Fatou Components of Attracting Skew-Products

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188 [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.     

The Continuity Method to Deform Cone Angle

The continuity method is used to deform the cone angle of a weak conical Kähler–Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson’s Openness Theorem on deforming cone angle Donaldson (Essays in Mathematics and Its Applications, 2012) by combining it with the regularity result of Guenancia–P?un (arXiv:1307.6375 2013) and Chen–Wang (arXiv:1405.1201 2014). This continuity method uses relatively less regularity of the metric (only weak conical Kähler–Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a weak conical Kähler–Einstein metric along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).     

Bounding Diameter of Conical Kähler Metric

In this paper we research the differential geometric and algebro-geometric properties of the noncollapsing limit in the conical continuity equation which generalize the theory in La Nave et al. in Bounding diameter of singular Kähler metric, arXiv:1503.03159v1 [23].     

A properly discontinuous free group of affine transformations

In this paper we prove that a fully irreducible outer automorphism relative to a non-exceptional free factor system acts loxodromically on the relative free factor complex as defined in Handel and Mosher (Relative free splitting and relative free factor complexes I: hyperbolicity, 2014. arXiv:1407.3508v1). We also prove a north-south dynamic result for the action of such outer automorphisms on the closure of relative outer space.     

On Focal Submanifolds of Isoparametric Hypersurfaces and Simons Formula

The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being \({\mathcal{A}}\) -manifolds in the sense of A.Gray but rarely Ricci-parallel, see Li et al. (Sci China Math 58, 2015), Qian et al. (Ann Glob Anal Geom 43:47–62, 2013), Tang and Yan (Isoparametric foliation and a problem of Besse on generalizations of Einstein condition arXiv:1307.3807, 2013). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that all the focal submanifolds with g ≥ 3 are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds.     

Relative exchangeability with equivalence relations

We describe an Aldous–Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman (Representations of \(\text {Aut}(\mathcal {M})\)-invariant measures: part I, 2015. arXiv:1509.06170) and Crane and Towsner (Relatively exchangeable structures, 2015) and hierarchical exchangeability results due to Austin and Panchenko (Probab Theory Relat Fields 159(3–4):809–823, 2014).     

Remarks on curvature dimension conditions on graphs

We show a connection between the \(CDE'\) inequality introduced in Horn et al. (Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for nonnegative curvature graphs. arXiv:1411.5087v2, 2014) and the \(CD\psi \) inequality established in Münch (Li–Yau inequality on finite graphs via non-linear curvature dimension conditions. arXiv:1412.3340v1, 2014). In particular, we introduce a \(CD_\psi ^\varphi \) inequality as a slight generalization of \(CD\psi \) which turns out to be equivalent to \(CDE'\) with appropriate choices of \(\varphi \) and \(\psi \). We use this to prove that the \(CDE'\) inequality implies the classical CD inequality on graphs, and that the \(CDE'\) inequality with curvature bound zero holds on Ricci-flat graphs.     

The NLS Limit for Bosons in a Quantum Waveguide

We consider a system of N bosons confined to a thin waveguide, i.e. to a region of space within an \({\epsilon}\)-tube around a curve in \({\mathbb{R}^3}\). We show that when taking simultaneously the NLS limit \({N \to \infty}\) and the limit of strong confinement \({\epsilon \to 0}\), the time-evolution of such a system starting in a state close to a Bose–Einstein condensate is approximately captured by a non-linear Schrödinger equation in one dimension. The strength of the non-linearity in this Gross–Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the “bending” and the “twisting” of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl (On the time-dependent Gross–Pitaevskii-and Hartree equation. arXiv:0808.1178, 2008).     

Deformations of Charged Axially Symmetric Initial Data and the Mass–Angular Momentum–Charge Inequality

We show how to reduce the general formulation of the mass–angular momentum–charge inequality, for axisymmetric initial data of the Einstein–Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass, angular momentum, and charge. This extends previous work by the authors (Cha and Khuri, Ann Henri Poincaré, doi: 10.1007/s00023-014-0332-6, arXiv:1401.3384, 2014), in which the role of charge was omitted. Lastly, we improve upon the hypotheses required for the mass–angular momentum–charge inequality in the maximal case.     

Uniqueness and symmetry of ground states for higher-order equations

We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schrödinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for higher-order approximations to the pseudo-relativistic ground state. Our proof adapts the strategy of Lenzmann (Anal PDE 2:1–27, 2009) using local uniqueness near the limit of ground states in a variational problem. However, in order to bypass difficulties from lack of symmetrization tools for higher-order differential operators, we employ the contraction mapping argument in our earlier work (Choi et al. 2017. arXiv:1705.09068) to construct radially symmetric real-valued solutions, as well as improving local uniqueness near the limit.     

Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

Let \({\textnormal {R}}\) be a real closed field, \(\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]\) finite subsets of polynomials, with the degrees of the polynomials in \(\mathcal{P}\) (resp., \(\mathcal{Q}\)) bounded by d (resp., d ₀). Let \(V \subset {\textnormal {R}}^{k}\) be the real algebraic variety defined by the polynomials in \(\mathcal{Q}\) and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family \(\mathcal{P}\) on V is bounded by

$\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),$

where \(s = \operatorname {card}\mathcal{P}\), and

$F_{d,d_0,k,k'}(j)=\binom{k+1}{k-k'+j+1} (2d_0)^{k-k'}d^j \max\{2d_0,d \}^{k'-j}+2(k-j+1).$

In case 2d ₀≤d, the above bound can be written simply as

$\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k}= (sd)^{k'} d_0^{k-k'} O(1)^k$

(in this form the bound was suggested by Matousek 2011). Our result improves in certain cases (when d ₀?d) the best known bound of

$\sum_{1 \leq j \leq k'}\binom{s}{j} 4^{j} d(2d-1)^{k-1}$

on the same number proved in Basu et al. (Proc. Am. Math. Soc. 133(4):965–974, 2005) in the case d=d ₀.

The distinction between the bound d ₀ on the degrees of the polynomials defining the variety V and the bound d on the degrees of the polynomials in \(\mathcal{P}\) that appears in the new bound is motivated by several applications in discrete geometry (Guth and Katz in arXiv:1011.4105v1 [math.CO], 2011; Kaplan et al. in arXiv:1107.1077v1 [math.CO], 2011; Solymosi and Tao in arXiv:1103.2926v2 [math.CO], 2011; Zahl in arXiv:1104.4987v3 [math.CO], 2011).     

A motivic study of generalized Burniat surfaces

Generalized Burniat surfaces are surfaces of general type with \(p_g=q\) and Euler number \(e=6\) obtained by a variant of Inoue’s construction method for the classical Burniat surfaces. I prove a variant of the Bloch conjecture for these surfaces. The method applies also to the so-called Sicilian surfaces introduced by Bauer et al. in (J Math Sci Univ Tokyo 22(2–15):55–111, 2015. arXiv:1409.1285v2). This implies that the Chow motives of all of these surfaces are finite-dimensional in the sense of Kimura.     

Vanishing of Rabinowitz Floer homology on negative line bundles

Following Frauenfelder (Rabinowitz action functional on very negative line bundles, Habilitationsschrift, Munich/München, 2008), Albers and Frauenfelder (Bubbles and onis, 2014. arXiv:1412.4360) we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. Ritter (Adv Math 262:1035–1106, 2014) showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem \(\mathrm {SH}=0\Leftrightarrow \mathrm {RFH}=0\) (Ritter in J Topol 6(2):391–489, 2013), does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak–Frauenfelder–Oancea long exact sequence Cieliebak et al. (Ann Sci Éc Norm Supér (4) 43(6):957–1015, 2010).     

A binomial Laurent phenomenon algebra associated with the complete graph

In this paper, we find the exchange graph of \(\mathcal {A}({\tau _n})\), the rank n binomial Laurent phenomenon algebra associated with the complete graph \(K_n\). More specifically, we prove that the exchange graph is isomorphic to that of \(\mathcal {A}(t_n)\), a rank n linear Laurent phenomenon algebra associated with the complete graph which is discussed in Lam and Pylyavskyy (Linear Laurent phenomenon algebras, arXiv:1206.2612v2, 2012).     

Character sums with smooth numbers

We use the large sieve inequality for smooth numbers due to Drappeau et al. (Smooth-supported multiplicative functions in arithmetic progressions beyond the \(x^{1/2}\)-barrier, Preprint, 2017. arXiv:1704.04831), together with some other arguments, to improve their bounds on the frequency of pairs \((q,\chi )\) of moduli q and primitive characters \(\chi \) modulo q, for which the corresponding character sums with smooth numbers are large.     

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow-up scenario for the 3D Euler equation proposed by Hou and Luo (Multiscale Model Simul 12:1722–1776, 2014) based on extensive numerical simulations. These models generalize the 1D Hou–Luo model suggested in Hou and Luo Luo and Hou (2014), for which finite-time blowup has been established in Choi et al. (arXiv preprint. arXiv:1407.4776, 2014). The main new aspects of this work are twofold. First, we establish finite-time blowup for a model that is a closer approximation of the three-dimensional case than the original Hou–Luo model, in the sense that it contains relevant lower-order terms in the Biot–Savart law that have been discarded in Hou and Luo Choi et al. (2014). Secondly, we show that the blow-up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou–Luo model. Such blow-up stability result may be useful in further work on understanding the 3D hyperbolic blow-up scenario.     

Monodromy and K-theory of Schubert curves via generalized jeu de taquin

We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves \(S(\lambda _\bullet )\), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In Levinson (One-dimensional Schubert problems with respect to osculating flags, 2016, doi: 10.4153/CJM-2015-061-1), it was shown that the real geometry of these curves is described by the orbits of a map \(\omega \) on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of \({\mathbb {RP}}^1\), with \(\omega \) as the monodromy operator. We provide a fast, local algorithm for computing \(\omega \) without rectifying the skew tableau and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong’s genomic tableaux (Pechenik and Yong in Genomic tableaux, 2016. arXiv:1603.08490), which enumerate the K-theoretic Littlewood–Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of \(S(\lambda _\bullet )\).     

Comparison and vanishing theorems for Kähler manifolds

In this paper, we consider orthogonal Ricci curvature \(Ric^{\perp }\) for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kähler manifolds with \(Ric^{\perp }>0\) is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233).     

Finite time blow up for a solution to system of the drift–diffusion equations in higher dimensions

We discuss the existence of a blow-up solution for a multi-component parabolic–elliptic drift–diffusion model in higher space dimensions. We show that the local existence, uniqueness and well-posedness of a solution in the weighted \(L^2\) spaces. Moreover we prove that if the initial data satisfies certain conditions, then the corresponding solution blows up in a finite time. This is a system case for the blow up result of the chemotactic and drift–diffusion equation proved by Nagai (J Inequal Appl 6:37–55, 2001) and Nagai et al. (Hiroshima J Math 30:463–497, 2000) and gravitational interaction of particles by Biler (Colloq Math 68:229–239, 1995), Biler and Nadzieja (Colloq Math 66:319–334, 1994, Adv Differ Equ 3:177–197, 1998). We generalize the result in Kurokiba and Ogawa (Differ Integral Equ 16:427–452, 2003, Differ Integral Equ 28:441–472, 2015) and Kurokiba (Differ Integral Equ 27(5–6):425–446, 2014) for the multi-component problem and give a sufficient condition for the finite time blow up of the solution. The condition is different from the one obtained by Corrias et al. (Milan J Math 72:1–28, 2004).     

Arithmetic differential equations on $$GL_n$$, III Galois groups

Differential equations have arithmetic analogues (Buium in Arithmetic differential equations, Mathematical Surveys and Monographs, vol 118. American Mathematical Society, Providence 2005) in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations, and the present paper is concerned with the “linear” ones. The equations themselves were introduced in a previous paper (Buium and Dupuy, in Arithmetic differential equations on \(GL_{n}\), II: arithmetic Lie–Cartan theory, arXiv:1308.0744). In the present paper we deal with the solutions of these equations as well as with the Galois groups attached to the solutions.