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Asymptotics of linear systems, with connections to line arrangementsMay 28 2017The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded negativity, and ... More

Regina Lectures on Fat PointsOct 14 2013These notes are a record of lectures given in the Workshop on Connections Between Algebra and Geometry at the University of Regina, May 29--June 1, 2012. The lectures were meant as an introduction to current research problems related to fat points for ... More

The resurgence of ideals of points and the containment problemJun 24 2009Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our previous paper ... More

Comparing powers and symbolic powers of idealsJun 25 2007Jun 24 2009We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure ... More

Containment results for ideals of various configurations of points in P^NSep 09 2011Jun 16 2013Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures ... More

Symbolic powers versus regular powers of ideals of general points in P^1 x P^1Jul 25 2011Feb 22 2012Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve asymptotic numerical ... More

Asymptotic resurgences for ideals of positive dimensional subschemes of projective spaceFeb 20 2012Mar 01 2012Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne defined a quantity called the resurgence to address this ... More

Free Resolutions of Fat Point Ideals on $P^2$Sep 01 1995Jun 18 1998By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme defined by a sheaf of integrally closed ideals one extends the notion of fat point subschemes to allow infinitely near points. With this notion of fat points, this preprint determines ... More

Seshadri constants and very ample divisors on algebraic surfacesMar 05 2001Apr 04 2001A broadly applicable geometric approach for constructing nef divisors on blow ups of algebraic surfaces at n general points is given; it works for all surfaces in all characteristics for any n. This construction is used to obtain substantial improvements ... More

Global aspects of the geometry of surfacesJul 23 2009Jul 30 2009These notes (prepared for the author's lectures at the Cracow Summer School on Linear Systems organized by S. Mueller-Stach and T. Szemberg, held March 23-27, 2009 at the Pedagogical University of Cracow under the sponsorship of the Deutsche Forschungsgemeinschaft) ... More

The Ideal Generation Problem for Fat PointsMar 27 1997This paper is concerned with determining the number of generators in each degree for minimal sets of homogeneous generators for saturated ideals defining fat point subschemes $Z=m_1p_1+ ... +m_rp_r$ for general sets of points $p_i$ of $P^2$. For thin ... More

Problems and progress: survey on fat points in P2Jan 12 2001This paper surveys certain problems involving numerical characters for ideals I(Z) defining fat points subschemes $Z=m_1p_1+...+m_np_n$ for general points $p_i\in {\bf P}^2$. It also presents some new results, and includes a suite of MACAULAY 2 scripts ... More

An Algorithm for Fat Points on P2Mar 26 1998Sep 16 1999Let $F$ be a line bundle on the blow-up $X$ of $P^2$ at $r$ general points $p_1, ..., p_r$ and let $L$ be the pullback to $X$ of the line bundle coming from a line on $P^2$. Under reasonable hypotheses that are conjectured always to hold if the points ... More

On Nagata's ConjectureSep 16 1999Modifying an approach of J. Roe, this paper gives an improved lower bound on the degrees d such that for general points p1,...,pn in P2 and m > 0 there is a plane curve of degree d vanishing at each point pi with multiplicity at least m. In certain cases, ... More

Anticanonical Rational SurfacesSep 01 1995Jan 24 1996A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically ... More

Generators for Symbolic Powers of Ideals Defining General Points of $P^2$Sep 01 1995Given distinct points $p_1,\cdots,p_r$ of the projective plane $P^2$ and a positive integer $m$, the homogeneous ideal defining the fat point subscheme $Z=m(p_1+\cdots+p_r)$ is the symbolic power $I^{(m)}$ of the homogeneous ideal $I$ defining the smooth ... More

The (unexpected) importance of knowing $α$Feb 19 2005In order to determine the Hilbert function of the ideal of a fat point subscheme of projective space, we show that it is enough to determine, both for the subscheme itself and the subschemes obtained from it by successively adjoining to it additional ... More

Single point Seshadri constants on rational surfacesJun 06 2017Dec 15 2017Motivated by a similar result of Dumnicki, K\"uronya, Maclean and Szemberg under a slightly stronger hypothesis, we exhibit irrational single-point Seshadri constants on a rational surface $X$ obtained by blowing up very general points of $\mathbb{P}^2_\mathbb{C}$, ... More

Are symbolic powers highly evolved?Mar 30 2011Sep 09 2011Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in projective N-space, we make conjectures which explain them, and we prove the conjectures in certain ... More

Real and complex supersolvable line arrangements in the projective planeJul 17 2019We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil) complex line ... More

A primer on Seshadri constantsOct 04 2008Jul 01 2010Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of ... More

Discrete behavior of Seshadri constants on surfacesSep 25 2007Working over C, we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constant for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is ... More

Linear systems with multiple base points in P2Jan 12 2001Mar 01 2003Given positive integers $m_1, m_2, ..., m_n$, and $n$ general points $p_i$ of ${\bf CP}^2$, bounds are given for the least degree $t$ among plane curves passing through each point $p_i$ with multiplicity at least $m_i$, and for the least $t$ such that ... More

Containment Counterexamples for ideals of various configurations of points in ${\bf P}^N$Jun 16 2013When $I$ is the radical homogeneous ideal of a finite set of points in projective $N$-space, ${\bf P}^N$, over a field $K$, it has been conjectured that $I^{(rN-N+1)}$ should be contained in $I^r$ for all $r\geq 1$. Recent counterexamples show that this ... More

Hilbert functions and resolutions for ideals of n = s^2 fat points in P2Apr 26 2001Conjectures for the Hilbert function of the m-th symbolic power of the ideal of n general points of P2 are verified for infinitely many m for each square n > 9, using an approach developed by the authors in a previous paper. In those cases that n is even, ... More

The Waldschmidt constant for squarefree monomial idealsAug 03 2015May 26 2016Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehat\alpha(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By ... More

Resolutions of Ideals of Uniform Fat Point Subschemes of P^2Jun 18 1999Let I be the ideal corresponding to a set of general points $p_1,...,p_n \in P^2$. There recently has been progress in showing that a naive lower bound for the Hilbert functions of symbolic powers $I^{(m)}$ is in fact attained when n>9. Here, for m sufficiently ... More

New constructions of unexpected hypersurfaces in $\mathbb{P}^n$Apr 05 2019In the paper we present new examples of unexpected varieties. The research on unexpected varieties started with a paper of Cook II, Harbourne, Migliore and Nagel and was continued in the paper of Harbourne, Migliore, Nagel and Teitler. Here we present ... More

On integral Zariski decompositions of pseudoeffective divisors on algebraic surfacesJul 27 2015In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be $(-1)$-curves, ... More

Resolutions of ideals of fat points with support in a hyperplaneFeb 19 2005Jul 11 2005Our results concern minimal graded free resolutions of fat point ideals for points in a hyperplane. Suppose, for example, that I(m,d) is the ideal defining r given points of multiplicity m in the projective space P^d. Assume that the given points lie ... More

The role of the cotangent bundle in resolving ideals of fat points in the planeJun 14 2007We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, ... More

Combinatorial bounds on Hilbert functions of fat points in projective spaceDec 10 2009Dec 11 2010We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in projective space, in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more ... More

Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz propertyAug 13 2010Jul 04 2011Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x_1,x_2,x_3], ... More

Betti numbers for fat point ideals in the plane: a geometric approachJun 18 2007We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of ... More

Inclics, galaxies, star configurations and Waldschmidt constantsApr 08 2013Jun 16 2013This paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). As examples, we study galaxies (these are constructed starting with a star configuration to which we add general points in a larger projective ... More

Fat lines in P^3: powers versus symbolic powersAug 26 2012We study the symbolic and regular powers of ideals I for a family of special configurations of lines in P^3. For this family, we show that I^(m) = I^m for all integers m if and only if I^(3) = I^3. We use these configurations to answer a question of Huneke ... More

On plane rational curves and the splitting of the tangent bundleFeb 05 2011Jun 16 2013Given an immersion $\phi: P^1 \to \P^2$, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as $\phi: P^1 \cong D \subset X \to P^2$, where ... More

Resolutions of fat point ideals involving 8 general points of P2Jan 12 2001The main result provides an algorithm for determining the minimal free resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8 general points with arbitrary multiplicities; the results hold over algebraically closed fields of any ... More

Line arrangements and configurations of points with an unusual geometric propertyFeb 06 2016Mar 02 2017The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in $\mathbb P^2$ of fixed degree $d$, and it is known to be true in many cases. We propose a new ... More

Very general monomial valuations of $\mathbb{P}^2$ and a Nagata type conjectureDec 19 2013Feb 05 2016It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\geq 9$ points. Tackling the problem in the language ... More

Bounded volume denominators and bounded negativityApr 16 2019In this paper we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections ... More

The Halphen cubics of order twoMar 14 2016For each $m\ge 1$, Roulleau and Urz\'ua give an implicit construction of a configuration of $4(3m^2-1)$ complex plane cubic curves. This construction was crucial for their work on surfaces of general type. We make this construction explicit by proving ... More

Line arrangements and configurations of points with an unusual geometric propertyFeb 06 2016The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in $\mathbb P^2$ of fixed degree $d$, and it is known to be true in many cases. We propose a new ... More

Linear subspaces, symbolic powers and Nagata type conjecturesJul 05 2012Sep 30 2012Prompted by results of Guardo, Van Tuyl and the second author for lines in projective 3 space, we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in projective ... More

Recent developments and open problems in linear seriesJan 23 2011Apr 16 2011In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, ... More

Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constantsSep 27 2016The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity ... More

Bounded Negativity and Arrangements of LinesJul 10 2014Nov 05 2014The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a lower bound is ... More

Resurgences for ideals of special point configurations in ${\bf P}^N$ coming from hyperplane arrangementsApr 19 2014Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers. Several invariants have been introduced and ... More

A matrixwise approach to unexpected hypersurfacesJul 10 2019The aim of this note is to give a generalization of some results concerning unexpected hypersurfaces. Unexpected hypersurfaces occur when the actual dimension of the space of forms satisfying certain vanishing data is positive and the imposed vanishing ... More

Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constantsSep 27 2016Oct 10 2017The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity ... More

Resolutions of ideals of six fat points in P^2Jun 30 2005Apr 13 2012The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P^2 are determined whenever Z is supported at any 6 or fewer distinct points. All results hold over an algebraically ... More

Computing multi-point Seshadri constants on P2Sep 04 2003Sep 26 2007Working over the complex field and formalizing and sharpening approaches introduced by several authors, we give a method for verifying when a divisor on a blow up of P^2 at general points is nef. The method is useful both theoretically and when doing ... More

Configuration types and cubic surfacesApr 13 2012This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers ... More

Negative curves on algebraic surfacesSep 09 2011Apr 04 2012We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of surfaces for which ... More

The Age of the UniverseMay 16 1996Globular clusters are the oldest objects in the Galaxy whose age may be accurately determined. As such globular cluster ages provide the best estimate for the age of the universe. The age of a globular cluster is determined by a comparison between theoretical ... More

Halo Star EvolutionSep 08 1995In this review, I will discuss a few problems which point to the need for improved stellar evolution models of halo stars. Current stellar evolution models do not match the observed $^7$Li abundance patterns, suggesting that the input physics and/or the ... More

Deviations from the Circular LawDec 01 2003Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the eigenvalues converges ... More

Multi-Epoch Observations of the Redwing Excess in the Spectrum of 3C279Nov 12 2012It has been previously determined that there is a highly significant correlation between the spectral index from 10 GHz to 1350 $\AA$ and the amount of excess luminosity in the red wing of quasar CIV $\lambda$1549 broad emission lines (BELs). Ostensibly, ... More

Models of the Compact Jet in GRS 1915+105Aug 23 2011In this article, models are constructed of the compact jet in GRS 1915+105 during an epoch of optimal data capture. On April 02, 2003, the object was observed in the hard X-ray/soft gamma ray band (INTEGRAL), hard X-ray band (RXTE), near IR (ESO/New Technology ... More

High Jet Efficiency and Simulations of Black Hole MagnetospheresDec 09 2010This article reports on a growing body of observational evidence that many powerful lobe dominated (FR II) radio sources likely have jets with high efficiency. This study extends the maximum efficiency line (jet power $\approx$ 25 times the thermal luminosity) ... More

X-ray Absorption in Type II Quasars: Implications for the Equatorial Paradigm of Broad Absorption Line QuasarsMay 01 2006In this article, the hydrogen column densities derived from X-ray observations of type II (hidden) quasars and broad absorption line quasars (BALQSOs) are compared. These column densities represent the amount of absorbing material between the X-ray source ... More

Improved Online Square-into-Square PackingJan 22 2014In this paper, we show an improved bound and new algorithm for the online square-into-square packing problem. This two-dimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. ... More

Singular differential equationsSep 22 2015Jan 18 2016The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential ... More

Regularity of minimal submanifolds and mean curvature flows with a common free boundarySep 26 2016Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,\mu}$ $n$-dimensional submanifolds-with-boundary $M_k \subset N$ with a common boundary $\Gamma$, then $\Gamma$ is smooth ... More

First betti numbers of Kähler manifolds with weakly pseudoconvex boundaryOct 20 2011We study K\"ahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), it has first betti number at least ... More

ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic SpaceOct 04 2016In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces ... More

Spline-Based Probability CalibrationSep 20 2018In many classification problems it is desirable to output well-calibrated probabilities on the different classes. We propose a robust, non-parametric method of calibrating probabilities called SplineCalib that utilizes smoothing splines to determine a ... More

Decompositions of the tensor products of irreducible sl(2)-modules in characteristic 3Sep 22 2008We completely describe the decompositions (into indecomposable submodules) of the tensor products of irreducible sl(2)-modules in characteristic 3. The answer resembles analogous decompositions for the Lie superalgebra sl(1|1).

On Lattice Barycentric TetrahedraJan 05 2004We say a lattice tetrahedron whose centroid is its only non-vertex lattice point is lattice barycentric. The notation T(a,b,c) describes the lattice tetrahedron with vertices {0, e_1, e_2, a e_1 + b e_2 + c e_3}. Our result is that all such T(a,b,c) are ... More

Association schemes, classical RCFT's, and centres of monoidal functor categoriesFeb 08 2007Aug 12 2011Here we describe three straightforward examples of what was called a graphic Fourier transformation in [4]. At least two of these examples may be viewed simply as monoidal comonads on suitable monoidal closed functor categories, but the third example, ... More

Applying Distributional Compositional Categorical Models of Meaning to Language TranslationNov 08 2018The aim of this paper is twofold: first we will use vector space distributional compositional categorical models of meaning to compare the meaning of sentences in Irish and in English (and thus ascertain when a sentence is the translation of another sentence) ... More

The $\square_b$ Heat Equation and Multipliers via the Wave EquationMay 09 2008Sep 10 2008Recently, Nagel and Stein studied the $\square_b$-heat equation, where $\square_b$ is the Kohn Laplacian on the boundary of a weakly-pseudoconvex domain of finite type in $\C^2$. They showed that the Schwartz kernel of $e^{-t\square_b}$ satisfies good ... More

Complete intersection P-partition ringsFeb 22 2018We present an alternate proof of a result of F\'eray and Reiner characterizing posets whose $P$-partition rings are complete intersections. This shortened proof relates the complete intersection property to a simple structural property of a graph associated ... More

Blow-up in the Parabolic Scalar Curvature EquationMay 25 2007Jun 05 2012The \textit{parabolic scalar curvature equation} is a reaction-diffusion type equation on an $(n-1)$-manifold $\Sigma$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\times\Sigma$ and a family of metrics $\gamma(r)$ ... More

Differential Equations with a Difference QuotientSep 22 2015Jan 02 2017The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential ... More

Generalized Kahler Taub-NUTs and Two Exceptional InstantonsFeb 19 2016Jun 17 2019We study the one-parameter family of twisted Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extend that should be sufficient for blow-up and gluing arguments. In particular ... More

Subdivision rules and the eight model geometriesJul 23 2012Cannon and Swenson have shown that each hyperbolic 3-manifold group has a natural subdivision rule on the space at infinity, and that this subdivision rule captures the action of the group on the sphere. Explicit subdivision rules have also been found ... More

Sharp Regularity for the Integrability of Elliptic StructuresOct 23 2018Nov 18 2018As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$ and $n$) in ... More

Limit linear series moduli stacks in higher rankMay 12 2014In order to prove new existence results in Brill-Noether theory for rank-2 vector bundles with fixed special determinant, we develop foundational definitions and results for limit linear series of higher-rank vector bundles. These include two entirely ... More

Extending Landau's Theorem on Dirichlet Series with Non-Negative CoefficientsSep 01 2010A classical theorem of Landau states that, if an ordinary Dirichlet series has non-negative coefficients, then it has a singularity on the real line at its abscissae of absolute convergence. In this article, we relax the condition on the coefficients ... More

Compact Embedded Minimal Surfaces of Positive Genus Without Area BoundsAug 22 2003Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily large area. This ... More

Integral-Input-Output to State StabilityJun 15 2001A notion of detectability for nonlinear systems is discussed. Within the framework of ``input to state stability'' (ISS), a dual notion of ``output to state stability'' (OSS), and a more complete detectability notion, ``input-output to state stability'' ... More

Topological Change in Mean Convex Mean Curvature FlowJul 23 2011Oct 28 2013Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking ... More

Deformations of covers, Brill-Noether theory, and wild ramificationDec 01 2004In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well ... More

Isometries of optimal pseudo-Riemannian metricsApr 26 2011Apr 29 2011We give a concise proof that large classes of optimal (constant curvature or Einstein) pseudo-Riemannian metrics are maximally symmetric within their conformal class.

Maximal Function Inequalities and a Theorem of BirchNov 12 2017Dec 05 2017In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous polynomial in $n$ ... More

Geometric Structures on Spaces of Weighted SubmanifoldsNov 02 2009In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold $(M,\omega)$, we construct ... More

*-Autonomous categories in quantum theoryMay 01 2006Nov 26 2011*-Autonomous categories were initially defined by M. Barr to describe a type of duality carried by many monoidal closed categories. Later they were generalised by the current author to include *-autonomous promonoidal categories. Together, these structures ... More

Localization on three-dimensional manifoldsAug 09 2016Oct 15 2016In this review article we describe the localization of three dimensional N=2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, S^3_b/Z_p, and S^2 x S^1. We describe how to write supersymmetric actions ... More

Absolute Ages of Globular Clusters and the Age of the UniverseDec 06 1994The main sequence turnoff luminosity is the best stellar `clock' which can be used to determine the absolute ages of globular clusters. This is due to the fact that it is generally assumed that the luminosity and lifetimes of main sequence globular cluster ... More

Mathematical toy model inspired by the problem of the adaptive origins of the sexual orientation continuumNov 11 2015Sep 15 2016Same-sex sexual behavior is ubiquitous in the animal kingdom, but its adaptive origins remain a prominent puzzle. Here I suggest the possibility that same-sex sexual behavior arises as a consequence of the competition between an evolutionary drive for ... More

Localization on three-dimensional manifoldsAug 09 2016Aug 14 2016In this review article we describe the localization of three dimensional N=2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, S^3_b/Z_p, and S^2 x S^1. We describe how to write supersymmetric actions ... More

Experimental signatures of 3d fractional topological insulatorsMay 09 2012In this work we explore experimental signatures of fractional topological insulators in three dimensions. These are states of matter with a fully gapped bulk that host exotic gapless surface states and fractionally charged quasiparticles. They are partially ... More

Renyi entropy, mutual information, and fluctuation properties of Fermi liquidsJul 27 2010I compute the leading contribution to the ground state Renyi entropy $S_{\alpha}$ for a region of linear size $L$ in a Fermi liquid. The result contains a universal boundary law violating term simply related the more familiar entanglement entropy. I also ... More

Entanglement Entropy at Finite Density from Extremal Black HolesAug 12 2009Aug 19 2009I compute the entanglement entropy of a strongly coupled 2+1d quantum field theory containing fermions at finite density using gauge/gravity duality. The dual geometry is an extremal black hole in 3+1d Einstein-Maxwell theory. This system was recently ... More

Interplay between short and long-range entanglement in symmetry protected phasesSep 04 2012We study a variety of questions related to entanglement in symmetry protected phases, especially those introduced in arXiv:1106.4772 (Chen et al., 2011). These phases are analogous to topological insulators in that they are short range entangled states ... More

Entanglement Renormalization and HolographyMay 08 2009I show how recent progress in real space renormalization group methods can be used to define a generalized notion of holography inspired by holographic dualities in quantum gravity. The generalization is based upon organizing information in a quantum ... More

Longitudinal Spin Physics at RHIC and a Future eRHICSep 30 2015Recent highlights from the spin program at the Relativistic Heavy Ion Collider (RHIC), focusing on the gluon contribution to the proton spin and the polarization of the light flavor sea, are presented. The impact of these data on recent global fits by ... More

Hierarchical Clustering using Randomly Selected SimilaritiesJul 19 2012The problem of hierarchical clustering items from pairwise similarities is found across various scientific disciplines, from biology to networking. Often, applications of clustering techniques are limited by the cost of obtaining similarities between ... More

External Sources of Poynting Flux in MHD Simulations of Black Hole ErgospheresOct 31 2005This article investigates the physics that is responsible for creating the outgoing Poynting flux emanating from the ergosphere of a rotating black hole in the limit that the magnetic energy density greatly exceeds the plasma rest mass density (magnetically ... More