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Counting $S_5$-fields with a power saving error termOct 08 2013Oct 09 2013We show how the Selberg $\Lambda^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic ... More

A proof of the Andre-Oort conjecture for A_gJun 04 2015Dec 01 2015We give a proof of the Andr\'e-Oort conjecture for $\mathcal{A}_g$ - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged' version of the Colmez conjecture yields lower bounds for Galois ... More

The Existence of an Abelian Variety over the Algebraic Numbers isogenous to no JacobianOct 11 2010We prove the existence of an Abelian variety $A$ of dimension $g$ over $\Qa$ which is not isogenous to any Jacobian, subject to the necessary condition $g>3$. Recently, C.Chai and F.Oort gave such a proof assuming the Andr\'e-Oort conjecture. We modify ... More

Brauer-Siegel for Arithmetic Tori and lower bounds for Galois orbits of special pointsMar 29 2011Jun 12 2011In \cite{S}, Shyr derived an analogue of Dirichlet's class number formula for arithmetic Tori. We use this formula to derive a Brauer-Siegel formula for Tori, relating the Discriminant of a torus to the product of its regulator and class number. We apply ... More

Unlikely Intersections in Finite CharacteristicOct 11 2016We present a heuristic argument based on Honda-Tate theory against many conjectures in `unlikely intersections' over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian, ... More

Multiplicative relations among singular moduliDec 29 2014We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic ... More

The Ax-Schanuel conjecture for variations of Hodge structuresDec 14 2017We extend the Ax-Schanuel theorem recently proven for Shimura varieties by Mok-Pila-Tsimerman to all varieties supporting a pure polarized integral variation of Hodge structures. The essential new ingredient is a volume bound on Griffiths transverse subvarieties ... More

Independence of CM points in Elliptic CurvesJul 05 2019We prove a result which describes, for each $n\ge 1$, all linear dependencies among $n$ images in elliptic curves of special points in modular or Shimura curves under parameterizations (or correspondences). Our result unifies and improves in certain aspects ... More

Non-split Sums of Coefficients of GL(2)-Automorphic FormsJun 06 2011Jun 07 2011Given a cuspidal automorphic form $\pi$ on $\GL_2$, we study smoothed sums of the form $\sum_{n\in\mathbb{N}} a_{\pi}(n^2+d)W(\frac{n}{Y})$. The error term we get is sharp in that it is uniform in both $d$ and $Y$ and depends directly on bounds towards ... More

The geometric torsion conjecture for abelian varieties with real multiplicationApr 08 2015The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian ... More

Sum-product estimates for rational functionsFeb 12 2010Apr 25 2011We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is given for the ... More

Ax-Lindemann for \mathcal{A}_gJun 12 2012Nov 18 2013We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$, and affirm the Andr\'e-Oort conjecture unconditionally for $\mathcal{A}_{g}$ for $g\le 6$.

The Andre-Oort conjecture for the moduli space of Abelian SurfacesJun 20 2011We provide an unconditional proof of the Andr\'e-Oort conjecture for the coarse moduli space $\mathcal{A}_{2,1}$ of principally polarized Abelian surfaces, following the strategy outlined by Pila-Zannier.

The Kodaira dimension of complex hyperbolic manifolds with cuspsMar 19 2015Apr 10 2015We prove a bound relating the volume of a curve near a cusp in a hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. ... More

Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and TsimermanJul 10 2019We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic ... More

Ax-Schanuel for the j-functionDec 29 2014Jan 14 2015In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions ... More

Equidistribution on the space of rank two vector bundles over the projective lineJul 31 2013Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of hyperelliptic curves ... More

How Large is $A_g(\mathbb{F}_q)$?Nov 06 2015Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p.$ Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ dimensional abelian varieties over the finite ... More

On the Frey-Mazur conjecture over low genus curvesSep 25 2013Nov 17 2015The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes ... More

Constructing elliptic curves from Galois representationsAug 08 2017Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces ... More

Cohen Lenstra Heuristics for Étale Group Schemes and Symplectic PairingsOct 28 2016We generalize the Cohen-Lenstra heuristics over function fields to \'{e}tale group schemes $G$ (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg-Venkatesh-Westerland, we make progress ... More

An analysis of a war-like card gameJan 07 2010In his book "Mathematical Mind-Benders", Peter Winkler poses the following open problem, originally due to the first author: "[In the game Peer Pressure,] two players are dealt some number of cards, initially face up, each card carrying a different integer. ... More

P-torsion monodromy representations of elliptic curves over geometric function fieldsMar 27 2014May 03 2016Given a complex quasiprojective curve $B$ and a non-isotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation $\rho_\mathcal{E}[p]:\pi_1(B)\rightarrow \mathrm{GL}_2(\mathbb{F}_p)$. ... More

Equations resolving a conjecture of Rado on partition regularityDec 08 2008Jan 25 2009A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k-1)-regular but not k-regular. ... More

Metaplectic Ramanujan conjecture over function fields with applications to quadratic formsAug 03 2010Nov 16 2012We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an effective solution ... More

Unlikely Intersections in Finite CharacteristicOct 11 2016Oct 18 2016We present a heuristic argument based on Honda-Tate theory against many conjectures in `unlikely intersections' over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian. ... More

A note on Fourier coefficients of Poincaré seriesAug 13 2010We give a short and "soft" proof of the asymptotic orthogonality of Fourier coefficients of Poincar\'e series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.

o-minimal GAGA and a conjecture of GriffithsNov 29 2018Jan 26 2019We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with ... More

Ax-Schanuel for Shimura varietiesNov 06 2017Sep 20 2018We prove the Ax-Schanuel theorem for a general (pure) Shimura variety.

Local spectral equidistribution for Siegel modular forms and applicationsOct 18 2010Aug 24 2011We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson's ... More

Tensor Rank: Some Lower and Upper BoundsFeb 01 2011The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent ... More

Tame topology of arithmetic quotients and algebraicity of Hodge lociOct 01 2018In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given ... More

On the Davenport-Heilbronn theorems and second order termsMay 05 2010Jun 20 2012We give simple proofs of the Davenport--Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having ... More

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curvesJan 10 2017We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_{\epsilon}(|{\rm Disc}(K)|^{1/2+\epsilon})$ by Brauer--Siegel). This yields ... More

Topological classification of time-asymmetry in unitary quantum processesMar 07 2017Effective gauge fields have allowed the emulation of matter under strong magnetic fields leading to the realization of Harper-Hofstadter, Haldane models, and led to demonstrations of one-way waveguides and topologically protected edge states. Central ... More

Bost-Connes type systems for function fieldsFeb 24 2006May 01 2012We describe a construction which associates to any function field $k$ and any place $\infty$ of $k$ a $C^*$-dynamical system $(C_{k,\infty},\sigma_t)$ that is analogous to the Bost-Connes system associated to $\QQ$ and its archimedian place. Our construction ... More

Test Sensitivity in the Computer-Aided Detection of Breast Cancer from Clinical Mammographic Screening: a Meta-analysisFeb 06 2013Objectives: To assess evaluative methodologies for comparative measurements of test sensitivity in clinical mammographic screening trials of computer-aided detection (CAD) technologies. Materials and Methods: This meta-analysis was performed by analytically ... More

A Statistical Significance Simulation Study for the General ScientistSep 29 2011When a scientist performs an experiment they normally acquire a set of measurements and are expected to demonstrate that their results are "statistically significant" thus confirming whatever hypothesis they are testing. The main method for establishing ... More

Semiclassical resolvent bounds in dimension twoApr 13 2016In this note we give an elementary proof of weighted resolvent bounds for semiclassical Schr\"odinger operators in dimension two. We require mild decay conditions on the potential. The resolvent norm grows exponentially in the inverse semiclassical parameter, ... More

Morse Matchings on a HypersimplexNov 07 2012We present a family of complete acyclic Morse matchings on the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. In a future paper we will utilize these matchings to ... More

Derived Algebraic Geometry V: Structured SpacesMay 04 2009In this paper, we describe a general theory of "spaces with structure sheaves." Specializations of this theory include the classical theory of schemes, the theory of Deligne-Mumford stacks, and their derived generalizations.

Stanley-Wilf limits are typically exponentialOct 31 2013For a permutation $\pi$, let $S_{n}(\pi)$ be the number of permutations on $n$ letters avoiding $\pi$. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that $L(\pi)= \lim_{n \to \infty} S_n(\pi)^{1/n}$ exists and is finite. Backed by numerical ... More

A bound on the mutual information, and properties of entropy reduction, for quantum channels with inefficient measurementsDec 01 2004Jan 24 2006The Holevo bound is a bound on the mutual information for a given quantum encoding. In 1996 Schumacher, Westmoreland and Wootters [Schumacher, Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] derived a bound which reduces to the Holevo bound ... More

Topics in Quantum Measurement and Quantum NoiseOct 05 1998Oct 06 1998In this thesis we consider primarily the dynamics of quantum systems subjected to continuous observation. In the Schr\"{o}dinger picture the evolution of a continuously monitored quantum system, referred to as a `quantum trajectory', may be described ... More

Lower Bounds for non-Archimedean Lyapunov ExponentsOct 08 2015Oct 18 2016Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\textbf{P}^1$ denote the Berkovich projective line over $K$. The Lyapunov exponent for a rational map $\phi\in K(z)$ of degree $d\geq 2$ measures the exponential rate ... More

An Equidistribution Result For Dynamical Systems on the Berkovich Projective LineSep 16 2014Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\phi\in K(z)$ with $\textrm{deg}(\phi) \geq 2$. In this paper we consider the functions $\textrm{ordRes}_{\phi^n}(x)$ that measure the resultant of $\phi$ at points in ... More

Feedback Control Using Only Quantum Back-ActionApr 24 2009The traditional approach to feedback control is to apply forces to a system by modifying the Hamiltonian. Here we show that quantum systems can be controlled without any Hamiltonian feedback, purely by exploiting the random quantum back-action of a continuous ... More

Feedback control for communication with non-orthogonal statesJan 24 2006Communicating classical information with a quantum system involves the receiver making a measurement on the system so as to distinguish as well as possible the alphabet of states used by the sender. We consider the situation in which this measurement ... More

Some properties of the Higgs sector of the Next-to Minimal SuperSymmetric ModelJun 16 2011The problems of the standard model are briefly reviewed and the motivations for introducing supersymmetry are discussed. Two realistic supersymmetric models; the Minimal SuperSymmetric Model, MSSM, and its proposed extension NMSSM are introduced briefly ... More

A Bound on the Norm of Shortest Vectors in Lattices Arising from CM Number FieldsOct 29 2012This paper partially addresses the problem of characterizing the lengths of vectors in a family of Euclidean lattices that arise from any CM number field. We define a modified quadratic form on these lattices, the weighted norm, that contains the standard ... More

Understanding ACT-R - an Outsider's PerspectiveJun 01 2013The ACT-R theory of cognition developed by John Anderson and colleagues endeavors to explain how humans recall chunks of information and how they solve problems. ACT-R also serves as a theoretical basis for "cognitive tutors", i.e., automatic tutoring ... More

Holography Inspired Stringy HadronsFeb 01 2016Jun 30 2016Holography inspired stringy hadrons (HISH) is a set of models that describe hadrons: mesons, baryons and glueballs as strings in four dimensional space time. The models are based on a "map" from stringy hadrons of holographic confining backgrounds. In ... More

Langevin process reflected on a partially elastic boundary IApr 12 2010Jul 22 2011Consider a Langevin process, that is an integrated Brownian motion, constrained to stay on the nonnegative half-line by a partially elastic boundary at 0. If the elasticity coefficient of the boundary is greater than or equal to a critical value (0.16), ... More

Jets in Nuclear Collisions: Status and PerspectiveMar 29 2005Apr 06 2005I review the status and future directions of jet-related measurements in high energy nuclear collisions and their application as a probe of QCD matter.

Measurements of High Density Matter at RHICNov 11 2002Nov 13 2002QCD predicts a phase transition between hadronic matter and a Quark Gluon Plasma at high energy density. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory is a new facility dedicated to the experimental study of matter under ... More

Projective Connections and the Algebra of DensitiesAug 20 2008Projective connections first appeared in Cartan's papers in the 1920's. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant quantisation. We recall ... More

Multiple elliptic gamma functions associated to conesSep 08 2016We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of $SL_r(\mathbb{Z})$-elements and prove that the generalized multiple ... More

Automatic Classifiers as Scientific Instruments: One Step Further Away from Ground-TruthDec 19 2018Automatic detectors of facial expression, gesture, affect, etc., can serve as scientific instruments to measure many behavioral and social phenomena (e.g., emotion, empathy, stress, engagement, etc.), and this has great potential to advance basic science. ... More

Measuring Compositionality in Representation LearningFeb 19 2019Many machine learning algorithms represent input data with vector embeddings or discrete codes. When inputs exhibit compositional structure (e.g. objects built from parts or procedures from subroutines), it is natural to ask whether this compositional ... More

The Calculus of Democratization and DevelopmentDec 12 2017In accordance with "Democracy's Effect on Development: More Questions than Answers", we seek to carry out a study in following the description in the 'Questions for Further Study.' To that end, we studied 33 countries in the Sub-Saharan Africa region, ... More

Duality for ConvexityNov 19 2009This paper studies convex sets categorically, namely as algebras of a distribution monad. It is shown that convex sets occur in two dual adjunctions, namely one with preframes via the Boolean truth values {0,1} as dualising object, and one with effect ... More

Lower bound for the remainder in the prime-pair conjectureJun 25 2008For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an explicit positive ... More

An elementary proof of the Briancon-Skoda theoremJul 01 2008Mar 26 2012We give a new elementary proof of the Brian\c{c}on-Skoda theorem, which states that for an $m$-generated ideal $\mathfrak{a}$ in the ring of germs of analytic functions at $0\in \C^n$, the $\nu$:th power of its integral closure is contained in $\mathfrak{a}$, ... More

Higher Topos TheoryAug 02 2006Jul 31 2008This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck topoi. A few ... More

The topological pigeonhole principle for ordinalsOct 09 2014Jul 12 2016Given a cardinal $\kappa$ and a sequence $\left(\alpha_i\right)_{i\in\kappa}$ of ordinals, we determine the least ordinal $\beta$ (when one exists) such that the topological partition relation \[\beta\rightarrow\left(top\,\alpha_i\right)^1_{i\in\kappa}\] ... More

Lens space surgeries and a conjecture of Goda and TeragaitoMay 06 2004Mar 30 2005Using work of Ozsvath and Szabo, we show that if a nontrivial knot in S^3 admits a lens space surgery with slope p, then p <= 4g+3, where g is the genus of the knot. This is a close approximation to a bound conjectured by Goda and Teragaito.

(Infinity,2)-Categories and the Goodwillie Calculus IMay 04 2009May 08 2009The bulk of this paper is devoted to the comparison of several models for the theory of (infinity,2)-categories: that is, higher categories in which all k-morphisms are invertible for k > 2 (the case of (infinity,n)-categories is also considered). Our ... More

Quasisymmetric Functions from Combinatorial Hopf Monoids and Ehrhart TheoryMar 31 2016We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class ... More

The Hopf monoid of coloring problemsNov 13 2016We study coloring problems, which are induced subposets P of a Boolean lattice, paired with an order ideal I from the poset of intervals, ordered by inclusion. We study a quasisymmetric function associated to coloring problems, called the chromatic quasisymmetric ... More

Tensors Masquerading as Matchgates: Relaxing Planarity Restrictions on Pfaffian CircuitsOct 06 2015Oct 30 2015Holographic algorithms, alternatively known as Pfaffian circuits, have received a great deal of attention for giving polynomial-time algorithms of $\#\mathsf{P}$-hard problems. Much work has been done to determine the extent of what this machinery can ... More

The homology of moduli stacks of complexesJul 07 2019We compute the $E$-homology of the moduli stack $\mathcal{M}$ of objects in the derived category of a smooth complex projective variety $X$, where $E$ is a complex-oriented homology theory with rational coefficient ring. For curves, surfaces, and some ... More

Regarding a uniqueness property of singly-periodic Scherk surfacesMar 18 2013May 13 2013Inspired by an argument of Ros [15] -- we use the L\'{o}pez-Ros deformation to give another proof of the fact -- due to Meeks and Wolf [13] -- that the only smooth, connected, singly-periodic minimal surfaces in $\Real^3$ with the area growth of two planes ... More

Generating Specials: The Zorro AlgorithmAug 14 2006The concept of a configuration graph associated to a primitive, aperiodic substitution is introduced in [1] as a convenient graphical representation of the infinite indeterminism of the shift space of the substitution. The main result of [1] is an algorithm ... More

Stable Infinity CategoriesAug 09 2006May 08 2009This paper is an expository account of the theory of stable infinity categories. We prove that the homotopy category of a stable infinity category is triangulated, and that the collection of stable infinity categories is closed under a variety of constructions. ... More

Derived Algebraic Geometry II: Noncommutative AlgebraFeb 11 2007Sep 19 2007In this paper, we present an infinity-categorical version of the theory of monoidal categories. We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits in each variable), ... More

Langevin process reflected on a partially elastic boundary IIMar 15 2011A particle subject to a white noise external forcing moves like a Langevin process. Consider now that the particle is reflected at a boundary which restores a portion c of the incoming speed at each bounce. For c strictly smaller than the critical value ... More

Excursions of the integral of the Brownian motionJan 22 2009Jun 18 2009The integrated Brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a ... More

Spin transport in nanocontacts and nanowiresDec 10 2007In this thesis we study electron transport through magnetic nanocontacts and nanowires with ab initio quantum transport calculations. The aim is to gain a thorough understanding of the interplay between electrical conduction and magnetism in atomic-size ... More

Global Minimizers for Free Energies of Subcritical Aggregation Equations with Degenerate DiffusionSep 27 2010We prove the existence of non-trivial global minimizers of a class of free energies related to aggregation equations with degenerate diffusion on $\Real^d$. Such equations arise in mathematical biology as models for organism group dynamics which account ... More

Background Fluctuations in Heavy Ion Jet ReconstructionDec 10 2010Jan 03 2011We present a new study by the STAR Collaboration of background fluctuations in jet reconstruction in heavy ion collisions.

On the optimality of universal classifiers for finite-length individual test sequencesSep 23 2009Nov 07 2011We consider pairs of finite-length individual sequences that are realizations of unknown, finite alphabet, stationary sources in a clas M of sources with vanishing memory (e.g. stationary Markov sources). The task of a universal classifier is to decide ... More

A new version of an old modal incompleteness theoremFeb 15 2012Thomason \cite{Thomason74} showed that a certain modal logic $\mathbf{L}\subset \mathbf{S4}$ is incomplete with respect to Kripke semantics. Later Gerson \cite{Gerson75} showed that $\mathbf{L}$ is also incomplete with respect to neighborhood semantics. ... More

Hyper Normalisation and Conditioning for Discrete Probability DistributionsJul 10 2016Normalisation in probability theory turns a subdistribution into a proper distribution. It is a partial operation, since it is undefined for the zero subdistribution. This partiality makes it hard to reason equationally about normalisation. A novel description ... More

Constructing dense graphs with sublinear Hadwiger numberAug 24 2011Aug 26 2011Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices. Such graphs exist as a random graph almost surely has this property. This question and variants were popularized by ... More

Quantum measurement and the first law of thermodynamics: the energy cost of measurement is the work value of the acquired informationAug 08 2012Nov 03 2012The energy cost of measurement is an interesting fundamental question, and may have profound implications for quantum technologies. In the context of Maxwell's demon, it is often stated that measurement has no minimum energy cost, while information has ... More

The Second Law of Thermodynamics and Quantum Feedback Control: Maxwell's Demon with Weak MeasurementsJun 23 2009Aug 05 2009Recently Sagawa and Ueda [Phys. Rev. Lett. 100, 080403 (2008)] derived a bound on the work that can be extracted from a quantum system with the use of feedback control. They left open the question of whether this bound could be achieved for every measurement ... More

A Mathematical Account of Soft Evidence, and of Jeffrey's `destructive' versus Pearl's `constructive' updatingJul 15 2018Evidence in probabilistic reasoning may be `hard' or `soft', that is, it may be of yes/no form, or it may involve a strength of belief, in the unit interval [0,1]. Reasoning with soft, $[0,1]$-valued evidence is important in many situations but may lead ... More

Reduced Ideals in Pure Cubic FieldsMay 01 2019Reduced ideals have been defined in the context of integer rings in quadratic number fields, and they are closely tied to the continued fraction algorithm. The notion of this type of ideal extends naturally to number fields of higher degree. In the case ... More

Reduced Ideals in Pure Cubic FieldsMay 01 2019Jun 02 2019Reduced ideals have been defined in the context of integer rings in quadratic number fields, and they are closely tied to the continued fraction algorithm. The notion of this type of ideal extends naturally to number fields of higher degree. In the case ... More

Algorithm to Detect Periodicity by Interleaving SequencesMay 16 2019We define an algorithm which begins with an sequence of sequences, and produces a single sequence, with following property: If at least one of the original sequences has a tail that is periodic, then the output sequence has a periodic tail, and conversely. ... More

Perfect Morse functions and exotic S^2 x S^2'sMay 25 2010Oct 13 2010The main theorem of the paper shows that a smooth manifold which is homeomorphic to S^2xS^2 and has nonvanishing Ozsvath-Szabo invariant does not admit a perfect Morse function. I am withdrawing the paper because it is unclear to me if such a manifold ... More

Khovanov-Rozansky homology of two-bridge knots and linksAug 25 2005Aug 15 2006We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.

Khovanov's invariant for closed surfacesFeb 24 2005We compute the Khovanov-Jacobsson number of an embedded torus in R^4. The answer is always 2, regardless of the embedding.

Lower Bounds for non-Archimedean Lyapunov ExponentsOct 08 2015Jul 21 2017Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $\textbf{P}^1$ denote the Berkovich projective line over $K$. The Lyapunov exponent for a rational map $\phi\in K(z)$ of degree $d\geq 2$ measures the exponential rate ... More

A Mathematical Account of Soft Evidence, and of Jeffrey's `destructive' versus Pearl's `constructive' updatingJul 15 2018Mar 03 2019Evidence in probabilistic reasoning may be `hard' or `soft', that is, it may be of yes/no form, or it may involve a strength of belief, in the unit interval [0,1]. Reasoning with soft, [0,1]-valued evidence is important in many situations but may lead ... More

A new degree bound for local unitary and $n$-qubit SLOCC InvariantsJun 02 2017Jun 09 2017Deep connections between invariant theory and entanglement have been known for some time and been the object of intense study. This includes the study of local unitary equivalence of density operators as well as entanglement that can be observed in stochastic ... More

The Briancon-Skoda number of analytic irreducible planar curvesJan 16 2012The Briancon-Skoda number of a ring R is defined as the smallest integer k, such that for any ideal I\subset R and r\geq 1, the integral closure of I^{k+r-1} is contained in I^r. We compute the Briancon-Skoda number of the local ring of any analytic irreducible ... More

Lattices in R^2 and finite subsets of a circleNov 27 1999Dec 09 1999An elementary geometric construction is used to relate the space of lattices in a plane to the space exp_3(S^1) of the subsets of a circle of cardinality at most 3. As a consequence we obtain new proofs of a theorem of Bott which says that exp_3(S^1) ... More

On the Classification of Topological Field TheoriesMay 04 2009This paper provides an informal sketch of a proof of the Baez-Dolan cobordism hypothesis, which provides a classification for extended topological quantum field theories.

Derangements with Ascending and Descending BlocksAug 23 2009Aug 29 2009We continue the work of Eriksen, Freij, and Wastlund [3], who study derangements that descend in blocks of prescribed lengths. We generalize their work to derangements that ascend in some blocks and descend in others. In particular, we obtain a generating ... More