Showing papers in "International Mathematics Research Notices in 2015"

Bounding the Smallest Singular Value of a Random Matrix Without Concentration

Abstract: Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N e_i$ be the matrix whose rows are $\frac{X_1}{\sqrt{N}},\dots, \frac{X_N}{\sqrt{N}}$. We obtain sharp probabilistic lower bounds on the smallest singular value $\lambda_{\min}(\Gamma)$ in a rather general situation, and in particular, under the assumption that $X$ is an isotropic random vector for which $\sup_{t\in S^{n-1}}{\mathbb{E}}| |^{2+\eta} \leq L$ for some $L,\eta>0$. Our results imply that a Bai-Yin type lower bound holds for $\eta>2$, and, up to a log-factor, for $\eta=2$ as well. The bounds hold without any additional assumptions on the Euclidean norm $\|X\|_{\ell_2^n}$. Moreover, we establish a nontrivial lower bound even without any higher moment assumptions (corresponding to the case $\eta=0$), if the linear forms satisfy a weak `small ball' property.

Associated Varieties of Modules Over Kac–Moody Algebras and C2-Cofiniteness of W-Algebras

Abstract: First, we establish the relation between the associated varieties of modules over Kac-Moody algebras \\hat{g} and those over affine W-algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact we show that the associated variates of G-integrable admissible representations are irreducible G-invariant subvarieties of the nullcone of g, by determining them explicitly. Third, we prove the C_2-cofiniteness of a large number of simple W-algebras, including all minimal series principal W-algebras and the exceptional W-algebras recently discovered by Kac-Wakimoto.

Categorical resolutions of irrational singularities

Abstract: We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity.

Local Universality of Zeroes of Random Polynomials

Abstract: In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n c_i \xi_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde \xi_i z^i$, where the $\xi_i$ and $\tilde \xi_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $\xi_i, \tilde \xi_i$, that the correlation functions of the zeroes of $f$ and $\tilde f$ are approximately the same. As an application, we give some answers to the classical question `"How many zeroes of a random polynomials are real?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions $f$ and $\tilde f$ if their log magnitudes $\log |f|, \log|\tilde f|$ are close in distribution, and if some non-concentration bounds are obeyed.

A Mean Curvature Type Flow in Space Forms

Abstract: In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounded convex domain in R n+1, the quermassintegrals evolve monotonically along the flow which allows us to prove a class of Alexandrov-Fenchel inequalities of quermassintegrals.

Uniqueness Theorems for Free Boundary Minimal Disks in Space Forms

Abstract: We show that a minimal disk satisfying the free boundary condition in a constant curvature ball of any dimension is totally geodesic. We weaken the condition to parallel mean curvature vector in which case we show that the disk lies in a three dimensional constant curvature submanifold and is totally umbilic. These results extend to higher dimensions earlier three dimensional work of J. C. C. Nitsche and R. Souam.

Small Ball Probabilities for Linear Images of High-Dimensional Distributions

Abstract: Author(s): Rudelson, Mark; Vershynin, Roman | Abstract: We study concentration properties of random vectors of the form $AX$, where $X = (X_1, ..., X_n)$ has independent coordinates and $A$ is a given matrix. We show that the distribution of $AX$ is well spread in space whenever the distributions of $X_i$ are well spread on the line. Specifically, assume that the probability that $X_i$ falls in any given interval of length $T$ is at most $p$. Then the probability that $AX$ falls in any given ball of radius $T \|A\|_{HS}$ is at most $(Cp)^{0.9 r(A)}$, where $r(A)$ denotes the stable rank of $A$ and $C$ is an absolute constant.

The Skeleton of the Jacobian, the Jacobian of the Skeleton, and Lifting Meromorphic Functions From Tropical to Algebraic Curves

Abstract: Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of these results, we deduce that \Lambda-rational principal divisors on \Gamma, in the sense of tropical geometry, are exactly the retractions of principal divisors on X. We actually prove a more precise result which says that, although zeros and poles of divisors can cancel under the retraction map, in order to lift a \Lambda-rational principal divisor on \Gamma to a principal divisor on X it is never necessary to add more than g extra zeros and g extra poles. Our results imply that a continuous function F:\Gamma -> R is the restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X if and only if F is a \Lambda-rational tropical meromorphic function, and we use this fact to prove that there is a rational map f : X --> P^3 whose tropicalization, when restricted to \Gamma, is an isometry onto its image.

Discrete Time q-TASEPs

Abstract: We introduce two new exactly solvable (stochastic) interacting particle systems which are discrete time versions of q-TASEP. We call these geometric and Bernoulli discrete time q-TASEP. We obtain concise formulas for expectations of a large enough class of observables of the systems to completely characterize their fixed time distributions when started from step initial condition. We then extract Fredholm determinant formulas for the marginal distribution of the location of any given particle. Underlying this work is the fact that these expectations solve closed systems of difference equations which can be rewritten as free evolution equations with k − 1t wobody boundary conditions—discrete q-deformed versions of the quantum delta Bose gas. These can be solved via a nested contour integral ansatz. The same solutions also arise in the study of Macdonald processes, and we show how the systems of equations our expectations solve are equivalent to certain commutation relations involving the Macdonald first difference operator.

The Elliptic Law

Abstract: A fundamental problem in random matrix theory is to determine the limiting distribution of the ESD as the size of the matrix tends to infinity. In certain cases when the entries have special distribution, such as Gaussian, the joint distribution of the eigenvalues can be given explicitly, and so the limiting distribution can be derived directly. However, these explicit formulas are not available for many random matrix ensembles, and so the problem of finding the limiting distribution becomes much more difficult. On the other hand, the well-known universality phenomenon in random matrix theory predicts that the limiting distribution should not depend on the distribution of the entries. We give two famous examples below. In the 1950s, Wigner studied the limiting ESD for a large class of random Hermitian matrices whose entries on or above the diagonal are independent [52]. In particular, Wigner showed that, under some additional moment

Rational curves on hyperkahler manifolds

Abstract: Let M be an irreducible holomorphically symplectic mani- fold. We show that all faces of the Kahler cone of M are hy- perplanes Hi orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kahler cone is a connected component of a complement of the positive cone to the union of all Hi. We provide several characterizations of the MBM-classes. We show the invari- ance of MBM property by deformations, as long as the class in question stays of type (1,1). For hyperkahler manifolds with Picard group generated by a negative class z, we prove that ±z is Q-effective if and only if it is an MBM class. We also prove some results towards the Morrison-Kawamata cone conjecture for hyperkahler manifolds.

Sharp hardy-littlewood-sobolev inequality on the upper half space

Abstract: There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent λ = n−α (that is for the case of α > n). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.

Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks

Abstract: The short-distance expansion of the tau function of the radial sine-Gordon/Painlev\\'e III equation is given by a convergent series which involves irregular $c=1$ conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.

Reversed Hardy–Littewood–Sobolev Inequality

Abstract: In this paper, we obtain a reversed Hardy-Littlewood-Sobolev in- equality: for 0 0, such that

-Algebras, False Theta Functions and Quantum Modular Forms, I

Abstract: In this paper, we study certain partial and false theta functions in connection to vertex operator algebras and conformal eld theory. We prove a variety of results concerning the asymptotics of modied characters of irreducible modules of certain W-algebras of singlet type, which allows us in particular to determine their (analytic) quantum dimensions. Our results are fully consistent with the previous conjectures on fusion rings for these vertex algebras. More importantly, we prove quantum modularity ( a la Zagier) of the numerator part of irreducible characters of singlet algebra modules, thus demonstrating that quantum modular forms naturally appear in many \suciently nice" irrational vertex algebras. It is interesting that quantum modularity persists on the whole set of rationals as in the original Zagier’s example coming from Kontsevich’s \strange series". In the last part, slightly independent of all this, we also discuss Nahm-type q-hypergometric series in connection to tails of colored Jones polynomials of certain torus knots and characters of modules for the (1;p)-singlet algebra.

Mod-φ Convergence

Abstract: Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.

Enumeration of Grothendieck's Dessins and KP Hierarchy

Abstract: Branched covers of the complex projective line ramified over $0,1$ and $\infty$ (Grothendieck's {\em dessins d'enfant}) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over $\infty$ and given numbers of preimages of $0$ and $1$ are considered. The generating function for the numbers of such covers is shown to satisfy a PDE that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the KP (Kadomtsev-Petviashvili) hierarchy. A specification of this generating function for certain values of parameters generates the numbers of {\em dessins} of given genus and degree, thus providing a fast algorithm for computing these numbers.

Stability Results for Logarithmic Sobolev and Gagliardo–Nirenberg Inequalities

Abstract: This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit stability measurements, i.e., bound with explicit constants distances to the manifold of optimal functions. Various results are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincare inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in diffusion equations (fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained.

Toric Actions on b-Symplectic Manifolds

Abstract: We study Hamiltonian actions on b-symplectic manifolds with a focus on the eective case of half the dimension of the manifold. In particular, we prove a Delzant-type theorem that classies these man- ifolds using polytopes that reside in a certain enlarged and decorated version of the dual of the Lie algebra of the torus. At the end of the paper we suggest further avenues of study, including an example of a toric action on a b 2 -manifold and applications of our ideas to integrable systems on b-manifolds.

Shimura Varieties in the Torelli Locus via Galois Coverings

Abstract: Given a family of Galois coverings of the projective line we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety in A_g. By a computer program we get the list of all families in genus up to 8 satisfying our condition. There is no family in genus 8, all of them are in genus at most 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples found by Moonen-Oort. We get 7 new non-abelian examples.

Bounded Negativity and Arrangements of Lines

Abstract: This thesis consists of six papers in algebraic geometry –all of which have close connections to combinatorics. In Paper A we consider complete smooth toric embeddings X ↪ P^N such that for a fixed positive integer k the t-th osculating space at every point has maximal dimension if and only if t ≤ k. Our main result is that this assumption is equivalent to that X ↪ P^N is associated to a Cayley polytope of order k having every edge of length at least k. This result generalizes an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalizing a result of Atsushi Ito. In Paper B we introduce H-constants that measure the negativity of curves on blow-ups of surfaces. We relate these constants to the bounded negativity conjecture. Moreover we provide bounds on H-constants when restricting to curves which are a union of lines in the real or complex projective plane. In Paper C we study Gauss maps of order k for k > 1, which maps a point on a variety to its k-th osculating space at that point. Our main result is that as in the case k = 1, the higher order Gauss maps are finite on smooth varieties whose k-th osculating space is full-dimensional everywhere. Furthermore we provide convex geometric descriptions of these maps in the toric setting. In Paper D we classify fat point schemes on Hirzebruch surfaces whose initial sequence are of maximal or close to maximal length. The initial degree and initial sequence of such schemes are closely related to the famous Nagata conjecture. In Paper E we introduce the package LatticePolytopes for Macaulay2. The package extends the functionality of Macaulay2 for compuations in toric geometry and convex geometry. In Paper F we compute the Seshadri constant at a general point on smooth toric surfaces satisfying certain convex geometric assumptions on the associated polygons. Our computations relate the Seshadri constant at the general point with the jet seperation and unnormalised spectral values of the surfaces at hand.

Movable Curves and Semistable Sheaves

Abstract: Daniel Greb1, Stefan Kebekus2, and Thomas Peternell3 1Essener Seminar fur Algebraische Geometrie und Arithmetik, Fakultat fur Mathematik, Universitat Duisburg-Essen, 45117 Essen, Germany, 2Mathematisches Institut, Albert-Ludwigs-Universitat Freiburg, Eckerstrase 1, 79104 Freiburg im Breisgau, Germany and University of Strasbourg Institute for Advanced Study (USIAS), Strasbourg, France, and 3Mathematisches Institut, Universitat Bayreuth, 95440 Bayreuth, Germany

Mixing Properties and the Chromatic Number of Ramanujan Complexes

Abstract: Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application we prove that non-partite Ramanujan complexes have ’high girth’ and high chromatic number, generalizing a well known result about Ramanujan graphs.

A Generalization of Reilly's Formula and its Applications to a New Heintze–Karcher Type Inequality

Abstract: In this paper, we prove a generalization of Reilly’s formula in [10]. We apply such general Reilly’s formula to give alternative proofs of the Alexandrov’s Theorem and the Heintze-Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly’s formula to prove a new Heintze-Karcher inequality for Riemannian manifolds with boundary and sectional curvature bounded below.

Large-N Asymptotic Expansion for Mean Field Models with Coulomb Gas Interaction

Abstract: Gaetan Borot123 and Alice Guionnet3 and Karol K. Kozlowski4 1 Section de Mathematiques, Universite de Geneve, 2-4 rue du Lievre, 1211 Geneve 4, Switzerland. 2 Max Planck Institut fur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. 3 Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA. 4 Institut de Mathematiques de Bourgogne, CNRS UMR 5584, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France.

Flow Polytopes of Signed Graphs and the Kostant Partition Function

Abstract: We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an interesting special family of flow polytopes, we study the Chan–Robbins–Yuen (CRY) polytopes. Motivated by the volume formula $\Pi^{n-2}_{k=1}\ {\mathrm Cat}(k)$ for the type A n version, where Cat(k) is the kth Catalan number, we introduce type C n+1 and D n+1 CRY polytopes along with intriguing conjectures about their volumes.

Sign of green's function of paneitz operators and the q curvature

Abstract: In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities between the Green's functions of the conformal Laplacian operator and the Paneitz operator.