Results for "Scott Manifold"
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Splitting the Curvature of the Determinant Line BundleDec 21 1998It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from ... More The Hilbert Schemes of Degree Three Curves are ConnectedMar 14 1996In this paper we show that the Hilbert scheme $H(3,g)$ of locally Cohen-Macaulay curves in $\Pthree$ of degree three and genus $g$ is connected. In contrast to $H(2,g)$, which is irreducible, $H(3,g)$ generally has many irreducible components (roughly ... More Oracles Are Subtle But Not MaliciousApr 12 2005Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from ... More Quantum Lower Bound for Recursive Fourier SamplingSep 09 2002Dec 18 2004One of the earliest quantum algorithms was discovered by Bernstein and Vazirani, for a problem called Recursive Fourier Sampling. This paper shows that the Bernstein-Vazirani algorithm is not far from optimal. The moral is that the need to "uncompute" ... More How much can we learn about the physics of inflation?Mar 25 2014Mar 28 2014The recent BICEP2 measurement of B-modes in the polarization of the cosmic microwave background suggests that inflation was driven by a field at an energy scale of $2\times 10^{16}$ GeV. I explore the potential of upcoming CMB polarization experiments ... More UV Perturbations in Brane Gas CosmologyFeb 02 2004We consider the effect of the ultraviolet (UV) or short wavelength modes on the background of Brane Gas Cosmology. We find that the string matter sources are negligible in the UV and that the evolution is given primarily by the dilaton perturbation. We ... More SU(3) Ghosts with SpinJun 15 2007A new Lorentz-covariant gauge is presented for SU(3). In this gauge, both the ghosts and the gauge fields in the (4, 5, 6, 7) gauge directions acquire half-integral spin. As a result, the ghosts in these directions have the correct relationship between ... More Symmetry Breaking in BRST Quantization of SU(3)Jan 21 2007New BRST-invariant states for SU(3) gauge field theory are presented. The states have finite norms and unlike the states that are usually used to derive path integrals, they break SU(3) symmetry by choosing preferred gauge directions. This symmetry breaking ... More New Ghost States in SU(3) Gauge Field TheoryDec 26 2006The ghost sector of SU(3) gauge field theory is studied, and new BRST-invariant states are presented that do not have any analog in other SU(N) field theories. The new states come in either ghost doublets or triplets, and they appear exclusively in SU(3) ... More PT-Invariance and Indefinite MetricOct 27 2009A new proof is given for why the non-Hermitian, PT-Invariant cubic oscillator with imaginary coupling has real eigenvalues. The proof consists of two steps. In the first step, it is shown that for many PT-Invariant Hamiltonians, one can define corresponding ... More Viscosity at RHIC: Theory and PracticeSep 01 2008Hydrodynamic behavior and the associated discussions of viscosity at RHIC has inspired a r enaissance in modeling viscous hydrodynamics. An explanation of Israel-Stewart hydrodynamics is presented here, with an emphasis on the tangible benefits compared ... More Formulating Viscous Hydrodynamics for Large Velocity GradientsNov 25 2007Viscous corrections to relativistic hydrodynamics, which are usually formulated for small velocity g radients, have recently been extended from Navier-Stokes formulations to a class of treatments based on Israel-Stewart equations. Israel-Stewart treatments, ... More The Long Slow Death of the HBT PuzzleDec 26 2008Femtoscopic measurements at RHIC have been hailed as a source of insight into the bulk properties of QCD matter. However, hydrodynamic models, which have been successful in reproducing other observables have failed to satisfactorily explain femtoscopic ... More Correlations and Fluctuations, A Summary of Quark Matter 2002Aug 06 2003Results for correlations and fluctuations presented at Quark Matter 2002 are summarized. These results include Hanbury-Brown Twiss interferometry of a wide variety of species, large scale fluctuations and correlations in $p_t$ and multiplicity, and charge ... More A family of critically finite maps with symmetryJul 03 2003May 17 2005The symmetric group S_n acts as a reflection group on CP^{n-2} (for $n\geq 3$) . Associated with each of the $\binom{n}{2}$ transpositions in S_n is an involution on CP^{n-2} that pointwise fixes a hyperplane--the mirrors of the action. For each such ... More A selective version of Lin's TheoremAug 14 2018Aug 24 2018We prove a selective version of Lin's Theorem for nearly commuting operators. This is accomplished by establishing the tracial stability of a certain class of graph products of $C^*$-algebras. This result involves the development of the "pincushion class" ... More Rank of Submatrices of the Pascal MatrixFeb 10 2017In a previous paper, we derived necessary and sufficient conditions for the invertibility of square submatrices of the Pascal upper triangular matrix. To do so, we established a connection with the two-point Birkhoff interpolation problem. In this paper, ... More Kergin Approximation in Banach SpacesOct 01 2008We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges. A note on Cauchy integrabilitySep 23 2014We show that for any bounded function $f:[a,b]\rightarrow{\mathbb R}$ and $\epsilon>0$ there is a partition $P$ of $[a,b]$ with respect to which the Riemann sum of $f$ using right endpoints is within $\epsilon$ of the upper Darboux sum of $f$. This leads ... More CP-odd Phases in Slepton Pair ProductionMar 23 1998Mar 27 1998The effects of CP-odd supersymmetric phases on slepton pair production are considered. It is shown that CP-even observables in $e^+ e^-$ and $e^- e^-$ collisions, such as the total selectron cross section, can depend on CP-odd supersymmetric phases through ... More The braid group surjects onto $G_2$ tensor spaceJul 01 2009Let V be the 7-dimensional irreducible representation of the quantum group U_q(g_2). For each n, there is a map from the braid group B_n to the endomorphism algebra of the n-th tensor power of V, given by R-matrices. We can extend this linearly to a map ... More Quasi-arithmeticity of lattices in PO(n,1)Dec 16 2014Jun 17 2015We show that the non-arithmetic lattices in PO(n,1) of Belolipetsky and Thomson (2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic ... More The hole probability for Gaussian random SU(2) polynomialsOct 23 2006We show that for Gaussian random SU(2)polynomials of a large degree $N$ the probability that there are no zeros in the disk of radius $r$ is less than $e^{-c_{1,r} N^2}$, and is also greater than $e^{-c_{2,r} N^2}$. Enroute to this result, we also derive ... More Encoding Data for HTM SystemsFeb 18 2016Hierarchical Temporal Memory (HTM) is a biologically inspired machine intelligence technology that mimics the architecture and processes of the neocortex. In this white paper we describe how to encode data as Sparse Distributed Representations (SDRs) ... More The evolution of galaxy formationDec 01 2011Our history of understanding galaxy formation could be traced through the development of individual ideas. A cynic might be tempted to suggest that new catchphrases are developed at a faster rate than genuine progress is made. CMB ANISOTROPIES: AN OVERVIEWFeb 02 1995A brief outline of the current status of CMB anisotropies and what they might mean, heavily biased towards the perspective of Berkeley theorists. Based on a talk presented at the 17th Texas Symposium on Relativistic Astrophysics held in Munich, December ... More Dynamics of a soccer ballApr 11 2014Exploiting the symmetry of the regular icosahedron, Peter Doyle and Curt McMullen constructed a solution to the quintic equation. Their algorithm relied on the dynamics of a certain icosahedral equivariant map for which the icosahedron's twenty face-centers--one ... More Solving the quintic by iteration in three dimensionsMar 09 1999Oct 06 1999The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the quintic, is that of the symmetric group S_5. Induced by its five-dimensional linear permutation representation is a three-dimensional projective action. ... More Relative Zeta Determinants and the Quillen MetricOct 27 1999We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over the boundary ... More The statistical mechanics of planet orbitsApr 05 2015Jun 10 2015The final "giant-impact" phase of terrestrial planet formation is believed to begin with a large number of planetary "embryos" on nearly circular, coplanar orbits. Mutual gravitational interactions gradually excite their eccentricities until their orbits ... More The Complexity of AgreementJun 30 2004A celebrated 1976 theorem of Aumann asserts that honest, rational Bayesian agents with common priors will never "agree to disagree": if their opinions about any topic are common knowledge, then those opinions must be equal. Economists have written numerous ... More Quantum Lower Bound for the Collision ProblemNov 20 2001The collision problem is to decide whether a function X:{1,..,n}->{1,..,n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Theta(n^{1/5}) on the number of queries needed by a quantum computer to solve this problem ... More The Equivalence of Sampling and SearchingSep 26 2010In a sampling problem, we are given an input x, and asked to sample approximately from a probability distribution D_x. In a search problem, we are given an input x, and asked to find a member of a nonempty set A_x with high probability. (An example is ... More Book Review: 'A New Kind of Science'Jun 13 2002Jul 30 2002This is a critical review of the book 'A New Kind of Science' by Stephen Wolfram. We do not attempt a chapter-by-chapter evaluation, but instead focus on two areas: computational complexity and fundamental physics. In complexity, we address some of the ... More A Fundamental Scale for Acceleration from the Holographic PrincipleMay 25 2005Jun 26 2005From the Eddington-Weinberg relationship, which may be explained by the holographic principle and the cosmic coincidence in a flat Universe, it follows that the characteristic gravitational acceleration aN associated with the nucleon and its Compton wavelength ... More An Inertial Reaction to Cosmological AccelerationsSep 01 2003May 17 2005Mach's "fixed stars" are actually not fixed at all. The distant clusters of galaxies are not only receding from each observer but they are also accelerating since the rate of cosmological expansion is not constant. If the distant cosmic masses in someway ... More Stars and the holographic upper bound on gravitational actionApr 14 2007Dec 22 2008The holographic upper bound on entropy is applied to the gravitational action associated with the non-relativistic contraction of a nebula. A critical radius is identified, as a function of the initial radius and mass, for which the number of bits associated ... More The Real Problem with MONDDec 06 2011Gravitational potentials in the cosmos are deeper than expected from observed visible objects, a phenomenon usually attributed to dark matter, presumably in the form of a new fundamental particle. Until such a particle is observed, the jury remains out ... More Cross-Correlating Probes of Primordial Gravitational WavesJan 27 2010One of the most promising ways of detecting primordial gravitational waves generated during inflation is to observe B-modes of polarization, generated by Thomson scattering after reionization, in the cosmic microwave background (CMB). Large scale foregrounds ... More Moduli Stabilization with the String Higgs EffectApr 23 2004May 02 2004We review the notion of the Higgs effect in the context of string theory. We find that by including this effect in time dependent backgrounds, one is led to a natural mechanism for stabilizing moduli at points of enhanced gauge symmetry. We consider this ... More An Exposition on Inflationary CosmologyApr 29 2000May 27 2000This paper is intended to offer a pedagogical treatment of cosmological modeling and inflationary cosmology. In recent years, inflation has become accepted as a standard scenario making predictions that are testable by observations of the cosmic background. ... More Szemerédi's Regularity Lemma for matrices and sparse graphsOct 04 2010Nov 08 2010Szemer\'edi's Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we ... More Universal Flow in the Early Stages at RHICMar 09 2009Pre-thermal flow plays an important role in the final state evolution of heavy ion collisions at RHIC. We show that flow has universal features for a wide range of models. This significantly reduces the uncertainty in initializing hydrodynamic models ... More Refractive Distortions of Two-Particle CorrelationsNov 02 2005Using optical model calculations it has recently been shown that refractive phenomena from the collective mean field can significantly alter the sizes inferred from two-pion correlations. We demonstrate that such effects can be accounted for in classical ... More Canonical and Microcanonical Distributions for Fermi SystemsMay 26 1999Oct 14 1999Recursion relations are presented that allow exact calculation of canonical and microcanonical partition functions of degenerate Fermi systems, assuming no explicit two-body interactions. Calculations of the level density, sorted by angular momentum, ... More Searches for Particle Dark Matter: An IntroductionOct 12 2011The identity of dark matter is one of the key outstanding problems in both particle and astrophysics. In this thesis, I describe a number of complementary searches for particle dark matter. I discuss how the impact of dark matter on stars can constrain ... More Classifying Spinor StructuresJun 10 2001I begin by explaining how Riemannian geometry can be understood in terms of principal fibre bundles and connections thereon. I then introduce and motivate the definition of a spinor structure in terms of familiar geometrical ideas. The central result ... More Vogan Duality for ~Spin(p,q)Aug 13 2009The main purpose of this paper is to describe a symmetry in the set genuine parameters for even rank nonlinear Spin groups in type B at certain half-integral infinitesimal characters. This symmetry is used to establish a duality of the corresponding generalized ... More Pricing for Large Positions in Contingent ClaimsFeb 17 2012Dec 11 2013Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models. It is shown that as the position size approaches ... More Supersymmetry in the Very Early UniverseJun 16 1995Jun 24 1995Supersymmetric flat directions can have a number of important consequences in the very early universe. Depending on the form of the SUSY breaking potential arising from the finite energy density at early times, coherent production of scalar condensates ... More Electromagnetic Contributions to the Schiff MomentFeb 07 1994The Schiff moment, $\smij$, is a parity and time reversal violating fermion-fermion coupling. The nucleus-electron Schiff moment generically gives the most important contribution to the electric dipole moments of atoms and molecules with zero net intrinsic ... More Elementary treatment of $p^a \pm p^b + 1 = x^2$Aug 31 2006We give a shorter simpler proof of a result of Szalay on the equation $2^a + 2^b + 1 = x^2$. We give an elementary proof of a result of Luca on the equation of the title for prime $p > 2$. The elementary treatment is made possible by a lemma which is ... More Augmented Homotopical Algebraic GeometryNov 07 2017We develop the framework for augmented homotopical algebraic geometry. This is an extension of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. To do so, we define the notion of augmentation categories, ... More Torus Invariant CurvesApr 13 2013Jul 30 2013Using the language of T-varieties, we study torus invariant curves on a complete normal variety $X$ with an effective codimension-one torus action. In the same way that the $T$-invariant Weil divisors on $X$ are sums of "vertical" divisors and "horizontal" ... More 51 constructions of the Moonshine moduleJul 10 2017Jul 18 2017We show using Borcherds products that for any fixed-point free automorphism of the Leech lattice satisfying a "no massless states" condition, the corresponding cyclic orbifold of the Leech lattice vertex operator algebra is isomorphic to the Monster vertex ... More Generalized Moonshine IV: Monstrous Lie algebrasAug 30 2012Aug 20 2016For each element of the Fischer-Griess Monster sporadic simple group, we construct an infinite dimensional Lie algebra equipped with a projective action of the centralizer of that element. Our construction is given by a string-theoretic "add a spacetime ... More Sobolev extensions of Lipschitz mappings into metric spacesAug 02 2016Aug 02 2017Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the Lipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map from a subset of $\mathbb{R}^m$ ... More Supercharacters of unipotent groups defined by involutionsNov 07 2013Dec 15 2014We construct supercharacter theories of finite unipotent groups in the orthogonal, symplectic and unitary types. Our method utilizes group actions in a manner analogous to that of Diaconis and Isaacs in their construction of supercharacters of algebra ... More The odd couple: quasars and black holesJul 23 2014Quasars emit more energy than any other objects in the universe, yet are not much bigger than the solar system. We are almost certain that quasars are powered by giant black holes of up to $10^{10}$ times the mass of the Sun, and that black holes of between ... More On the origin of irregular structure in Saturn's ringsNov 07 2002We suggest that the irregular structure in Saturn's B ring arises from the formation of shear-free ring-particle assemblies of up to ~100 km in radial extent. The characteristic scale of the irregular structure is set by the competition between tidal ... More A Counterexample to the Generalized Linial-Nisan ConjectureOct 27 2011In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that ... More Why Philosophers Should Care About Computational ComplexityAug 08 2011Aug 14 2011One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue ... More Are Quantum States Exponentially Long Vectors?Jul 26 2005I'm grateful to Oded Goldreich for inviting me to the 2005 Oberwolfach Meeting on Complexity Theory. In this extended abstract, which is based on a talk that I gave there, I demonstrate that gratitude by explaining why Goldreich's views about quantum ... More Limits on Efficient Computation in the Physical WorldDec 20 2004Feb 15 2005More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the ... More Limitations of Quantum Advice and One-Way CommunicationFeb 15 2004Oct 01 2004Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages ... More Is Quantum Mechanics An Island In Theoryspace?Jan 12 2004Mar 17 2004This recreational paper investigates what happens if we change quantum mechanics in several ways. The main results are as follows. First, if we replace the 2-norm by some other p-norm, then there are no nontrivial norm-preserving linear maps. Second, ... More The Ghost in the Quantum Turing MachineJun 02 2013Jun 07 2013In honor of Alan Turing's hundredth birthday, I unwisely set out some thoughts about one of Turing's obsessions throughout his life, the question of physics and free will. I focus relatively narrowly on a notion that I call "Knightian freedom": a certain ... More Lower Bounds for Local Search by Quantum ArgumentsJul 21 2003Feb 05 2004The problem of finding a local minimum of a black-box function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1}^n, we show a lower bound of Omega(2^{n/4}/n) on the number of ... More Quantum Computing and Dynamical Quantum ModelsMay 11 2002A dynamical quantum model assigns an eigenstate to a specified observable even when no measurement is made, and gives a stochastic evolution rule for that eigenstate. Such a model yields a distribution over classical histories of a quantum state. We study ... More Action in a fractal universe and the holographic upper boundFeb 28 2008Aug 13 2008The basic scaling laws for structures in a fractal universe require that the characteristic quantity of action associated with astronomical bodies should be of order near the maximum possible action allowed by the holographic upper bound. That conclusion ... More The fundamental scales of structures from first principlesApr 15 2008May 29 2008Five fundamental scales of mass follow from holographic limitations, a self-similar law for angular momentum and the basic scaling laws for a fractal universe with dimension 2. The five scales correspond to the observable universe, clusters, galaxies, ... More Coherent Phase Argument for InflationSep 05 2003Cosmologists have developed a phenomenally successful picture of structure in the universe based on the idea that the universe expanded exponentially in its earliest moments. There are three pieces of evidence for this exponential expansion -- {\it inflation} ... More