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Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3). Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems., 50 pages. Originally part of arXiv:1703.09945 . We decided to split that paper following the recommendations of a referee. Updated subsequent to acceptance by Transactions of the American Mathematical Society

Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)

A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke., to appear in Trans. Amer. Math. Soc

In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata., The paper has been augmented considerably and the second version is now 81 pages long. This version has been accepted for publication in Transactions of the American Mathematical Society

Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric., Comment: 50 pages, 14 figures. Final version

For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of $M(X,r)$ as a regular submonoid in the semidirect product $A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module finite normal extension of a commutative affine subalgebra. In particular, $K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension bounded by $|X|$. We also characterize, in ring-theoretical terms of $K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $M(X,r)$. These results allow us to control the prime spectrum of the algebra $K[M(X,r)]$ and to describe the Jacobson radical and prime radical of $K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra $K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times \mathrm{M}_{v_m}(K[G_m])$., A subtle mistake in the proof of Theorem 4.4 has been corrected (will appear in a corrigendum et addendum, TAMS). In the latter paper we also strengthen some of the results by removing the "square free'' condition in Section 5 and in this paper we also prove new homological equivalences in Theorem 4.4

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper., Minor change basically on the proof of Theorem 1.2 in Section 5

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper., Comment: 44 pages

In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups., Accepted to Transactions of the American Mathematical Society

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages

A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given Fomin and Zelevinsky.) The universal objects are closely related to a fan F_B called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of F_B for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces, and use it to construct universal geometric coefficients for these surfaces., Comment: 39 pages, 24 figures. Version 2: Stated explicitly several results that are implicit in the earlier version. Also minor expository changes. Version3: Expository changes (including additional figures) and changes to correct an error from arXiv:1209.3987 (see comments to arXiv:1209.3987v3)

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function., Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled., Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest Vinberg

Let $X$ be a K\"ahler manifold which is fibered over a complex manifold $Y$ such that every fiber is a Calabi-Yau manifold. Let $\omega$ be a fixed K\"ahler form on $X$. By Yau's theorem, there exists a unique Ricci-flat K\"ahler form $\rho\vert_{X_y}$ for each fiber, which is cohomologous to $\omega\vert_{X_y}$. This family of Ricci-flat K\"ahler forms $\rho\vert_{X_y}$ induces a smooth $(1,1)$-form $\rho$ on $X$ with a normalization condition. In this paper, we prove that the direct image of $\rho^{n+1}$ is positive on the base $Y$. We also discuss several byproducts, among them the local triviality of families of Calabi-Yau manifolds., Comment: 25 pages. Minor changes

In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category., 18 pages, 6 figures, to appear in Trans. Amer. Math. Soc

Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non-Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and $S$-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem., 29 pages; some references have been added; to appear in Trans. Amer. Math. Soc

Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and R\"omer., Comment: 20 pages

In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion work. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time., Comment: 17 pages, 6 figures

In this paper we consider the problem of finding entrance laws at the origin for self-similar Markov processes in R d \mathbb {R}^d , killed upon hitting the origin. Under suitable assumptions, we show the existence of an entrance law and the convergence to this law when the process is started close to the origin. We obtain an explicit description of the process started from the origin as the time reversal of the original self-similar Markov process conditioned to hit the origin.

We study the distribution of the sandpile group of random d-regular graphs. For the directed model, we prove that it follows the Cohen-Lenstra heuristics, that is, the limiting probability that the $p$-Sylow subgroup of the sandpile group is a given $p$-group $P$, is proportional to $|Aut(P)|^{-1}$. For finitely many primes, these events get independent in the limit. Similar results hold for undirected random regular graphs, where for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne. This answers an open question of Frieze and Vu whether the adjacency matrix of a random regular graph is invertible with high probability. Note that for directed graphs this was recently proved by Huang. It also gives an alternate proof of a theorem of Backhausz and Szegedy., Comment: We improved the presentation of the paper. More details are given in several proofs

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $(\bar G, \bar g)$. Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-K\"ahler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in \cite{EFV} for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures., Comment: 25 pages. Revised version. To appear in TAMS

Let $\Gamma$ be a lattice of a semisimple Lie group $L$. Suppose that one parameter Ad-diagonalizable subgroup $\{g_t\}$ of $L$ acts ergodically on $L/\Gamma$ with respect to the probability Haar measure $\mu$. For certain proper subgroup $U$ of the unstable horospherical subgroup of $\{g_t\}$ we show that given $x\in L/\Gamma$ for almost every $u\in U$ the trajectory $\{g_tux: 0\le t\le T\}$ is uniformly distributed with respect to $\mu$ as $T\to \infty$., Comment: The detailed proof is modified a lot to make the paper easy to read

In this paper, we find a composition series of GKZ-systems with semisimple successive quotients. We also study the composition series of the corresponding perverse sheaves and compare these two composition series under the Riemann-Hilbert correspondence., 32 pages, appear to Transactions of AMS

Let $B^{\sigma}_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $\alpha$-regular measure $\mu$. The {\it critical exponent} $\sigma^*$ is the supremum of the $\sigma$ such that $B^{\sigma}_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{\sigma^*}_{2, \infty}$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on the p.c.f. sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied., Comment: 34 pages,15 figures

In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if E E is any arithmetic progression, the set of primes, or the set of squares { n 2 } n ∈ N \{n^2\}_{n \in \mathbb {N}} , then the continued fractions whose digits lie in E E have full dimension spectrum, which we denote by D S ( C F E ) DS(\mathcal {CF}_E) . Moreover we prove that if E E is an infinite set of consecutive powers, then the dimension spectrum D S ( C F E ) DS(\mathcal {CF}_E) always contains a non-trivial interval. We also show that there exists some E ⊂ N E \subset \mathbb {N} and two non-trivial intervals I 1 , I 2 I_1, I_2 , such that D S ( C F E ) ∩ I 1 = I 1 DS(\mathcal {CF}_E) \cap I_1=I_1 and D S ( C F E ) ∩ I 2 DS(\mathcal {CF}_E) \cap I_2 is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets.

In this paper, motivated by recent works due to Frank-Lewin-Lieb-Seiringer and Frank-Sabin, we study the Strichartz inequality on torus with the orthonormal system input and obtain sharp estimates in a certain sense. In particular, we will reveal the tradeoff relation between Sobolev regularity and Schatten exponent gain where the 1 / p 1/p derivative-loss Strichartz inequality plays an important role as in the context on compact manifold due to Burq-Gérard-Tzvetkov. An application of the inequality shows the local well-posedness to the periodic Hartree equation describing the infinitely many quantum particles interacting with the power type potential.

The purpose of this paper is to study and classify singular solutions of the Poisson problem { L μ s u = f in Ω ∖ { 0 } , u = 0 in R N ∖ Ω \begin{equation*} \left \{ \begin {aligned} \mathcal {L}^s_\mu u = f \quad \ \text {in}\ \, \Omega \setminus \{0\},\\ u =0 \quad \ \text {in}\ \, \mathbb {R}^N \setminus \Omega \ \end{aligned} \right . \end{equation*} for the fractional Hardy operator L μ s u = ( − Δ ) s u + μ | x | 2 s u \mathcal {L}_\mu ^s u= (-\Delta )^s u +\frac {\mu }{|x|^{2s}}u in a bounded domain Ω ⊂ R N \Omega \subset \mathbb {R}^N ( N ≥ 2 N \ge 2 ) containing the origin. Here ( − Δ ) s (-\Delta )^s , s ∈ ( 0 , 1 ) s\in (0,1) , is the fractional Laplacian of order 2 s 2s , and μ ≥ μ 0 \mu \ge \mu _0 , where μ 0 = − 2 2 s Γ 2 ( N + 2 s 4 ) Γ 2 ( N − 2 s 4 ) > 0 \mu _0 = -2^{2s}\frac {\Gamma ^2(\frac {N+2s}4)}{\Gamma ^2(\frac {N-2s}{4})}>0 is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case μ = μ 0 \mu = \mu _0 which requires more subtle estimates than the case μ > μ 0 \mu >\mu _0 .

Motivated by the relationship between numerical Grothendieck groups induced by the embedding of a smooth anticanonical elliptic curve into a del Pezzo surface, we define the notion of a quasi del Pezzo homomorphism between pseudolattices and establish its basic properties. The primary aim of the paper is then to prove a classification theorem for quasi del Pezzo homomorphisms, using a pseudolattice variant of the minimal model program. Finally, this result is applied to the classification of a certain class of genus one Lefschetz fibrations over discs., Comment: v2: Minor revisions at the request of the referee. Remark 3.24 is new, as is the discussion of the noncommutative setting in the final section

We study the existence of zero-cycles of degree one on varieties that are defined over a function field of a curve over a complete discretely valued field. In particular, we show that local-global principles hold for such zero-cycles provided that local-global principles hold for the existence of rational points over extensions of the function field. This assertion is analogous to a known result concerning varieties over number fields. We also show that our results hold more generally in the henselian case., 23 pages. Some explanations were expanded for greater clarity throughout the paper. Some results in Section 2 in the earlier version in the complete case were generalized here to the henselian case. Subsection 3.3 in the earlier version was broken into two subsections, and there was some renumbering of results in the new Subsection 3.4 due to an added remark

In this paper we consider diffusions on the half line ( 0 , ∞ ) (0,\infty ) such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity which takes finite values at positive times. We study the behavior of hitting times of large barriers and, in a dual way, the behavior of the process starting at infinity for small time. In particular, we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion.

We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $3$-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible $3$-manifolds containing a genus $1$ fibred knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibred knot in an integer homology $3$-sphere admits a co-oriented taut foliation and has left-orderable fundamental group, even if the surgered manifold does not, and that the same holds for many branched covers of satellite knots with braided patterns. A key fact used in our proofs is that the Euler class of a universal circle representation associated to a co-oriented taut foliation coincides with the Euler class of the foliation's tangent bundle. Though known to experts, no proof of this important result has appeared in the literature. We provide such a proof in the paper., v3: 40 pages, minor revisions and corrections, improved exposition, accepted for publication in the Trans. Amer. Math. Soc

The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a longstanding open problem and has been solved so far only in a certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in which the waves propagate with near sonic speed, have large wave length, and carry low energy. In this paper we derive a similar result in a complementary asymptotic regime related to fast and strongly localized waves with high energy. In particular, we show that the spectrum of the linearized FPUT operator contains asymptotically no unstable eigenvalues except for the neutral ones that stem from the shift symmetry and the spatial discreteness. This ensures that high-energy waves are linearly stable in some orbital sense, and the corresponding nonlinear stability is granted by the general, non-asymptotic part of the seminal Friesecke-Pego result and the extension by Mizumachi. Our analytical work splits into two principal parts. First we refine two-scale techniques that relate high-energy wave to a nonlinear asymptotic shape ODE and provide accurate approximation formulas. In this way we establish the existence, local uniqueness, smooth parameter dependence, and exponential localization of fast lattice waves for a wide class of interaction potentials with algebraic singularity. Afterwards we study the crucial eigenvalue problem in exponentially weighted spaces, so that there is no unstable essential spectrum. Our key argument is that all proper eigenfunctions can asymptotically be linked to the unique bounded and normalized solution of the linearized shape ODE, and this finally enables us to disprove the existence of unstable eigenfunctions in the symplectic complement of the neutral ones., additional stability result in energy space; also some minor corrections; 56 pages, 10 figures

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$., To appear on Transactions of the American Mathematical Society. Comments are welcome. arXiv admin note: text overlap with arXiv:1607.00289

We determine the Gross–Hopkins duals of certain higher real K K –theory spectra. More specifically, let p p be an odd prime, and consider the Morava E E –theory spectrum of height n = p − 1 n=p-1 . It is known, in expert circles, that for certain finite subgroups G G of the Morava stabilizer group, the homotopy fixed point spectra E n h G E_n^{hG} are Gross–Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups G G which contain p p –torsion. This generalizes previous results for n = 2 n=2 and p = 3 p=3 .

Given a measure $\nu$ on a regular planar domain $D$, the Gaussian multiplicative chaos measure of $\nu$ studied in this paper is the random measure ${\widetilde \nu}$ obtained as the limit of the exponential of the $\gamma$-parameter circle averages of the Gaussian free field on $D$ weighted by $\nu$. We investigate the dimensional and geometric properties of these random measures. We first show that if $\nu$ is a finite Borel measure on $D$ with exact dimension $\alpha>0$, then the associated GMC measure ${\widetilde \nu}$ is non-degenerate and is almost surely exact dimensional with dimension $\alpha-\frac{\gamma^2}{2}$, provided $\frac{\gamma^2}{2}, Comment: 33 pages, 1 figure

In this paper we initiate the study of racks from the combined perspective of combinatorics and finite group theory. A rack R is a set with a self-distributive binary operation. We study the combinatorics of the partially ordered set {\cal R}(R) of all subracks of R with inclusion as the order relation. Groups G with the conjugation operation provide an important class of racks. For the case R = G we show that -> the order complex of {\cal R}(R) has the homotopy type of a sphere, -> the isomorphism type of {\cal R}(R) determines if G is abelian, nilpotent, supersolvable, solvable or simple, -> {\cal R}(R) is graded if and only if G is abelian, G = S_3, G = D_8 or G = Q_8. In addition, we provide some examples of subracks R of a group G for which {\cal R}(R) relates to well studied combinatorial structures. In particular, the examples show that the order complex of {\cal R}(R) for general R is more complicated than in the case R = G.

In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, we prove that if the mean curvature $H$ of the boundary obeys the condition $$ - \bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} \leq \frac{H}{n-1} \leq \bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} , $$ then $\Omega$ is a round ball., Comment: 41 pages

Let G be a connected complex reductive group. A well known theorem of I. Losev's says that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties., v1: 33 pages. v2: 36 pages, improved exposition, updated references and citations

In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper, we generalize the theorems of Dougherty and Mycielski about $S_\infty$ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the Cofinal Strong Amalgamation Property. As an application we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set.

In this paper, we give necessary and sufficient conditions for weighted $L^2$ estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form: \[ \| T(\mathbf{W} f)\|_{L^2(\mathbf{V})} \le C\|f\|_{L^2(\mathbf{W})} \] where $T$ is formally an integral operator with additional structure, $\mathbf{W}, \mathbf{V}$ are matrix measures, and the underlying measure space possesses a filtration. The characterization we obtain is of Sawyer-type; in particular we show that certain natural testing conditions obtained by studying the operator and its adjoint on indicator functions suffice to determine boundedness. Working in both the matrix weighted setting and the setting of measure spaces with arbitrary filtrations requires novel modifications of a T1 proof strategy; a particular benefit of this level of generality is that we obtain polynomial estimates on the complexity of certain Haar shift operators., Comment: v1: 27 pages

The aim of this paper is to develop a general theory for the class of skip-free Markov chains on denumerable state space. This encompasses their potential theory via an explicit characterization of their potential kernel expressed in terms of family of fundamental excessive functions, which are defined by means of the theory of Martin boundary. We also describe their fluctuation theory generalizing the celebrated fluctuations identities that were obtained by using the Wiener-Hopf factorization for the specific skip-free random walks. We proceed by resorting to the concept of similarity to identify the class of skip-free Markov chains whose transition operator has only real and simple eigenvalues. We manage to find a set of sufficient and easy-to-check conditions on the one-step transition probability for a Markov chain to belong to this class. We also study several properties of this class including their spectral expansions given in terms of Riesz basis, derive a necessary and sufficient condition for this class to exhibit a separation cutoff, and give a tighter bound on its convergence rate to stationarity than existing results., 41 pages. To appear Trans. Amer. Math. Soc

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp., Comment: 25 pages Some some corrections are made in this final version

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rarita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformal invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.

Let $M$ be a positive integer and $q \in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\{0,\ldots, M\}$. In particular, we study the set $\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\mathbf{U}_q$ of corresponding sequences. It was shown in (Komornik et al, 2017 Adv. Math.) that the function $H$, which associates to each $q\in(1, M+1]$ the topological entropy of $\mathcal{U}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the bifurcation set $\mathcal E$ of $q$'s where the function $H$ is not locally constant. Moreover, we show that $\mathcal E$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $(\mathbf{V}_q, \sigma),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of $\mathcal{U}_q$ coincide for all $q\in(1,M+1]$., Comment: 56 pages, 7 figures. To appear in Trans. Amer. Math. Soc

We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that, still, almost all real numbers are normal. Our theorem states that almost all real numbers are normal when at least slightly more than $\log \log n$ consecutive digits with indices starting at position $n$ are independent. As the main application, we consider the Toeplitz set $T_P$, which is the set of all sequences $a_1a_2 \ldots $ of symbols from $\{0, \ldots, b-1\}$ such that $a_n$ is equal to $a_{pn}$, for every $p$ in $P$ and $n=1,2,\ldots$. Here $b$ is an integer base and $P$ is a finite set of prime numbers. We show that almost every real number whose base $b$ expansion is in $T_P$ is normal to base $b$. In the case when $P$ is the singleton set $\{2\}$ we prove that more is true: almost every real number whose base $b$ expansion is in $T_P$ is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set $T_P$ and we characterize the normal sequences whose Toeplitz transform is normal as well., Comment: 21 pages. This paper will appear in Trans. AMS

In this paper we study the singular set of Dirichlet-minimizing $Q$-valued maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single valued case, we prove that the target $\mathcal{N}$ being non-positively curved but not simply connected does not imply continuity of the map., Comment: 43 pages

For certain quasi-split reductive groups $G$ over a general field $F$, we construct an automorphism $\iota_G$ of $G$ over $F$, well-defined as an element of ${\rm Aut}(G)(F)/jG(F)$ where $j:G(F) \rightarrow {\rm Aut}(G)(F)$ is the inner-conjugation action of $G(F)$ on $G$. The automorphism $\iota_G$ generalizes (although only for quasi-split groups) an involution due to Moeglin-Vigneras-Waldspurger in [MVW] for classical groups which takes any irreducible admissible representation $\pi$ of $G(F)$ for $G$ a classical group and $F$ a local field, to its contragredient $\pi^\vee$. The paper also formulates a conjecture on the contragredient of an irreducible admissible representation of $G(F)$ for $G$ a reductive algebraic group over a local field $F$ in terms of the (enhanced) Langlands parameter of the representation., Comment: arXiv admin note: substantial text overlap with arXiv:1512.04347