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Set mapping reflectionJan 28 2005In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2 and that L(P(omega_1)) satisfies ... More

Fast growth in Folner sets for Thompson's group FMay 08 2009Aug 08 2012The purpose of this note is to prove a lower bound on the Folner function for Thompson's groups F.

The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals HypothesisJan 28 2005The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, ... More

A solution to the L space problem and related ZFC constructionsJan 28 2005In this paper I will construct a non-separable hereditarily Lindelof space (L space) without any additional axiomatic assumptions. I will also show that there is a function f from [omega_1]^2 to omega_1 such that if A,B, subsets of omega_1, are uncountable ... More

A five element basis for the uncountable linear ordersJan 28 2005In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five ... More

The method of forcingFeb 08 2019The purpose of this article is to give a presentation of the method of forcing aimed at someone with a minimal knowledge of set theory and logic. The emphasis will be on how the method can be used to prove theorems in ZFC.

A Boolean action of C(M,U(1)) without a spatial model and a re-examination of the Cameron-Martin theoremJan 18 2012Aug 08 2012We will demonstrate that if M is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from M to U(1) on a separable probability algebra which preserves the measure and yet does not admit a point ... More

A finitely presented group of piecewise projective homeomorphismsAug 20 2013Aug 01 2014In this article we will describe a finitely presented subgroup of Monod's group of piecewise projective homeomorphisms of R. This in particular provides a new example of a finitely presented group which is nonamenable and yet does not contain a nonabelian ... More

There may be no minimal non $σ$-scattered linear ordersJul 18 2017In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non $\sigma$-scattered. This shows that a theorem of Laver, which asserts that ... More

Parametrized $\diamondsuit$ principlesAug 25 2006We will present a collection of guessing principles which have a similar relationship to $\diamond$ as cardinal invariants of the continuum have to $\CH$. The purpose is to provide a means for systematically analyzing $\diamond$ and its consequences. ... More

Groups of fast homeomorphisms of the interval and the ping-pong argumentJan 28 2017We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $\mathrm{Homeo}_+(I)$ ... More

Countable Toronto spacesFeb 11 1999A space X is called an alpha-Toronto space if X is scattered of Cantor-Bendixson rank alpha and is homeomorphic to each of its subspaces of same rank. We answer a question of Steprans by constructing a countable alpha-Toronto space for each alpha<=omega. ... More

A note on Shavgulidze's papers concerning the amenability problem for Thompson's group $F$Feb 03 2011Feb 09 2011This paper examines Shavgulidze's postings to the ArXiv and publications which concern his argument that Thompson's group F is amenable. In particular I list specific places where there are errors in each of the postings and publications and give a public ... More

Hindman's Theorem, Ellis's Lemma, and Thompson's group FJun 23 2011Sep 11 2012The purpose of this article is to formulate generalizations of Hindman's Theorem and Ellis's Lemma for non associative groupoids. A relation between these conjectures is proved and it is shown that they imply the amenability of Thompson's group F. In ... More

Amenability and Ramsey TheoryJun 16 2011Oct 20 2011The purpose of this article is to connect the notion of the amenability of a discrete group with a new form of structural Ramsey theory. The Ramsey theoretic reformulation of amenability constitutes a considerable weakening of the Folner criterion. As ... More

Forcing Axioms and the Continuum Hypothesis, part II: Transcending ω_1-sequences of real numbersOct 23 2011Aug 03 2012The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah. The corresponding completely proper forcing which can be constructed ... More

Nonassociative Ramsey Theory and the amenability of Thompson's groupSep 10 2012Oct 01 2012The purpose of this article is prove that Thompson's group F is amenable. The methods developed will then be used to prove a generalization of Hindman's theorem for the free nonassociative binary system on one generator.

An Attempt towards Interpretable Audio-Visual Video CaptioningDec 07 2018Automatically generating a natural language sentence to describe the content of an input video is a very challenging problem. It is an essential multimodal task in which auditory and visual contents are equally important. Although audio information has ... More

Linear Optical Quantum Computing in a Single Spatial ModeMay 15 2013Nov 21 2014We present a scheme for linear optical quantum computing using time-bin encoded qubits in a single spatial mode. We show methods for single-qubit operations and heralded controlled phase (CPhase) gates, providing a sufficient set of operations for universal ... More

Generalisations of Rozansky-Witten invariantsDec 19 2001Oct 15 2002We survey briefly the definition of the Rozansky-Witten invariants, and review their relevance to the study of compact hyperkahler manifolds. We consider how various generalisations of the invariants might prove useful for the study of non-compact hyperkahler ... More

Blazars in Context in the Fermi EraMar 20 2013Blazars are the most plentiful gamma-ray source at GeV energies, and despite detailed study, there is much that is not known about these sources. In this review I explore some recent results on blazars, including the controversy of the "blazar sequence", ... More

Sheaves, Cosheaves and ApplicationsMar 13 2013Dec 17 2014This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, ... More

Norm inequalities in generalized Morrey spacesMay 29 2012May 14 2014We prove that Calder\'on-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey ... More

Topological Data Analysis and CosheavesNov 03 2014Mar 04 2015This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level-set persistence is presented using sheaves and cosheaves.

The Mezard-Parisi equation for matchings in pseudo-dimension d>1Sep 09 2014We establish existence and uniqueness of the solution to the cavity equation for the random assignment problem in pseudo-dimension $d>1$, as conjectured by Aldous and Bandyopadhyay (Annals of Applied Probability, 2005) and W\"astlund (Annals of Mathematics, ... More

The interpolation method for random graphs with prescribed degreesApr 26 2014We consider large random graphs with prescribed degrees, such as those generated by the configuration model. In the regime where the empirical degree distribution approaches a limit $\mu$ with finite mean, we establish the systematic convergence of a ... More

Brace Bar-Cobar DualitySep 11 2013Sep 12 2013Using Kadeishvili's formulas with appropriate signs, we show that the classical cobar construction from coalgebras to algebras \Omega: CoAlg -> Alg can be enhanced to a functor from Hopf algebras to E_2 algebras (for a certain choice of E_2 operad) \Omega ... More

Invertible sums of matricesMar 22 2016Apr 19 2016We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli. The proof yields a polynomial identity, expressing the determinant of a large sum of matrices in terms of determinants ... More

Time-Optimal Interactive Proofs for Circuit EvaluationApr 13 2013Jul 27 2013Recently, researchers have been working toward the development of practical general-purpose protocols for verifiable computation. These protocols enable a computationally weak verifier to offload computations to a powerful but untrusted prover, while ... More

Desingularization of a steady vortex pair in the lake equationNov 17 2017We construct a family of steady solutions of the lake model perturbed by some small Coriolis force, that converge to a singular vortex pair. The desingularized solutions are obtained by maximization of the kinetic energy over a class of rearrangements ... More

Fibrations on four-folds with trivial canonical bundlesApr 01 2009Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and ... More

The Thompson-Lyons transfer lemma for fusion systemsMar 24 2013Apr 25 2014In this note, a generalization of the Thompson transfer lemma and its various extensions, most recently due to Lyons, is proven in the context of saturated fusion systems. A strengthening of Alperin's fusion theorem is also given in this setting, following ... More

Mono: an algebraic study of torus closuresOct 12 2017Given an ideal I in a polynomial ring, we consider the largest monomial subideal contained in I, denoted mono(I). We study mono as an interesting operation in its own right, guided by questions that arise from comparing the Betti tables of I and mono(I). ... More

Topology of eigenspace posets for unitary reflection groupsAug 09 2012Apr 02 2013The eigenspace theory of unitary reflection groups, initiated by Springer and Lehrer, suggests that the following object is worthy of study: the poset of eigenspaces of elements of a unitary reflection group, for a fixed eigenvalue, ordered by the reverse ... More

Generating mapping class groups with elements of fixed finite orderOct 12 2017We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We additionally prove ... More

Unramified geometric class field theory and Cartier dualityOct 08 2017We prove a generalized Albanese property for the Picard stack of a smooth projective curve, which in particular implies Deligne's unramified geometric class field theory. This Albanese property also specializes to the Cartier self-duality of the Picard ... More

Nearby cycles of Whittaker sheaves on Drinfeld's compactificationFeb 14 2017In this article we study the perverse sheaf on Drinfeld's compactification obtained by applying the geometric Jacquet functor (alias nearby cycles) to a nondegenerate Whittaker sheaf. Namely, we describe its restrictions along the defect stratification ... More

The big projective module as a nearby cycles sheafNov 30 2014Aug 22 2016We give a new geometric construction of the big projective module in the principal block of the BGG category $\mathscr{O}$, or rather the corresponding $\mathscr{D}$-module on the flag variety. Namely, given a one-parameter family of nondegenerate additive ... More

Up-down asymmetric tokamaksNov 21 2016Bulk toroidal rotation has proven capable of stabilising both dangerous MHD modes and turbulence. In this thesis, we explore a method to drive rotation in large tokamaks: up-down asymmetry in the magnetic equilibrium. We seek to maximise this rotation ... More

Every totally real algebraic integer is a tree eigenvalueFeb 18 2013Sep 04 2014Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite graph is a ... More

Compton Dominance and the Blazar SequenceDec 04 2012Does the "blazar sequence" exist, or is it a result of a selection effect, due to the difficulty in measuring the redshifts of blazars with both high synchrotron peak frequencies (\gtrsim 10^{15} Hz) and luminosities (\gtrsim 10^{46} erg s^{-1})? We explore ... More

Intrinsic square functions on functions spaces including weighted Morrey spacesMay 01 2012Apr 16 2013We prove that the intrinsic square functions including Lusin area integral and Littlewood-Paley $g^{\ast}_{\lambda}$-function as defined by Wilson, are bounded in a class of function spaces include weighted Morrey spaces. The corresponding commutators ... More

Nilpotence in the symplectic bordism ringOct 14 2014Oct 23 2014We prove a folklore result which gives a sequence of necessary and sufficient conditions for a stably symplectic manifold to define a nilpotent element in the symplectic bordism ring.

An Alternative to Particle Dark MatterAug 29 2014Dec 11 2014We propose an alternative to particle dark matter that borrows ingredients of MOdified Newtonian Dynamics (MOND) while adding new key components. The first new feature is a dark matter fluid, in the form of a scalar field with small equation of state ... More

Dark Matter SuperfluidityMay 26 2016In this talk I summarize a novel framework that unifies the stunning success of MOND on galactic scales with the triumph of the $\Lambda$CDM model on cosmological scales. This is achieved through the rich and well-studied physics of superfluidity. The ... More

Geodesic deviation at higher orders via covariant bitensorsJul 25 2014We review a simple but instructive application of the formalism of covariant bitensors, to use a deviation vector field along a fiducial geodesic to describe a neighboring worldline, in an exact and manifestly covariant manner, via the exponential map. ... More

Modeling Fermi Large Area Telescope and Multiwavelength Data from BlazarsFeb 18 2016Blazars are active galactic nuclei with relativistic jets pointed at the Earth, making them extremely bright at essentially all wavelengths, from radio to gamma rays. I review the modeling of this broadband spectral energy distributions of these objects, ... More

The cavity method for counting spanning subgraphs subject to local constraintsMar 16 2011Using the theory of negative association for measures and the notion of random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically tree-like graphs. ... More

Amalgamation Classes with $\exists$-ResolutionsDec 15 2015Jan 13 2016Let $K_d$ denote the class of all finite graphs and, for graphs $A\subseteq B$, say $A \leq_d B$ if distances in $A$ are preserved in $B$; i.e. for $a, a' \in A$ the length of the shortest path in $A$ from $a$ to $a'$ is the same as the length of the ... More

Full Amalgamation Classes with Intrinsic TranscendentalsDec 12 2015We develop some basic results about full amalgamation classes with intrinsic trascendentals. These classes have generics whose models may have finite subsets whose intrinsic closure is not contained in its algebraic closure. We will show that under fairly ... More

On the Negative $K$-theory of Singular VarietiesJun 16 2013Let $X$ be an $n$-dimensional variety over a field $k$ of characteristic zero, regular in codimension 1 with singular locus $Z$. In this paper we study the negative $K$-theory of $X$, showing that when $Z$ is sufficiently nice, $K_{1-n}(X)$ is an extension ... More

Quantum Corrections to Diffusion in StarsSep 29 2011Quantum corrections can be important for diffusion and the melting temperature of dense plasmas in compact astrophysical objects, particulary white dwarfs and neutron stars. Typically ions in these systems are modeled classically, but Daligault et al. ... More

Local ill-posedness of the 1D Zakharov systemFeb 08 2006Aug 24 2006Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system for any dimension $d$, in the inhomogeneous Sobolev spaces $(u,n)\in H^k(\mathbb{R}^d)\times H^s(\mathbb{R}^d)$ for a range of exponents $k$, $s$ depending on $d$. Here we ... More

Moduli spaces of sheaves on K3 surfacesMar 02 2016In this survey article we describe moduli spaces of simple, stable, and semistable sheaves on K3 surfaces, following the work of Mukai, O'Grady, Huybrechts, Yoshioka, and others. We also describe some recent developments, including applications to the ... More

The Bernstein Center of a p-adic Unipotent GroupMar 16 2011Mar 18 2011Francois Rodier proved that it is possible to view smooth representations of certain totally disconnected abelian groups (the underlying additive group of a finite-dimensional p-adic vector space, for example) as sheaves on the Pontryagin dual group. ... More

Derived equivalence of holomorphic symplectic manifoldsApr 20 2004We use twisted Fourier-Mukai transforms to study the relation between an abelian fibration on a holomorphic symplectic manifold and its dual fibration. Our reasoning leads to an equivalence between the derived category of coherent sheaves on one space ... More

Measurement of gamma and 2 beta + gammaDec 20 2004Jan 26 2005We report on the initial measurements of the angle gamma and the sum of angles 2 beta + gamma of the Unitarity Triangle. When compared with indirect information on the value of gamma from other measurements of CKM parameters, the measurement of these ... More

A bound on the second Betti number of hyperkähler manifolds of complex dimension sixNov 29 2015Let M be an irreducible compact hyperk\"ahler manifold of complex dimension six. We prove that the second Betti number of M is at most 23.

Perturbative expansion of Chern-Simons theoryApr 24 2005Mar 13 2009An overview of the perturbative expansion of the Chern--Simons path integral is given. The main goal is to describe how trivalent graphs appear: as they already occur in the perturbative expansion of an analogous finite-dimensional integral, we discuss ... More

Efficient estimation of the error distribution function in heteroskedastic nonparametric regression with missing dataOct 27 2016A residual-based empirical distribution function is proposed to estimate the distribution function of the errors of a heteroskedastic nonparametric regression with responses missing at random based on completely observed data, and this estimator is shown ... More

Rozansky-Witten invariants of hyperkähler manifoldsApr 20 2004We investigate invariants of compact hyperk{\"a}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and ... More

Classical 6j-symbols and the tetrahedronDec 15 1998Mar 22 1999A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j-symbol, which ... More

Free Energy of Multiple Systems of Spherical Spin Glasses with Constrained OverlapsJun 26 2018The free energy of multiple systems of spherical spin glasses with constrained overlaps was first studied in arXiv:math/0604082. The authors proved an upper bound of the constrained free energy using Guerra's interpolation. In this paper, we prove this ... More

Topology of eigenspace posets for imprimitive reflection groupsAug 22 2012Apr 02 2013This paper studies the poset of eigenspaces of elements of an imprimitive unitary reflection group, for a fixed eigenvalue, ordered by the reverse of inclusion. The study of this poset is suggested by the eigenspace theory of Springer and Lehrer. The ... More

Pseudo-Anosov homeomorphisms and the lower central series of a surface groupFeb 21 2007Let Gamma_k be the lower central series of a surface group Gamma of a compact surface S with one boundary component. A simple question to ponder is whether a mapping class of S can be determined to be pseudo-Anosov given only the data of its action on ... More

Fourier-Mukai transforms, mirror symmetry, and generalized K3 surfacesSep 14 2012We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, ... More

On Lagrangian fibrations by Jacobians IMar 07 2008Sep 16 2011Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X ... More

A resolution of singularities for Drinfeld's compactification by stable mapsJun 05 2016Drinfeld's relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting ... More

Source-LDA: Enhancing probabilistic topic models using prior knowledge sourcesJun 02 2016A popular approach to topic modeling involves extracting co-occurring n-grams of a corpus into semantic themes. The set of n-grams in a theme represents an underlying topic, but most topic modeling approaches are not able to label these sets of words ... More

Summary: Acoustic Detection of EHE NeutrinosNov 15 2006Neutrino astronomy was initiated primarily to search for TeV to PeV neutrinos from Active Galactic Nuclei, and the optical Cherenkov technique is well suited for this energy range. Interest has grown recently in detecting EeV neutrinos, particularly the ... More

Products of functions in $\BMO$ and $\H^{1}$ spaces on spaces of homogeneous typeFeb 18 2009We give an extension to certain \textit{RD-space} $\X$, i.e space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property, of the definition and various properties of the product of functions in $\BMO(\X)$ and $\H^{1}(\X)$, ... More

On the Density of Happy NumbersOct 17 2011Mar 01 2015The happy function $H: \mathbb{N} \rightarrow \mathbb{N}$ sends a positive integer to the sum of the squares of its digits. A number $x$ is said to be happy if the sequence $\{H^n(x)\}^\infty_{n=1}$ eventually reaches one. A basic open question regarding ... More

Thermoacoustic Tomography in Elastic MediaMay 04 2011Oct 11 2011We investigate the problem of recovering the initial displacement f for a solution u of a linear, isotropic, non-homogeneous elastic wave equation, given measurements of u on [0,T] x \partial \Omega, where \Omega\subset\R^3 is some bounded domain containing ... More

Chameleon Field TheoriesJun 18 2013Chameleons are light scalar fields with remarkable properties. Through the interplay of self-interactions and coupling to matter, chameleon particles have a mass that depends on the ambient matter density. The manifestation of the fifth force mediated ... More

Comparisons of polychromatic and monochromatic Ramsey theoryMar 06 2012May 17 2012We compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the ... More

Structured Learning via Logistic RegressionJul 03 2014A successful approach to structured learning is to write the learning objective as a joint function of linear parameters and inference messages, and iterate between updates to each. This paper observes that if the inference problem is "smoothed" through ... More

On the discriminant locus of a Lagrangian fibrationJul 21 2006Let $X\to\P^n$ be an irreducible holomorphic symplectic manifold of dimension $2n$ fibred over $\P^n$. Matsushita proved that the generic fibre is a holomorphic Lagrangian abelian variety. In this article we study the discriminant locus $\Delta\subset\P^n$ ... More

Deformations of holomorphic Lagrangian fibrationsSep 09 2005Let $X\to\P^n$ be a $2n$-dimensional projective holomorphic symplectic manifold admitting a Lagrangian fibration over $\P^n$. Matsushita proved that the fibration can be deformed in a codimension one family in the moduli space $\mathrm{Def}(X)$ of deformations ... More

Twisted Fourier-Mukai transforms for holomorphic symplectic fourfoldsSep 09 2005We apply the methods of C{\u{a}}ld{\u{a}}raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain four-fold ... More

Kirby calculus in manifolds with boundaryDec 15 1998Suppose there are two framed links in a compact, connected 3-manifold (possibly with boundary, or non-orientable) such that the associated 3-manifolds obtained by surgery are homeomorphic (relative to their common boundary, if there is one.) How are the ... More

Grothendieck classes of quiver cycles as iterated residuesOct 14 2013In the case of Dynkin quivers we establish a formula for the Grothendieck class of a quiver cycle as the iterated residue of a certain rational function, for which we provide an explicit combinatorial construction. Moreover, we utilize a new definition ... More

The initial-boundary value problem for the 1D nonlinear Schroedinger equation on the half-lineFeb 08 2006We prove, by adapting the method of Colliander-Kenig (2002), local well-posedness of the initial-boundary value problem for the one-dimensional nonlinear Schroedinger equation on the half-line under low boundary regularity assumptions.

On the Cohomology of the Lie Algebra Arising from the Lower Central Series of a p-GroupMar 26 2003We study the cohomology H*(A) = Ext_A(k,k) of a locally finite, connected, cocommutative Hopf algebra A over k = F_p. Specifically, we are interested in those algebras A for which H*(A) is generated as an algebra by H^1(A) and H^2(A). We shall call such ... More

Asymptotics and 6j-symbolsJan 18 2002Oct 15 2002Recent interest in the Kashaev-Murakami-Murakami hyperbolic volume conjecture has made it seem important to be able to understand the asymptotic behaviour of certain special functions arising from representation theory -- for example, of the quantum 6j-symbols ... More

Interpolating factorizations for acyclic Donaldson--Thomas invariantsJul 05 2018Jul 15 2018We prove a family of factorization formulas for the combinatorial Donaldson--Thomas invariant for an acyclic quiver. A quantum dilogarithm identity due to Reineke, later interpreted by Rim\'anyi by counting codimensions of quiver loci, gives two extremal ... More

On Lagrangian fibrations by Jacobians IISep 19 2011Let Y->P^n be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that X is a Beauville-Mukai integrable system if n=3, 4, or 5, and the curves are irreducible ... More

Abelian fibred holomorphic symplectic manifoldsApr 20 2004We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and ... More

Les Houches Lectures on Physics Beyond the Standard Model of CosmologyDec 06 2013In these Lectures, I review various extensions of the Lambda-Cold Dark Matter model, characterized by additional light degrees of freedom in the dark sector. In order to reproduce the successful phenomenology of GR in the solar system, these fields must ... More

Maximum Likelihood Learning With Arbitrary Treewidth via Fast-Mixing Parameter SetsSep 30 2015Oct 30 2015Inference is typically intractable in high-treewidth undirected graphical models, making maximum likelihood learning a challenge. One way to overcome this is to restrict parameters to a tractable set, most typically the set of tree-structured parameters. ... More

Unbounded and dominating reals in Hechler extensionsJan 13 2012We give results exploring the relationship between dominating and unbounded reals in Hechler extensions, as well as the relationships among the extensions themselves. We show that in the standard Hechler extension there is an unbounded real which is dominated ... More

Weak Values are Interference PhenomenaOct 03 2014Feb 26 2015Weak values arise experimentally as conditioned averages of weak (noisy) observable measurements that minimally disturb an initial quantum state, and also as dynamical variables for reduced quantum state evolution even in the absence of measurement. These ... More

The current status of neutrino mixingJul 19 2011A brief review of the experimental status of neutrino mixing. The model of neutrino oscillations has now been established with high confidence, with many of the model parameters measured to an accuracy of a few per cent. However, some parameters still ... More

Up-down asymmetric tokamaksNov 21 2016Nov 23 2016Bulk toroidal rotation has proven capable of stabilising both dangerous MHD modes and turbulence. In this thesis, we explore a method to drive rotation in large tokamaks: up-down asymmetry in the magnetic equilibrium. We seek to maximise this rotation ... More

A characterization of the 2-fusion system of L_4(q)Mar 24 2013Oct 25 2014We study a saturated fusion system F on a finite 2-group S having a Baumann component based on a dihedral 2-group. Assuming F is 2-perfect with no nontrivial normal 2-subgroups, and the centralizer of the component is a cyclic 2-group, it is shown that ... More

The T-algebra spectral sequence: Comparisons and applicationsAug 27 2013Apr 22 2014In previous work with Niles Johnson the author constructed a spectral sequence for computing homotopy groups of spaces of maps between structured objects such as G-spaces and E_n-ring spectra. In this paper we study special cases of this spectral sequence ... More

Isotrivial elliptic K3 surfaces and Lagrangian fibrationsJun 04 2014A fibration is said to be isotrivial if all of its smooth fibres are isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that are isotrivial, and use them to construct Lagrangian fibrations that are isotrivial. We then modify the ... More

Lagrangian fibrations on Hilbert schemes of points on K3 surfacesSep 09 2005Oct 07 2005Let $\mathrm{Hilb}^gS$ be the Hilbert scheme of $g$ points on a K3 surface $S$. Suppose that $\mathrm{Pic}S\cong\Z C$ where $C$ is a smooth curve with $C^2=2(g-1)n^2$. We prove that $\mathrm{Hilb}^gS$ is a Lagrangian fibration.

The Undecidability of the Definability of Principal SubcongruencesJan 22 2013Jul 24 2014For each Turing machine T, we construct an algebra A'(T) such that the variety generated by A'(T) has definable principal subcongruences if and only if T halts, thus proving that the property of having definable principal subcongruences is undecidable ... More

Beal's Conjecture vs. "Positive Zero", FightJan 07 2015This article seeks to encourage a mathematical dialog regarding a possible solution to Beals Conjecture. It breaks down one of the worlds most difficult math problems into laymans terms and encourages people to question some of the most fundamental rules ... More