Large time behavior and Lyapunov functionals for a nonlocal differential equation

A new approach is used to describe the large time behavior of the nonlocal differential equation initially studied in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016). Our approach is based upon the existence of infinitely many Lyapunov functionals and allows us to extend the analysis performed in T.-N. Nguyen (On the \({\omega}\)-limit set of a nonlocal differential equation: application of rearrangement theory. Differ. Integr. Equ. arXiv:1601.06491, 2016).     

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).     

Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces

We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time-dependent energy functionals in both settings. In particular, in the case when each underlying space satisfies a lower Ricci curvature bound in the sense of Lott, Sturm and Villani, we provide time-discrete approximations of the time-dependent heat flows introduced in Kopfer and Sturm (Heat flows on time-dependent metric measure spaces and super-Ricci flows, 2017. arXiv:1611.02570).     

Remainder Estimates for the Long Range Behavior of the van der Waals Interaction Energy

The van der Waals–London’s law, for a collection of atoms at large separation, states that their interaction energy is pairwise attractive and decays proportionally to one over their distance to the sixth. The first rigorous result in this direction was obtained by Lieb and Thirring (Phys Rev A 34(1):40–46, 1986), by proving an upper bound which confirms this law. Recently the van der Waals–London’s law was proven under some assumptions by Anapolitanos and Sigal (arXiv:1205.4652v2). Following the strategy of Anapolitanos and Sigal (arXiv:1205.4652v2) and reworking the approach appropriately, we prove estimates on the remainder of the interaction energy. Furthermore, using an appropriate test function, we prove an upper bound for the interaction energy, which is sharp to leading order. For the upper bound, our assumptions are weaker, the remainder estimates stronger and the proof is simpler. The upper bound, for the cases it applies, improves considerably the upper bound of Lieb and Thirring. Their bound holds in a much more general setting, however. Here we consider only spinless Fermions.     

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).     

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.     

Instanton Floer homology and contact structures

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or without boundary—defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by Baldwin and Sivek (arXiv:1403.1930, 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.     

Complex Outliers of Hermitian Random Matrices

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in Benaych-Georges and Rochet (Probab Theory Relat Fields, 2015), as well as possible correlations between outliers at macroscopic distance as in Knowles and Yin (Ann Probab 42(5):1980–2031, 2014) and Benaych-Georges and Rochet (2015). We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in Benaych-Georges and Nadakuditi (Adv Math 227(1):494–521, 2011) and Belinschi et al. (Outliers in the spectrum of large deformed unitarily invariant models 2012, arXiv:1207.5443v1). In the particular case where the perturbation matrix is Hermitian, our results complete the work of Benaych-Georges et al. (Electron J Probab 16(60):1621–1662, 2011), as we consider fluctuations of outliers lying in “holes” of the limit support, which happen to exhibit surprising correlations.     

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.     

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.     

Congruences between word length statistics for the finitary alternating and symmetric groups

Bacher and de la Harpe (arxiv:1603.07943, 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and \(p(n)_\mathbf{e }\), a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe (arxiv:1603.07943, 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function \(p_{k}(n)\), we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes \(\ell \ge 5\).     

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).     

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.     

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.     

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.     

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.