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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 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 brief introduction to amenable equivalence relationsSep 03 2018This is a short survey of amenable equivalence relations.

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

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

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

Idempotent means on free binary systems do not existJul 15 2018Free binary systems are shown to not admit idempotent means. This refutes a conjecture of the author. It is also shown that the extension of Hindman's theorem to nonassociative binary systems formulated and conjectured by the author is false.

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

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

Hindman's Theorem, Ellis's Lemma, and Thompson's group $F$Jun 23 2011Jul 16 2018The purpose of this article is to formulate conjectural generalizations of Hindman's Theorem and Ellis's Lemma for nonassociative binary systems and relate them to the amenability problem for Thompson's group $F$. Partial results are obtained for both ... 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.

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 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

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

Complexity among the finitely generated subgroups of Thompson's groupNov 29 2017We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $\epsilon_0 +1$. All except the maximum element of this family (which is $F$ ... 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

Domain Wall Motion and Interfacial Dzyaloshinskii-Moriya Interactions in Pt/Co/Ir$(t_\mathrm{Ir})$/Ta MultilayersOct 08 2018The interfacial Dzyaloshinskii-Moriya interaction (DMI) is important for chiral domain walls (DWs) and for stabilizing magnetic skyrmions. We study the effects of introducing increasing thicknesses of Ir, from zero to 2 nm, into a Pt/Co/Ta multilayer ... 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

Matroids: A Macaulay2 packageNov 14 2015We give an overview of the Macaulay2 package Matroids, which contains functionality to create and compute with matroids. Examples highlighting the use of all major functions in the package are provided, along with explanations of some of the algorithms. ... More

Exotic Branes from del Pezzo SurfacesApr 10 2019We revisit a correspondence between toroidal compactifications of M-theory and del Pezzo surfaces, in which rational curves on the del Pezzo are related to ${1\over 2}$-BPS branes of the corresponding compactification. We argue that curves of higher genus ... 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

A family of structure isomorphisms for the Cohomological Hall Algebra of an acyclic quiverAug 01 2019For any acyclic quiver, we establish a family of structure isomorphisms for its cohomological Hall algebra (CoHA). The domain of each isomorphism is a tensor product of subalgebras in which each factor is isomorphic to the CoHA of the quiver with a single ... More

Seeking the Principles of Sustainable Software EngineeringMay 18 2014Jul 22 2014Like other engineering disciplines, software engineering should also have principles to guide the construction of sustainable computer applications. Tangible properties include a) unlimited scalability, b) maximal reproducibility, and c) optimizable energy ... More

Ulam Sphere Size Analysis for Permutation and Multipermutation Codes Correcting Translocation ErrorsDec 11 2017Jan 08 2019Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation sphere sizes in the Ulam metric ... More

Lagrangian fibrations by Prym varietiesMay 08 2019We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties.

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

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

Semi-Streaming Algorithms for Annotated Graph StreamsJul 13 2014Aug 10 2015Considerable effort has been devoted to the development of streaming algorithms for analyzing massive graphs. Unfortunately, many results have been negative, establishing that a wide variety of problems require $\Omega(n^2)$ space to solve. One of the ... 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

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

The Two Dimensional Kondo Model with Rashba Spin-Orbit CouplingAug 17 2007Sep 11 2007We investigate the effect that Rashba spin-orbit coupling has on the low energy behaviour of a two dimensional magnetic impurity system. It is shown that the Kondo effect, the screening of the magnetic impurity at temperatures T < T_K, is robust against ... More

Unusual formulae for the Euler characteristicJan 18 2002Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, ... More

Almost sure global well posedness for the BBM equation with infinite $L^{2}$ initial dataJan 12 2019We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus $\mathbb{T}$ with initial data below $L^{2}(\mathbb{T})$. With respect to random initial data of strictly negative Sobolev regularity, ... More

Learning Graphical Model Parameters with Approximate Marginal InferenceJan 15 2013Likelihood based-learning of graphical models faces challenges of computational-complexity and robustness to model mis-specification. This paper studies methods that fit parameters directly to maximize a measure of the accuracy of predicted marginals, ... More

Acoustic detection of astrophysical neutrinos in South Pole iceDec 30 2011When high-energy particles interact in dense media to produce a particle shower, most of the shower energy is deposited in the medium as heat. This causes the medium to expand locally and emit a shock wave with a medium-dependent peak frequency on the ... More

Dualities between Cellular Sheaves and CosheavesJul 23 2016This paper affirms a conjecture of MacPherson: that the derived category of cellular sheaves is equivalent to the derived category of cellular cosheaves. We give a self-contained treatment of cellular sheaves and cosheaves and note that certain classical ... 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

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

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

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

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

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

Learning Convex Inference of MarginalsJun 13 2012Graphical models trained using maximum likelihood are a common tool for probabilistic inference of marginal distributions. However, this approach suffers difficulties when either the inference process or the model is approximate. In this paper, the inference ... 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

Up-down asymmetric tokamaksNov 21 2016Nov 22 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

Topological quantum field theory and hyperkähler geometrySep 25 2000Rozansky and Witten proposed a 3-dimensional sigma-model whose target space is a hyperk\"ahler manifold. They conjectured that this theory has an associated TQFT, with Hilbert spaces given by certain cohomology groups of the hyperk\"ahler manifold. On ... More

2-subnormal quadratic offenders and Oliver's p-group conjectureDec 07 2011Bob Oliver conjectures that if $p$ is an odd prime and $S$ is a finite $p$-group, then the Oliver subgroup $\X(S)$ contains the Thompson subgroup $J_e(S)$. A positive resolution of this conjecture would give the existence and uniqueness of centric linking ... 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

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.

Rationality of motivic zeta-functions for curves with finite abelian group actionsMar 10 2011Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,\sigma)$, where $X$ is a variety over $k$ and $\sigma$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes ... 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

Some Comments on Branes, G-flux, and K-theoryDec 01 2000This is a summary of a talk at Strings2000 explaining three ways in which string theory and M-theory are related to the mathematics of K-theory.

Finite in All DirectionsMay 25 1993We study toroidal compactifications of string theories which include compactification of a timelike coordinate. Some new features in the theory of toroidal compactifications arise. Most notably, Narain moduli space does not exist as a manifold since the ... More

K-Theory from a physical perspectiveApr 02 2003This is an expository paper which aims at explaining a physical point of view on the K-theoretic classification of D-branes. We combine ideas of renormalization group flows between boundary conformal field theories, together with spacetime notions such ... More

Binding Energies of Charged Metal Nanoparticle ConfigurationsAug 03 2010Oct 14 2010The electrostatic interaction between metal spheres is an influential component in the assembly of many nanoscale materials in chemistry. Here we derive a method to calculate the energy and polarizations of metal spheres in arbitrary configurations to ... More

Attractors and ArithmeticJul 08 1998Jul 11 2003We consider attractor varieties arising in the construction of dyonic black holes in Calabi-Yau compactifications of IIB string theory. We show that the attractor varieties are constructed from products of elliptic curves with complex multiplication for ... More

String duality, automorphic forms, and generalized Kac-Moody algebrasOct 27 1997Apr 03 1998This is a summary of a review talk at the Trieste conference on duality symmetries in string theory, April 1997. We review some work that has been done on the relation of BPS states, automorphic forms and the geometry of the quantum ground states of string ... More

2D Yang-Mills Theory and Topological Field TheorySep 09 1994Contribution to the Proceedings of the International Congress of Mathematicians 1994. We review recent developments in the physics and mathematics of Yang-Mills theory in two dimensional spacetimes. This is a condensed version of a forthcoming review ... More

Gravitational Phase Transitions and the Sine-Gordon ModelMar 22 1992We consider the Sine-Gordon model coupled to 2D gravity. We find a nonperturbative expression for the partition function as a function of the cosmological constant, the SG mass and the SG coupling constant. At genus zero, the partition function exhibits ... More

Rooted Graph Minors and Reducibility of Graph PolynomialsApr 15 2017In 2009, Brown gave a set of conditions which when satisfied imply that a Feynman integral evaluates to a multiple zeta value. One of these conditions is called reducibility, which loosely says there is an order of integration for the Feynman integral ... More

Homology representations of unitary reflection groupsMar 21 2013Apr 02 2013This paper continues the study of the poset of eigenspaces of elements of a unitary reflection group (for a fixed eigenvalue), which was commenced in [6] and [5]. The emphasis in this paper is on the representation theory of unitary reflection groups. ... 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

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

Inflationary Quantum Cosmology: General Framework and Exact Bianchi I SolutionJul 29 2004Sep 11 2004Using the methods of loop quantum gravity, we derive a framework for describing an inflationary, homogeneous universe in a purely quantum theory. The classical model is formulated in terms of the Ashtekar-Sen connection variables for a general subclass ... 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

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

H-infinity is not E-infinityOct 19 2009Apr 19 2014We provide an example of a spectrum over S^0 with an H_\infty structure which does not rigidify to an E_3 structure. It follows that in the category of spectra over S^0 not every H_\infty ring spectrum comes from an underlying E_\infty ring spectrum. ... 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

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

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

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

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

Matroids: A Macaulay2 packageNov 14 2015Aug 31 2018We give an overview of the Macaulay2 package Matroids, which contains functionality to create and compute with matroids. Examples highlighting the use of all major functions in the package are provided, along with explanations of some of the algorithms. ... More

Maximal unbalanced familiesSep 11 2012A family of subsets of the set {1,2,...,n} is said to be unbalanced if the convex hull of its characteristic vectors misses the diagonal in the n-cube.The purpose of this article is to develop the combinatorics of maximal unbalanced families. Specifically, ... More

The Variety Generated by A(T) -- Two CounterexamplesApr 17 2013Jul 24 2014We show that V(A(T)) does not have definable principal subcongruences or bounded Maltsev depth. When the Turing machine T halts, V(A(T)) is an example of a finitely generated semilattice based (and hence congruence meet-semidistributive) variety with ... More

Quelques applications de l'Ansatz de Bethe (Some applications of the Bethe Ansatz)Oct 11 1998The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and ... More

Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger EquationAug 18 2013Oct 01 2013We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by N^{\beta -1}V(N^{\beta}\cdot) where \int V\leqslant 0. We develop new techniques in treating the N-body Hamiltonian so that we overcome ... More