Search

In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mariş in 2013. However, such result is valid only in dimension 3 and higher. In this paper we first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3. With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3, recovering the aforementioned result by Mariş. The planar case turns out to be more intricate and the compactness argument works only under an additional assumption on the vortex set of the approximating solutions. [ABSTRACT FROM AUTHOR]

In this paper, we construct a higher rank Euler system for Gm over a totally real field by using Stickelberger elements. A key ingredient of the construction is to generalize the notion of the characteristic ideal. Under certain technical assumptions, we prove that all higher Fitting ideals of a certain p-ramified Iwasawa module are described by analytic invariants canonically associated with Stickelberger elements. [ABSTRACT FROM AUTHOR]

In this paper I present a new and unified method of proving character formulas for discrete series representations of connected Lie groups by applying a Chern character-type construction to the matrix factorizations of Freed-Hopkins-Teleman. In the case of a compact group I recover the Kirillov formula, thereby exhibiting the work of Freed-Teleman as a categorification of the Kirillov correspondence. In the case of a real semisimple group I recover the Rossman character formula with only a minimal amount of analysis. The appeal of this method is that it relies almost entirely on highest-weight theory, which is a far more ubiquitous phenomenon than the varied techniques that were previously used to prove such formulas. [ABSTRACT FROM AUTHOR]

In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in H3(R3). We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case. [ABSTRACT FROM AUTHOR]

For the spherical mean operators At in Rd, d = 2, we consider the maximal functions MEf =suptE |Atf|, with dilation sets E [1,2]. In this paper we give a surprising characterization of the closed convex sets which can occur as closure of the sharp Lp improving region of ME for some E. This region depends on the Minkowski dimension of E, but also other properties of the fractal geometry such as the Assouad spectrum of E and subsets of E. A key ingredient is an essentially sharp result onME for a class of sets called (quasi-)Assouad regular which is new in two dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum of the complex Hill operator L = d2 dx2 +q(x;τ) in L2(R,C) with the Darboux-Treibich-Verdier potential q(x;τ):= - 3 k=0 nk(nk +1)℘x+z0 + ωk 2;τ, where nk ∈ Z≥0 with maxnk ≥ 1 and z0 ∈ C is chosen such that q(x;τ) has no singularities on R. For any fixed τ ∈ iR>0, we give a necessary and sufficient condition on (n0,n1,n2,n3) to guarantee that the spectrum σ(L) is σ(L) = -∞,E2g∪E2g-1,E2g-2∪· · ·∪[E1,E0], Ej ∈ R, and hence generalizes Ince's remarkable result in 1940 for the Lam'e potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval (E2j-1,E2j-2), which generalizes the recent result by Haese-Hill et al., who studied the Lamé case n1 = n2 = n3 = 0. [ABSTRACT FROM AUTHOR]

A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every ϵ > 0 it is measurably conjugate to the direct product of a transformation with entropy < ϵ and a Bernoulli shift. In a recent breakthrough, Tim Austin proved that every ergodic transformation satisfies this property. Moreover, the natural analog for amenable group actions is also true. By contrast, this paper provides a counterexample in which the group Γ is a non-abelian free group and the notion of entropy is sofic entropy. The counterexample is a limit of hardcore models on random regular graphs. In order to prove that it does not have the WPP, this paper introduces new measure conjugacy invariants based on the growth of homology of the model spaces of the action. The main result is obtained by showing that any action with the WPP has subexponential homology growth in dimension 0, while the counterexample has exponential homology growth in dimension 0. [ABSTRACT FROM AUTHOR]

In this paper we give a complete and explicit description of the Howe correspondence of unipotent characters for a finite reductive dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization. That is, the conjecture by Aubert-Michel-Rouquier is confirmed. [ABSTRACT FROM AUTHOR]

In the early 1940s, P. A. Smith showed that if a finite p-group G acts on a finite dimensional complex X that is mod p acyclic, then its space of fixed points, XG, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a G-space X is a equivariant homotopy retract of the p-localization of a based finite G-C.W. complex, given H < G and n, what is the smallest r such that if XH is acyclic in the (n+r)th Morava K-theory, then XG must be acyclic in the nth Morava K-theory? Barthel et. al. then answered this when G is abelian, by finding general lower and upper bounds for these "blue shift" numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equiv- alent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod p homology, and thus applies to all finite dimensional G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for K(n)*-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava K-theories of all real Grassmanians. [ABSTRACT FROM AUTHOR]

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem. The problem of classifying stable (or minimal) homogeneous solutions in dimensions ≥ 3 is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open. The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of "large solutions" for the fractional Laplacian, which blow up on the free boundary. On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions ≤ 5 is one-dimensional, independently of the parameter s ∈ (0,1). [ABSTRACT FROM AUTHOR]

We correct a mistake in our proof of Lemma 2.17. Although we have to strengthen the assumptions therein and, accordingly, in Theorem 2.22, all of the results on the existence and properties of the IMCF are not affected. Minor changes, with no influence elsewhere in the paper, regard Lemma 3.3, Proposition 4.3 and Lemma 5.3. [ABSTRACT FROM AUTHOR]

This is the first of a series of papers in which we study the plurigenera, the Kodaira dimension, and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost complex manifolds. We show that plurigenera and the Kodaira dimension are birational invariants in almost complex category, at least in dimension 4, where a birational morphism is defined to be a degree one pseudoholomorphic map. However, they are no longer deformation invariants, even in dimension 4 or under tameness assumption. On the way to establish the birational invariance, we prove the Hartogs extension theorem in the almost complex setting by the foliation-by-disks technique. Some interesting phenomena of these invariants are shown through examples. In particular, we construct non-integrable compact almost complex manifolds with large Kodaira dimensions. Hodge numbers and plurigenera are computed for the standard almost complex structure on the six sphere S6, which are different from the data of a hypothetical complex structure. [ABSTRACT FROM AUTHOR]

Given a finite collection of C¹ vector fields on a C² manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are Cs+1 for s ∈ (1, ∞], where Cs denotes the Zygmund space of order s. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the second part in a three part series of papers. The first part, joint with Stovall, addressed the same question, though the results were not sharp, and showed how such coordinate charts can be viewed as scaling maps in sub-Riemannian geometry. When viewed in this light, these results can be seen as strengthening and generalizing previous works on the quantitative theory of sub-Riemannian geometry, initiated by Nagel, Stein, and Wainger, and furthered by Tao and Wright, the author, and others. In the third part, we prove similar results concerning real analyticity. [ABSTRACT FROM AUTHOR]

In this paper, we prove an averaged version of an algebraicity conjecture of Gross and Zagier concerning the values of higher Green's function at CM points. Furthermore, we give the factorization of the ideal generated by such algebraic value in the spirit of the famous work of Gross and Zagier on singular moduli. [ABSTRACT FROM AUTHOR]

In this paper we construct families of bounded domains Ωε and solutions uε of such that, for any integer k ≥ 2, uε admits at least k maximum points for small enough. The domain Ωε is "not far" to be convex in the sense that it is starshaped, the curvature of ∂Ωε vanishes at exactly two points and the minimum of the curvature of ∂Ωε goes to 0 as ε→0. [ABSTRACT FROM AUTHOR]

In this paper we determine the automorphism groups of the profinite braid groups with four or more strings in terms of the profinite Grothendieck-Teichmuller group. [ABSTRACT FROM AUTHOR]

Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, Lp estimates, which are a dual formulation of the Kunze-Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of SL(2,R) and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new L² estimate of dispersive type for the free Schrödinger equation in Rn. [ABSTRACT FROM AUTHOR]

Let X be a topological space with Noetherian mod p cohomology and let C*(X;Fp) be the commutative ring spectrum of Fp-valued cochains onX. The goal of this paper is to exhibit conditions under which the category of module spectra on C*(X;Fp) is stratified in the sense of Benson, Iyengar, Krause, providing a classification of all its localizing subcategories. We establish stratification in this sense for classifying spaces of a large class of topological groups including Kac-Moody groups as well as whenever X admits an H-space structure. More generally, using Lannes' theory we prove that stratification for X is equivalent to a condition that generalizes Chouinard's theorem for finite groups. In particular, this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory. [ABSTRACT FROM AUTHOR]

Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces. First, these include many countable groups G acting properly nonelementarily by isometries on a proper CAT(0) space X. More precisely, proper proximality holds in the presence of rank one isometries or when X is a locally thick affine building with a minimal G-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary CAT(0) cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a 2-dimensional piecewise Euclidean CAT(0) simplicial complex. Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana, and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph. In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions. [ABSTRACT FROM AUTHOR]

In this paper we show the existence of weak solutions w: M R of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of w and for the mean curvature of its level sets, which are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the p-Laplace equation, and relies on new gradient and decay estimates for p-harmonic capacity potentials, notably for the kernel Gp of Δp. These bounds, stable as p1, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of w. [ABSTRACT FROM AUTHOR]

In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with C² boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. The second is for biholomorphisms of domains who have an invariant Kähler metric with bounded sectional curvature. This second result applies to holomorphic homogeneous regular domains and appears to be the first boundary Schwarz lemma that makes no assumptions on the regularity of the boundary. [ABSTRACT FROM AUTHOR]

Given a domain I ⊂ C and an integer N > 0, a function f : I → C is said to be entrywise positivity preserving on positive semidefinite N x N matrices A = (ajk) ∈ INxN, if the entrywise application f[A] = (f(ajk)) of f to A is positive semidefinite for all such A. Such preservers in all dimensions have been classified by Schoenberg and Rudin as being absolutely monotonic. In fixed dimension N, results akin to work of Horn and Loewner show that the first N non-zero Maclaurin coefficients of any positivity preserver f are positive; and the last N coefficients are also positive if I is unbounded. However, very little was known about the higher-order coefficients: the only examples to date for unbounded domains I were absolutely monotonic, hence work in all dimensions; and for bounded I examples of non-absolutely monotonic preservers were very few (and recent). In this paper, we provide a complete characterization of the sign patterns of the higher-order Maclaurin coefficients of positivity preservers in fixed dimension N, over bounded and unbounded domains I = (0, ρ). In particular, this shows that the above Horn-Loewner-type conditions cannot be improved upon. As a further special case, this provides the first examples of polynomials which preserve positivity on positive semidefinite matrices in INxN but not in I(N+I)x(N+I). Our main tools in this regard are the Cauchy-Binet formula and lower and upper bounds on Schur polynomials. We also obtain analogous results for real exponents, using the Harish-Chandra-Itzykson-Zuber formula in place of bounds on Schur polynomials. We then go from qualitative existence bounds--which suffice to understand all possible sign Patterns--to exact quantitative bounds. This is achieved using a Schur positivity result due to Lam, Postnikov, and Pylyavskyy, and in particular provides a second proof of the existence of threshold bounds for tuples of integer and real powers. As an application, we extend our previous qualitative and quantitative results to understand preservers of total non-negativity in fixed dimension--including their sign patterns. We deduce several further applications, including extending a Schur polynomial conjecture of Cuttler, Greene, and Skandera to obtain a novel characterization of weak majorization for real tuples. [ABSTRACT FROM AUTHOR]

We prove dynamical dichotomy into scattering and blow-up (in a weak sense) for all radial solutions of the Zakharov system in the energy space of four spatial dimensions that have less energy than the ground state, which is written using the Aubin-Talenti function. The dichotomy is characterized by the critical mass of the wave component of the ground state. The result is similar to that by Kenig and Merle for the energy-critical nonlinear Schrödinger equation (NLS). Unlike NLS, however, the most difficult interaction in the proof stems from the free wave component. In order to control it, the main novel ingredient we develop in this paper is a uniform global Strichartz estimate for the linear Schrödinger equation with a potential of subcritical mass solving a wave equation. This estimate, as well as the proof, may be of independent interest. For the scattering proof, we follow the idea by Dodson and Murphy. [ABSTRACT FROM AUTHOR]

This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences, several generalizations of circle pattern theorem are obtained. Moreover, our approach suggests a computational method to find the desired circle patterns. [ABSTRACT FROM AUTHOR]

Given an irrational number α ∈ (0,1), the Sudler product is defined by PN (α) = QNr=1 2|sinπrα|. Answering a question of Grepstad, Kaltenbock and Neumüller we prove aöasymptotic formula for distorted Sudler products when α is the golden ratio ( √ 5+1)/2 and establish that in this case limsup N→∞ PN (α)/N < ∞. We obtain similar results for quadratic irrationals α with continued fraction expansion α = [a,a,a,...] for some integer a ≥ 1, and give a full characterisation of the values of a for which liminf N→∞ PN (α) > 0 and limsup N→∞ PN (α)/ N < ∞ hold, respectively. We establish that there is a (sharp) transition point at a = 6, and resolve as a by-product a problem of the first author, Larcher, Pillichshammer, Saad Eddin, and Tichy. [ABSTRACT FROM AUTHOR]

We establish structure theorems for a smooth projective variety X with semi-positive holomorphic sectional curvature. We first prove that X is rationally connected ifX has no truly flat tangent vectors at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive). This result solves Yau's conjecture on positive holomorphic sectional curvature in a strong form. Moreover, we prove that X admits a locally trivial morphism X =Y such that the fiber F is rationally connected and the image Y has a finitetale cover A=Y by an abelian variety A. We also show that the universal cover of X is biholomorphic and isometric to the product Cm×F of the complex Euclidean space Cm with the flat metric and the rationally connected fiber F with the induced Kahler metric. Our structure theorem is a natural generalization of the structure theorem established by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. [ABSTRACT FROM AUTHOR]

For almost all Riemannian metrics (in the C∞ Baire sense) on a compact manifold with boundary (Mn+1,∂M), 3 ≤ (n+1)≤ 7, we prove that, for any open subset V of ∂M, there exists a compact, properly embedded free boundary minimal hypersurface intersecting V. [ABSTRACT FROM AUTHOR]

We investigate the local distribution of roots of random functions of the form Fn(z) = Σi=1² ξiφi(z), where ξi are independent random variables and φi(z) are arbitrary analytic functions. Starting with the fundamental works of Kac and Littlewood-Offord in the 1940s, random functions of this type have been studied extensively in many fields of mathematics. We develop a robust framework to solve the problem by reducing, via universality theorems, the calculation of the distribution of the roots and the interaction between them to the case where ξi are Gaussian. In this special case, one can use the Kac-Rice formula and various other tools to obtain precise answers. Our framework has a wide range of applications, which include the most popular models of random functions, such as random trigonometric polynomials and all basic classes of random algebraic polynomials (Kac, Weyl, and elliptic). Each of these ensembles has been studied heavily by deep and diverse methods. Our method, for the first time, provides a unified treatment for all of them. Among the applications, we derive the first local universality result for random trigonometric polynomials with arbitrary coefficients. When restricted to the study of real roots, this result extends several recent results, proved for less general ensembles. For random algebraic polynomials, we strengthen several recent results of Tao and the second author, with significantly simpler proofs. As a corollary, we sharpen a classical result of Erdös and Offord on real roots of Kac polynomials, providing an optimal error estimate. Another application is a refinement of a recent result of Flasche and Kabluchko on the roots of random Taylor series. [ABSTRACT FROM AUTHOR]

Let (S,h) be a closed hyperbolic surface and M be a quasi-Fuchsian 3-manifold. We consider incompressible maps from S to M that are critical points of an energy functional F which is homogeneous of degree 1. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps--but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface inM. We prove the uniqueness of smooth minimizing maps from (S,h) to M in a given homotopy class. When (S,h) is fixed, smooth minimizing maps from (S,h) are described by a simple holomorphic data on S: a complex self-adjoint Codazzi tensor of determinant 1. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the monodromy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of F analoguous to the complex length. [ABSTRACT FROM AUTHOR]

In the past years, there has been a new light shed on the harmonic map problem with free boundary in view of its connection with nonlocal equations. Here we fully exploit this link, considering the harmonic map flow with free boundary for a function u: R2 + ×[0,T) R2. Here u0: R2 + R2 is a given smooth map and stands for orthogonality. We prove the existence of initial data u0 such that (0.1) blows up at finite time with a profile being the half-harmonic map. This answers a question raised by Chen and Lin. [ABSTRACT FROM AUTHOR]

We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the form on (0,T]t ×T3 x, where aij (t,x) and pi(x) are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as t 0+. These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the "singular hypersurface" {t = 0}. This is the first such result without imposing symmetry or analyticity. To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-t hypersurfaces. [ABSTRACT FROM AUTHOR]

We study holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of C2 of the form F(z,w) = (p(z),q(z,w)) that extend to holomorphic endomorphisms of P2 (C). We prove that stability in the sense of [Berteloot, Bianchi, and Dupont, 2018] preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family z2 + c. We also precisely describe the geometry of the bifurcation locus and current near the boundary of the parameter space. One of our tools is an asymptotic equidistribution property for the bifurcation current. This is established in the general setting of families of endomorphisms of Pk, and is the first equidistribution result of this kind for holomorphic dynamical systems in dimension larger than one. [ABSTRACT FROM AUTHOR]

For Fano manifolds T. Mabuchi introduced a generalization of the Kähler-Einstein metric, which is characterized as the critical point of the Ricci-Calabi functional. We show that a Fano manifold admits Mabuchi’s metric if and only if it is uniformly relatively D-stable. The idea of the proof includes some equivariant generalization of the recent developed variational approach to the Kahler-Einstein problem. [ABSTRACT FROM AUTHOR]

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on GSp4 whose archimedean components belong to either discrete series representations or spherical principal series representations. Our formula expresses the Petersson norms in terms of special values of adjoint L-functions and some elementary constants depending only on local representations. [ABSTRACT FROM AUTHOR]

We study certain automorphic descent constructions for symplectic groups, and obtain results related to branching problems of automorphic representations. As a byproduct of the construction, based on the knowledge of the global Vogan packets for Mp2 (A), we give a new approach to prove the result that for an automorphic cuspidal representation of GL2(A) of symplectic type, if there exists a quadratic twist with positive root number, then there exist quadratic twists with non-zero central L-values. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive group over a non-archimedean local field K, and assume that G splits over an unramified extension of K. We establish a local Langlands correspondence for irreducible unipotent representations of G. It comes as a bijection between the set of such representations (up to isomorphism) and the collection of enhanced L-parameters for G, which are trivial on the inertia subgroup of the Weil group of K. We show that this correspondence has many of the expected properties, for instance with respect to central characters, tempered representations, the discrete series, cuspidality and parabolic induction. The core of our strategy is the investigation of affine Hecke algebras on both sides of the local Langlands correspondence. When a Bernstein component of G-representations is matched with a Bernstein component of enhanced L-parameters, we prove a comparison theorem for the two associated affine Hecke algebras. This generalizes work of Lusztig in the case of adjoint K-groups. [ABSTRACT FROM AUTHOR]

We study Lelong numbers of currents of full mass intersection on a compact Kähler manifold in a mixed setting. Our main theorems cover some recent results due to Darvas-Di Nezza-Lu. The key ingredient in our approach is a new notion of products of pseudoeffective (1,1)-classes which captures some “pluripolar part” of the “total intersection” of given pseudoeffective (1,1)-classes. [ABSTRACT FROM AUTHOR]

Given a set of lines in ℝ³, a joint is a point passed through by three lines not all lying in some plane. The joints problem asks for the maximum number of joints formed by L lines. Using the polynomial method, Guth and Katz proved that the answer is O(L3/2), which is best possible up to a constant factor. We prove a new upper bound of (√2/3)L3/2, which matches, up to a 1+o(1) factor, the best known construction: place k generic planes, and use their pairwise intersections to form (k2) lines and their triple-wise intersections to form (k3) joints. Guth conjectured that this construction is optimal. Our result generalizes to arbitrary dimensions and fields. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint. [ABSTRACT FROM AUTHOR]

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrödinger equation with nonlinearity |u|6u. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on L²-based critical Sobolev norms imply scattering estimates. As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Strichartz norm, then the solution can be extended globally in time, with a corresponding scattering estimate. [ABSTRACT FROM AUTHOR]

We prove that the coefficients of a GL3×GL2 Rankin–Selberg L-function do not correlate with a wide class of trace functions of small conductor modulo primes, generalizing the corresponding result of Fouvry, Kowalski, and Michel for GL2 and of Kowalski, Lin, Michel, and Sawin for GL3. This result is inspired by a recent work of P. Sharma who discussed the case of a Dirichlet character of prime modulus. [ABSTRACT FROM AUTHOR]

The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions—one geometric, the other probabilistic in nature— are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann–Petty centroid inequality. [ABSTRACT FROM AUTHOR]

We prove that the affine space of dimension n ≥ 1 over an uncountable algebraically closed field k is determined, among connected affine varieties, by its automorphism group (viewed as an abstract group). The proof is based on a new result concerning algebraic families of pairwise commuting automorphisms. [ABSTRACT FROM AUTHOR]

We present an approach over arbitrary fields to bound the degree of intersection of families of varieties in terms of how these concentrate on algebraic sets of smaller codimension. This provides in particular a substantial extension of the method of degree-reduction that enables it to deal efficiently with higher-dimensional problems and also with high-degree varieties. We obtain sharp bounds that are new even in the case of lines in ℝn and show that besides doubly-ruled varieties, only a certain rare family of varieties can be relevant for the study of incidence questions. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for unirationality and rationality of Fano three-folds of geometric Picard rank 1 over an arbitrary field of zero characteristic. [ABSTRACT FROM AUTHOR]

We study semigroup C∗-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. These semigroup C∗-algebras turn out to have an interesting C∗-algebraic structure, giving access to many new examples of classifiable C∗-algebras and exhibiting phenomena which have not appeared before. Moreover, using K-theoretic invariants, we investigate how much information about the initial number-theoretic data is encoded in our semigroup C∗-algebras. [ABSTRACT FROM AUTHOR]

We classify global solutions of the Monge-Amp`ere equation det D²u = 1 on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global C2,α estimates for the non-degenerate Monge-Ampère equation in convex polygonal domains in R² provided a globally C², convex strict subsolution exists. [ABSTRACT FROM AUTHOR]

Using arithmetic jet spaces we attach perfectoid spaces to smooth schemes and we attach morphisms of perfectoid spaces to δ-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable δ-morphisms appearing in the theory such as the δ-characters of elliptic curves and the δ-period maps on modular curves. [ABSTRACT FROM AUTHOR]

We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity. [ABSTRACT FROM AUTHOR]