play basic roles in the study of a CDGA, such as the Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolutions for ordinary modules. In the paper, we show that sppj and ifij resolutions are powerful tools to study DG-modules.

The classical Szego polynomial approximation theorem states that the polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $\rho$ diverges. In this note we give a quantitative version of Szego's theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $\rho$ on a sufficiently rare subset of the circle., Comment: To appear in Israel Journal of Mathematics

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra K(sl2,k) associated to the integrable highest weight modules for the affine Kac—Moody algebra $$A_1^{(1)}$$ is the building block of the general parafermion vertex operator K( $$K(\mathfrak{g},k)$$ ,k) for any finite-dimensional simple Lie algebra $$\mathfrak{g}$$ and any positive integer k. We first classify the irreducible modules of ℤ2-orbifold of the simple affine vertex operator algebra of type $$A_1^{(1)}$$ and determine their fusion rules. Then we study the representations of the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k). The quantum dimensions, and more technically, fusion rules for the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k) are completely determined.

In this paper, we extend the construction of pressure metrics to Teichmuller spaces of surfaces with punctures. This construction recovers Thurston’s Riemannian metric on Teichmuller spaces. Moreover, we prove the real analyticity and convexity of Manhattan curves of finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.

In [4] Furstenberg, Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of ℝ2 of positive upper (Banach) density. A second proof of this result, as well as a stronger “pinned variant”, was given by Bourgain in [2] using Fourier analytic methods. In [8] the second author adapted Bourgain’s Fourier analytic approach to establish a result analogous to that of Furstenberg, Katznelson and Weiss for subsets ℤd provided d ≥ 5. We present a new direct proof of this discrete distance set result and generalize this to arbitrary trees. Using appropriate discrete spherical maximal function theorems we ultimately establish the natural “pinned variants” of these results.

Suppose that p is an odd prime and G is a finite group having no normal non-trivial p'-subgroup. We show that if a is an automorphism of G of p-power order centralizing a Sylow p-group of G, then a is inner. This answers a conjecture of Gross. An easy corollary is that if p is an odd prime and P is a Sylow p-subgroup of G, then the center of P is contained in the generalized Fitting subgroup of G. We give two proofs both requiring the classification of finite simple groups. For p=2, the result fails but Glauberman in 1968 proved that the square of a is inner. This answered a problem of Kourovka posed in 1999., Comment: There was a change of authors from the previous version and a considerable difference in the article

F. Albiac acknowledges the support of the Spanish Ministry for Economy and Competitivity Grants MTM2014-53009-P for Analisis Vectorial, Multilineal y Aplicaciones, and MTM2016-76808-P for Operators, lattices, and structure of Banach spaces as well as the Spanish Ministry for Science and Innovation under Grant PID2019-1077701GB-I00. J. L. Ansorena acknowledges the support of the Spanish Ministry for Economy and Competitivity Grant MTM2014-53009-P for Analisis Vectorial, Multilineal y Aplicaciones. M. Cuth has been supported by Charles University Research program No. UNCE/SCI/023 and by the Research grant GACR 17-04197Y. M. Doucha was supported by the GACR project 16-34860L and RVO: 67985840.

We show that every model filiform group $\mathbb{E}_{n}$ contains a measure zero set $N$ such that every Lipschitz map $f\colon \mathbb{E}_{n}\to \mathbb{R}$ is differentiable at some point of $N$. Model filiform groups are a class of Carnot groups which can have arbitrarily high step. Essential to our work is the question of whether existence of an (almost) maximal directional derivative $Ef(x)$ in a Carnot group implies differentiability of a Lipschitz map $f$ at $x$. We show that such an implication is valid in model Filiform groups except for a one-dimensional subspace of horizontal directions. Conversely, we show that this implication fails for every horizontal direction in the free Carnot group of step three and rank two., Comment: 42 pages. arXiv admin note: text overlap with arXiv:1505.07986

In this note we show that if $p$ is an odd prime and $G$ is a powerful $p$-group with $N\leq G^{p}$ and $N$ normal in $G$, then $N$ is powerfully nilpotent. An analogous result is proved for $p=2$ when $N\leq G^{4}$.

We establish the Hao–Ng isomorphism for generalized gauge actions of locally compact abelian groups on product systems over abelian lattice orders and we then use it to explore Takai duality in this context. As an application we generalize some recent work of Schafhauser.

We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box. The error term is expressed in terms of a certain function ν(Γ⊥, ·) of the dual lattice Γ⊥, and we carefully analyse the relation of this quantity with ν(Γ, ·). In particular, we show that ν(Γ⊥, ·) = ν(Γ, ·) for any unimodular lattice of rank 2, but that for higher ranks it is in general not possible to bound one function in terms of the other. This result relies on Beresnevich’s recent breakthrough on Davenport’s problem regarding badly approximable points on submanifolds of ℝn. Finally, we apply our counting theorem to establish asymptotics for the number of Diophantine approximations with bounded denominator as the denominator bound gets large.

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

The graph of a Hecke operator encodes all information about the action of this operator on automorphic forms over a global function field. These graphs were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators". After reviewing some general properties, we explain the connection to the Hall algebra of the function field. In the case of an elliptic function field, we can use structure results of Burban-Schiffmann and Fratila to develop an algorithm which explicitly calculate these graphs. We apply this algorithm to determine some structure constants and provide explicitly the rank two case in the last section., Comment: 38 pages, comments are welcome - minor typos were corrected in the second version

The Lubin-Tate moduli space X0rig is a p-adic analytic open unit polydisc which parametrizes deformations of a formal group H0 of finite height defined over an algebraically closed field of characteristic p. It is known that the natural action of the automorphism group Aut(H0) on X0rig gives rise to locally analytic representations on the topological duals of the spaces H0(X0rig , (ℳ0 )rig) of global sections of certain equivariant vector bundles (ℳ0 )rig over X0rig . In this article, we show that this result holds in greater generality. On the one hand, we work in the setting of deformations of formal modules over the valuation ring of a finite extension of ℚp. On the other hand, we also treat the case of representations arising from the vector bundles (ℳ )rig over the deformation spaces Xrig with Drinfeld level-m-structures. Finally, we determine the space of locally finite vectors in H0(Xrig , (ℳ )rig). Essentially, all locally finite vectors arise from the global sections of invertible sheaves over the projective space via pullback along the Gross-Hopkins period map.