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In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden classes of graphs, all of cardinality $\aleph_1$. Unfortunately, their proof contains a gap, and their result is incorrect. In this paper, we construct a third type of obstruction: an $\aleph_1$-sized graph without a normal spanning tree that contains neither of the two types described by Diestel and Leader as a minor. Further, we show that any list of forbidden minors characterising the graphs with normal spanning trees must contain graphs of arbitrarily large cardinality., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.02833

We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module., 15 pages

A Borel set B⊂Rn is visible from x∈Rn, if the radial projection of B with base point x has positive Hn−1 measure. I prove that if dimB>n−1, then B is visible from every point x∈Rn∖E, where E is an exceptional set with dimension dimE⩽2(n−1)−dimB. This is the sharp bound for all n⩾2. Many parts of the proof were already contained in a recent previous paper by P. Mattila and the author, where a weaker bound for dimE was derived as a corollary from a certain slicing theorem. Here, no improvement to the slicing result is obtained; in brief, the main observation of the present paper is that the proof method gives the optimal result, when applied directly to the visibility problem.

We determine relations between the Euler obstruction of a map f and the Chern obstruction of a collection of 1-forms associated to f . We obtain explicit applications in singularity theory. On one hand, in [10], Grulha defines a generalization of the Euler obstruction of a holomorphic function to the case of a map-germ f : (V, 0) → (C, 0) where (V, 0) is the germ of a reduced equidimensional complex analytic variety of dimension d ≥ p. The main result in that paper is a formula that computes the Euler obstruction of f in terms of the Euler obstruction for p-frames as defined by Brasselet, Seade and Suwa in [3]. On the other hand, Ebeling and Gusein-Zade introduced in [4] the notion of Chern obstruction for singular spaces using collections of differential 1-forms. This number is well defined for any germ of reduced equidimensional complex analytic space, and the authors provide a formula for the Chern obstruction in terms of intersection number. One of the main results of the present paper is that the Euler obstruction of a map can be written as a Chern obstruction, therefore as an intersection multiplicity.

For a flnite group G let i(G) denote the graph deflned on the non- identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph i(G) contains a Hamiltonian cycle for many flnite groups G. In the literature many deep results about flnite simple groups G can equivalently be stated as theorems about i(G). Three examples are given. Guralnick and Shalev (10) showed that for su-ciently large G the graph i(G) has diameter at most 2. Guralnick and Kantor (9) showed that there is no isolated vertex in i(G). Finally, Breuer, Guralnick, Kantor (4) showed that the diameter of i(G) is at most 2 for all G. In this paper those flnite groups G are considered for which i(G) contains a Hamiltonian cycle. The following proposition reduces the investigations to those non-solvable groups G for which G=N is cyclic for any non-trivial normal subgroup N of G. Proposition 1.1. Let G be a flnite solvable group that has at least 4 elements. Then the graph i(G) contains a Hamiltonian cycle if and only if G=N is cyclic for all non-trivial normal subgroups N of G. The three main results of this paper are Theorems 1.2, 1.3, and 1.4.

Twenty-five years ago, Conway and Norton published their remarkable paper `Monstrous Moonshine', proposing a completely unexpected relationship between finite simple groups and modular functions. This paper reviews the progress made in broadening and understanding that relationship., 33 pages. 10 references added; minor changes to text

In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals κ, there is an unexpected connection between invariants ae(κ), b(κ) and a certain cardinal invariant md(κ+) on κ+. As a corollary we get for example the following result. For a successor cardinal κ, even assuming that κ

In a celebrated paper, Borwein, Erd\'elyi, Ferguson and Lockhart constructed cosine polynomials of the form \[ f_A(x) = \sum_{a \in A} \cos(ax), \] with $A\subseteq \mathbb{N}$, $|A|= n$ and as few as $n^{5/6+o(1)}$ zeros in $[0,2\pi]$, thereby disproving an old conjecture of J.E. Littlewood. Here we give a sharp analysis of their constructions and, as a result, prove that there exist examples with as few as $C(n\log n)^{2/3}$ roots., Comment: 17 pages. A few typos fixed

Let $\Gamma$ be the hyperbola $\{(x,y)\in\mathbb R^2 : xy=1\}$ and $\Lambda_\beta$ be the lattice-cross defined by $\Lambda_\beta=\left(\mathbb Z\times\{0\}\right)\cup\left(\{0\}\times\beta\mathbb Z\right)$ in $\mathbb R^2,$ where $\beta$ is a positive real. A result of Hedenmalm and Montes-Rodriguez says that $\left(\Gamma,\Lambda_\beta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq1.$ In this paper, we show that for a rational perturbation of $\Lambda_\beta,$ namely \[\Lambda_\beta^\theta=\left((\mathbb Z+\{\theta\})\times\{0\}\right)\cup\left(\{0\}\times\beta\mathbb Z\right),\] where $\theta=1/{p},~\text{for some}~{p}\in\mathbb N$ and $\beta$ is a positive real, the pair $\left(\Gamma,\Lambda_\beta^\theta\right)$ is a Heisenberg uniqueness pair if and only if $\beta\leq{p}.$

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carath\'eodory pseudodistance on CR manifolds., Comment: 16 pages

In this paper, we investigate the weak-type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc we provide a proof that the weak-type behavior is of "$L\log L$" type. This result is likely known to the experts, but does not appear to be in the literature. For the Hartogs triangle we show that the operator is of weak-type $(4,4)$; settling the question of the behavior of the projection at this endpoint. At the other endpoint of interest, we show that the Bergman projection is not of weak-type $(\frac{4}{3}, \frac{4}{3})$ and provide evidence as to what the correct behavior at this endpoint might be., Comment: 15 pages

Peter Aczel has given a categorical construction for fixed points of normal functors, i.e. dilators which preserve initial segments. For a general dilator $X\mapsto T_X$ we cannot expect to obtain a well-founded fixed point, as the order type of $T_X$ may always exceed the order type of $X$. In the present paper we show how to construct a Bachmann-Howard fixed point of $T$, i.e. an order $\operatorname{BH}(T)$ with an "almost" order preserving collapse $\vartheta:T_{\operatorname{BH}(T)}\rightarrow\operatorname{BH}(T)$. Building on previous work, we show that $\Pi^1_1$-comprehension is equivalent to the assertion that $\operatorname{BH}(T)$ is well-founded for any dilator $T$., Comment: This version has been accepted for publication in the Bulletin of the London Mathematical Society

The well-known Siegel Lemma gives an upper bound $cU^{m/(n−m)}$ for the size of the smallest non-zero integral solution of a linear system of $m \ge 1$ equations in $n > m$ unknowns whose coefficients are integers of absolute value at most $U \ge 1$; here $c = c(m, n) \ge 1$. In this paper we show that a better upper bound $U^{m/(n−m)}/B$ is relatively rare for large $B \ge 1$; for example there are $\theta = \theta(m,n) > 0$ and $c′ = c′(m,n)$ such that this happens for at most $c′U^{mn}/B^\theta$ out of the roughly $(2U)^{mn}$ possible such systems.

In a recent paper by Das, Laishram and Saradha, it was shown that if there exists a rational solution of yl=(x+1)…(x+i−1)(x+i+1)…(x+k) for i not too close to k/2 and y≠0, then logl

Let $K$ be a number field, and let $d\geq 2$. A conjecture of Odoni (stated more generally for characteristic zero Hilbertian fields $K$) posits that there is a monic polynomial $f\in K[x]$ of degree $d$, and a point $x_0\in K$, such that for every $n\geq 0$, the so-called arboreal Galois group $Gal(K(f^{-n}(x_0))/K)$ is an $n$-fold wreath product of the symmetric group $S_d$. In this paper, we prove Odoni's conjecture when $d$ is even and $K$ is an arbitrary number field, and also when both $d$ and $[K:Q]$ are odd., 15 pages, 1 figure

In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is infinite, and shares many properties with the bungee set of a transcendental entire function. By way of contrast, we give examples of novel properties of this set in the quasiregular setting. In particular, we give an example of a quasiconformal map of the plane with a non-empty bungee set; this behaviour is impossible for an analytic homeomorphism.

This is the third, and last, of a series of papers dealing with oriented regular representations. Here we complete the classification of finite groups that admit an oriented regular representation (or ORR for short), and give a complete answer to a 1980 question of Laszlo Babai: "Which [finite] groups admit an oriented graph as a DRR?" It is easy to see and well-understood that generalised dihedral groups do not admit ORRs. We prove that, with 11 small exceptions (having orders ranging from 8 to 64), every finite group that is not generalised dihedral has an ORR.

In 1987, Dan Gordon defined an elliptic curve analogue to Carmichael numbers known as elliptic Carmichael numbers. In this paper, we prove that there are infinitely many elliptic Carmichael numbers. In doing so, we resolve in the affirmative the question of whether there exist infinitely square-free, composite integers $n$ such that for every prime $p$ that divides $n$, $p+1|n+1$.

In this paper, we prove BG-type inequality conjecture for threefolds in the title. In particular, there exist Bridgeland stability conditions on these threefolds., 15 pages, minor changes

In a paper in 1982, Said Sidki defined a 2-parameter family of finitely presented groups Y(m,n) that generalise the Carmichael presentation for a finite alternating group satisfied by its generating 3-cycles (1,2,t) for t⩾3. For m⩾2 and n⩾2, the group Y(m,n) is the abstract group generated by elements a1,a2,⋯,am subject to the defining relations ain=1 for 1⩽i⩽m and (aikajk)2=1 for 1⩽i

Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum $R$ whose graded homotopy ring $\pi_*R$ is concentrated in dimensions divisible by a natural number $N \geq 5$ and has homological dimension at most three, the homotopy category of $R$-modules is equivalent to the derived category of $\pi_*R$. The Johnson-Wilson spectrum $E(3)$ and the truncated Brown-Peterson spectrum $BP\langle 2 \rangle$ for any prime $p \geq 5$ are our main examples. If additionally the homological dimension of $\pi_*R$ is equal to two, then the homotopy category of $R$-modules and the derived category of $\pi_*R$ are triangulated equivalent. Here the main examples are $E(2)$ and $BP \langle 1 \rangle$ at $p \geq 5$. The last part of the paper discusses a triangulated equivalence between the homotopy category of $E(1)$-local spectra at a prime $p \geq 5$ and the derived category of Franke's model. This is a theorem of Franke and we fill a gap in the proof.

Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\geq2$ an integer. An irreducible representation of the finite group $G_{r}=\mathrm{GL}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+\mathfrak{p}^{r-1}\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p}^{r})$ consists of representations whose stabilisers modulo $K^{1}$ are centralisers of regular elements in $\mathrm{M}_{N}(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_{r}$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_{r}$.

Let G be a finite group and let g∈G. In 1896, Frobenius showed that the number of ways to express g as a commutator of elements of G is |G|FG(g), where FG(g)=∑χ∈ Irr (G)χ(g)/χ(1) is the Frobenius character sum. This sum received particular attention in the case where G is a (non-abelian) finite simple group, and some related conjectures were posed. In this paper we discuss these conjectures, refute one of them, and provide partial evidence in favor of another one.

In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let Δ⩾5, e>0, and let H be a graph on (1−e)n vertices and with maximum degree Δ. We show that a random graph Gn,p with high probability contains a copy of H, provided that p≫(n−1log1/Δn)2/(Δ+1). Our assumption on p is optimal up to the polylog factor. We note that this polylog term matches the conjectured threshold for the spanning case.

Two graphs G and H are hypomorphic if there exists a bijection φ:V(G)→V(H) such that G−v≅H−φ(v) for each v∈V(G). A graph G is reconstructible if H≅G for all H hypomorphic to G. It is well known that not all infinite graphs are reconstructible. However, the Harary–Schwenk–Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we construct a counterexample to the Harary–Schwenk–Scott Conjecture. Our example also answers four other questions of Nash-Williams, Halin and Andreae on the reconstruction of infinite graphs.

A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper, we show that the isomorphism class of the holomorphic Poisson structure occurring in this local model is independent of the choice of gauge equivalence, and is hence the unique local invariant of generalized complex manifolds. This completes the local classification of generalized complex structures. We use this result to prove that the complex locus of a generalized complex manifold naturally inherits the structure of a complex analytic space.

If a hyperbolic 3-manifold admits an exceptional Dehn filling, then the length of the slope of that Dehn filling is known to be at most six. However, the bound of six appears to be sharp only in the toroidal case. In this paper, we investigate slope lengths of other exceptional fillings. We construct hyperbolic 3-manifolds that have the longest known slopes for reducible fillings. As an intermediate step, we show that the problem of finding the longest such slope is equivalent to a problem on the maximal density horoball packings of planar surfaces, which should be of independent interest. We also discuss lengths of slopes of other exceptional Dehn fillings, and prove that six is not realized by a slope corresponding to a small Seifert fibered space filling., 18 page, 5 figures, 1 table. V3: Changes to section 2: Theorem 2.1 and its proof have been modified to correct an issue with the previous version. Additional minor changes to exposition. V4: Changes to section 2 to avoid coverings. Proof of Theorem 2.6 is now more direct. Additional wording changes

In this paper we consider 4-superlinear Schrodinger–Poisson systems. In contrast to most studies, we consider the case where the potential V is indefinite so that the Schrodinger operator −Δ+V possesses a finite-dimensional negative space. We obtain nontrivial solutions for the problem via Morse theory.

It is well known that every closed Riemann surface S of genus g⩾2, admitting a group G of conformal automorphisms so that S/G has triangular signature, can be defined over a finite extension of Q. It is interesting to know, in terms of the algebraic structure of G, if S can in fact be defined over Q. This is the situation if G is either abelian or isomorphic to A⋊Z2, where A is an abelian group. On the other hand, as shown by Streit and Wolfart, if G≅Zp⋊Zq, where p,q>3 are prime integers, then S is not necessarily definable over Q. In this paper, we observe that if G≅Z22⋊Zm with m⩾3, then S can be defined over Q. Moreover, we describe explicit models for S, the corresponding groups of automorphisms, and an isogenous decomposition of their Jacobian varieties as product of Jacobians of hyperelliptic Riemann surfaces.

Using the $E_\infty-$structure on singular cochains, we construct a homotopy coherent map from the cyclic bar construction of the differential graded algebra of cochains on a space to a model for the cochains on its free loop space. This fills a gap in the paper "Cyclic homology and equivariant homology" by John D.S. Jones.

The paper deal with the noncommutative Frechet ${}^*$-algebra $\mathcal{L}(s',s)$ of the so-called smooth operators, i.e. linear and continuous operators acting from the space $s'$ of slowly increasing sequences to the Frechet space $s$ of rapidly decreasing sequences. By a canonical identification, this algebra of smooth operators can be also seen as the algebra of the rapidly decreasing matrices. We give a full description of closed commutative ${}^*$-subalgebras of this algebra and we show that every closed subspace of $s$ with basis is isomorphic (as a Frechet space) to some closed commutative ${}^*$-subalgebra of $\mathcal{L}(s',s)$. As a consequence, we give some equivalent formulation of the long-standing Quasi-equivalence Conjecture for closed subspaces of $s$.

It is known that, in many cases, the field (known as the Hecke field) generated by all Hecke eigenvalues of an elliptic modular newform f is in fact generated by the pth Hecke eigenvalue ap of f for a single prime number p. In this paper, we study the density of such primes in a more general situation in terms of Galois representations. This is applied to the study of the fields of rationality of certain automorphic representations.

We prove that a double covering of P^3 branched along a very general sextic surface is not stably rational., A reference to the paper of Iliev-Katzarkov-Przyjalkowski added, and a small inaccuracy corrected

In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with ample generics embeds into a connected Polish group with ample generics and that full groups of type III hyperfinite ergodic equivalence relations have ample generics. We also sketch a proof of the following result: the full group of any type III ergodic equivalence relation has topological rank 2., Comment: New version mentioning the results Malicki obtained independently and simultaneously http://arxiv.org/abs/1503.03919, which also answer Kechris and Rosendal's question in a different way. Comments welcome!

Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an element is obtained by substitution in w is a class function. Thus, it has a presentation as a combination of irreducible characters, called its Fourier expansion. In this paper we present formulas regarding the Fourier expansion of words in which some letters appear twice. These formulas give simple proofs for classical results, as well as new ones.

Let $\Fth$ be a 2 graph generated by $m$ blue edges and $n$ red edges, and $\omega$ be the distinguished faithful state associated with its graph C*-algebra $\O_\theta$. In this paper, we characterize the factorness of the von Neumann algebra $\pi_\omega(\O_\theta)"$ induced from the GNS representation of $\omega$ under a certain condition. Moreover, when $\pi_\omega(\O_\theta)"$ is a factor, then it is of type III$_{m^{-\frac{1}{b}}}$ (or III$_{n^{-\frac{1}{a}}}$) if $\frac{\ln m}{\ln n}\in\bQ$, where $a,b\in\bN$ with $\gcd(a,b)=1$ satisfy $m^a=n^b$, and of type III$_1$ if $\frac{\ln m}{\ln n}\not\in\bQ$. In the case of $\theta$ being the identity permutation, our condition turns out to be redundant. On the way to our main results, we also obtain the structure of the fixed point algebra $\O_\theta^\sigma$ of the modular action $\sigma$ from $\omega$. This could be useful in proving the redundancy of our extra condition.

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta's Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational points on algebraic varieties. In particular, we show that Vojta's Main Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces., 10 pages

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise asymptotic estimates for rank and crank moments and their differences., 11 pages

This is a companion paper to math.AT/0609762. For a filtered colimit of commutative rings k=colim k_i, we prove that the homotopy theory of smooth and proper dg-algebras over k is the colimit of the homotopy theories of smooth and proper dg-algebras over k_i. As a consequence, we deduce that any smooth and proper dg-algebra can be defined over a commutative Z-algebra of finite type.

Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of {\em every} singularity with local algebra Q can be recovered from Ts_Q.

Recently, J. Roe proved by a direct computation that the Baum–Connes assembly map can be expressed as a boundary map associated to a certain short exact sequence of C*-algebras. In this paper we deduce the same result as a corollary of the author's characterisation of the Baum–Connes assembly map.