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Traversing three-manifold triangulations and spinesDec 06 2018Dec 20 2018A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of ... More

Hypernom: Mapping VR Headset Orientation to S^3Jul 21 2015Hypernom is a virtual reality game. The cells of a regular 4D polytope are radially projected to S^3, the sphere in 4D space, then stereographically projected to 3D space where they are viewed in the headset. The orientation of the headset is given by ... More

Non-euclidean virtual reality I: explorations of $\mathbb{H}^3$Feb 13 2017We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.

A generalisation of the deformation varietyApr 13 2009Jun 18 2012Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's ... More

On spun-normal and twisted squares surfacesOct 07 2008Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann ... More

Detection of incompressible surfaces in hyperbolic punctured torus bundlesOct 10 2006Oct 08 2008Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible ... More

Connectivity of triangulations without degree one edges under 2-3 and 3-2 movesMay 30 2016Sep 20 2016Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence ... More

Non-euclidean virtual reality II: explorations of $\mathbb{H}^2\times\mathbb{E}$Feb 16 2017We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.

The Quaternion Group as a Symmetry GroupApr 26 2014We briefly review the distinction between abstract groups and symmetry groups of objects, and discuss the question of which groups have appeared as the symmetry groups of physical objects. To our knowledge, the quaternion group (a beautiful group with ... More

Visualizing Hyperbolic HoneycombsNov 08 2015We explore visual representations of tilings corresponding to Schl\"afli symbols. In three-dimensions, we call these tilings "honeycombs". Schl\"afli symbols encode, in a very efficient way, regular tilings of spherical, euclidean and hyperbolic spaces ... More

Pseudo-Developing Maps for Ideal Triangulations I: Essential Edges and Generalised Hyperbolic Gluing EquationsJul 06 2011Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a generalisation of ... More

Sculptures in S^3Apr 23 2012May 17 2012We construct a number of sculptures, each based on a geometric design native to the three-dimensional sphere. Using stereographic projection we transfer the design from the three-sphere to ordinary Euclidean space. All of the sculptures are then fabricated ... More

Triple gearApr 25 2013A relatively common sight in graphic designs is a planar arrangement of three gears in contact. However, since neighboring gears must rotate in opposite directions, none of the gears can move. We give a non-planar, and non-frozen, arrangement of three ... More

Conformally correct tilingsDec 25 2016We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles.

Puzzling the 120-cellOct 14 2013Nov 20 2015We introduce Quintessence: a family of burr puzzles based on the geometry and combinatorics of the 120-cell. We discuss the regular polytopes, their symmetries, the dodecahedron as an important special case, the three-sphere, and the quaternions. We then ... More

Essential loops in taut ideal triangulationsFeb 08 2019In this note we combinatorialise a technique of Novikov. We use this to prove that, in a three-manifold equipped with a taut ideal triangulation, any vertical or normal loop is essential in the fundamental group.

Squares that Look Round: Transforming Spherical ImagesMay 04 2016We propose M\"obius transformations as the natural rotation and scaling tools for editing spherical images. As an application we produce spherical Droste images. We obtain other self-similar visual effects using rational functions, elliptic functions, ... More

Visualizing Hyperbolic HoneycombsNov 08 2015Nov 15 2016We explore visual representations of tilings corresponding to Schl\"afli symbols. In three dimensions, we call these tilings "honeycombs". Schl\"afli symbols encode, in a very efficient way, regular tilings of spherical, euclidean and hyperbolic spaces ... More

Incompressible surfaces in handlebodies and boundary reducible 3-manifoldsNov 01 2009Jan 04 2011We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. ... More

Non-geometric veering triangulationsJun 25 2014Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if ... More

Veering triangulations admit strict angle structuresNov 16 2010Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit ... More

Triangulations of 3-manifolds with essential edgesDec 01 2014Apr 29 2015We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to ... More

1-efficient triangulations and the index of a cusped hyperbolic 3-manifoldMar 21 2013In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic ... More

Triangulations of hyperbolic 3-manifolds admitting strict angle structuresNov 14 2011Jul 06 2012It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal ... More

Collection of abstracts of the Workshop on Triangulations in Geometry and Topology at CG Week 2014 in KyotoJun 02 2014This workshop about triangulations of manifolds in computational geometry and topology was held at the 2014 CG-Week in Kyoto, Japan. It focussed on computational and combinatorial questions regarding triangulations, with the goal of bringing together ... More

Applying Bayesian Inference to Galileon Solutions of the Muon ProblemSep 23 2016We derive corrections to atomic energy levels from disformal couplings in Galileon theories. Through Bayesian inference, we constrain the cut-off radii and Galileon scale via these corrections. To connect different atomic systems, we assume the various ... More

Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust OptimizationMay 30 2016We investigate the use of distributionally robust optimization (DRO) as a tractable tool to recover the asymptotic statistical guarantees provided by the Central Limit Theorem, for maintaining the feasibility of an expected value constraint under ambiguous ... More

Nuclear k_T in d+Au Collisions from Multiparticle Jet Reconstruction at STARNov 01 2005This paper presents the most recent nuclear k_T measurements from STAR derived from multiparticle jet reconstruction of d+Au and p+p collisions at sqrt(s)=200 GeV. Since jets reconstructed from multiple particles are relatively free of fragmentation biases, ... More

A conceptual breakthrough in sphere packingNov 05 2016This expository paper describes Viazovska's breakthrough solution of the sphere packing problem in eight dimensions, as well as its extension to twenty-four dimensions by Cohn, Kumar, Miller, Radchenko, and Viazovska.

A short proof of the simple continued fraction expansion of eJan 26 2006Feb 25 2006This note presents an especially short and direct variant of Hermite's proof of the simple continued fraction expansion e = [2,1,2,1,1,4,1,1,6,...] and explains some of the motivation behind it.

Solutions to Bestvina & Feighn's Exercises on Limit GroupsApr 06 2006This article gives solutions to the exercises in Bestvina and Feighn's paper on Sela's work on limit groups. We prove that all constructible limit groups are limit groups and give an account of the shortening argument of Rips and Sela.

Hall's Theorem for limit groupsMay 19 2006Jun 07 2007A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work ... More

Connections between Floer-type invariants and Morse-type invariants of Legendrian knotsNov 09 2009Sep 15 2010We define an algebraic/combinatorial object on the front projection $\Sigma$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating ... More

An Inverse Ackermannian Lower Bound on the Local Unconditionality Constant of the James SpaceMar 16 2015The proof that the James space is not locally unconditional appears to be non-constructive, since it makes use of an ultraproduct construction. Using proof mining, we extract a constructive proof and obtain a lower bound on the growth of the local unconditionality ... More

Constructive Gelfand duality for non-unital commutative C*-algebrasDec 05 2014Feb 03 2015We prove constructive versions of various usual results related to the Gelfand duality. Namely, that the constructive Gelfand duality extend to a duality between commutative nonunital C*-algebras and locally compact completely regular locales, that ideals ... More

New upper bounds on sphere packings IIOct 01 2001Jun 28 2002We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related results and conjectures.

Central limit theorem for supercritical binary homogeneous Crump-Mode-Jagers processesSep 22 2015Nov 18 2016We consider a supercritical general branching population where the lifetimes of individuals are i.i.d. with arbitrary distribution and each individual gives birth to new individuals at Poisson times independently from each others. The population counting ... More

Frobenius-Schur indicators for near-group and Haagerup-Izumi fusion categoriesOct 19 2015Jun 09 2017Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion ... More

Constructing Sequences One Step at a TimeSep 18 2016We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's Lemma (WKL), ... More

Epsilon Substitution for $ID_1$ via Cut-EliminationSep 01 2015The $\epsilon$-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory $ID_1$ using a variant of the cut-elimination formalism introduced ... More

Liouville Brownian Motion and Thick Points of the Gaussian Free FieldDec 04 2014We find a lower bound for the Hausdorff dimension that a Liouville Brownian motion spends in $\alpha$-thick points of the Gaussian Free Field, where $\alpha$ is not necessarily equal to the parameter used in the construction of the geometry. This completes ... More

CLTs for general branching processes related to splitting treesSep 22 2015We consider a general branching population where the lifetimes of individuals are i.i.d. with arbitrary distribution and each individual gives birth to new individuals at Poisson times independently from each others. The population counting process is ... More

Les indicateus avancés de l'inflation en RDCongoSep 22 2015This study aims to identify the leading of inflation indicators of monetary policy in DRC. The results reveal that the most relevant inflation indicators usually come from the monetary origin than the real sector. Variance decomposition analyzes place ... More

Robust Sensitivity Analysis for Stochastic SystemsMar 02 2013Jul 12 2015We study a worst-case approach to measure the sensitivity to model misspecification in the performance analysis of stochastic systems. The situation of interest is when only minimal parametric information is available on the form of the true model. Under ... More

Full Jet Reconstruction in d+Au and p+p Collisions at RHICMar 16 2004The STAR detector is well suited for investigating jet production at RHIC. It has a large acceptance for both charged particles and electromagnetic radiation, so that reconstructed jets contain a large fraction of the particles descending from an initial ... More

Packing, coding, and ground statesMar 16 2016These are the lecture notes from my 2014 PCMI graduate summer school lectures. In these lectures, we'll study simple models of materials from several different perspectives: geometry (packing problems), information theory (error-correcting codes), and ... More

Non-unital polygraphs form a presheaf categoriesNov 02 2017We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf category. More generally we develop a new criterion for ... More

Approximations of the allelic frequency spectrum in general supercritical branching populationsJan 25 2017We consider a general branching population where the lifetimes of individuals are i.i.d.\ with arbitrary distribution and where each individual gives birth to new individuals at Poisson times independently from each other. In addition, we suppose that ... More

On toposes generated by cardinal finite objectsMay 19 2015Apr 05 2016We give a characterizations of toposes which admit a generating family of objects which are internally cardinal finite (i.e. Kuratowski finite and decidable) in terms of "topological" conditions. The central result is that, constructively, a hyperconnected ... More

The convolution algebra of an absolutely locally compact toposDec 31 2016We introduce a class of toposes called "absolutely locally compact" toposes and of "admissible" sheaf of rings over such toposes. To any such ringed topos $(\mathcal{T},A)$ we attach an involutive convolution algebra $\mathcal{C}_c(\mathcal{T},A)$ which ... More

Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete $C^{*}$-categoriesJan 28 2015** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal ... More

An abstract elementary class non-axiomatizable in $L_{(\infty,κ)}$Dec 03 2018We show that for any uncountable cardinal $\lambda$, the category of sets of cardinality at least $\lambda$ and monomorphisms between them cannot appear as the category of point of a topos, in particular is not the category of models of a $L_{(\infty,\omega)}$-theory. ... More

Spectral instability for even non-selfadjoint anharmonic oscillatorsJan 22 2013Oct 17 2013We study the instability of the spectrum for a class of non-selfadjoint anharmonic oscillators, estimating the behavior of the instability indices (i. e. the norm of spectral projections) associated with the large eigenvalues of these oscillators. We ... More

The Bernstein-Sato b-function for the complement of the open $SL_n$-orbit on a triple flag varietyJun 30 2015We calculate Bernstein-Sato b-functions for $f_{G^3}^\lambda$, a $SL_n$-invariant section of a line bundle on $SL_n/B \times SL_n/B \times \mathbb{P}^{n - 1}$ whose zero-set is the complement of the open $G$-diagonal orbit. The proof uses a similar calculation ... More

Effect Of Running Shoes on Foot Impact During RunningSep 07 2016Running is part of almost every sport, and requires a great amount of stamina, endurance, mental toughness and overall strength. At every step, the foot experiences ground reaction forces necessary to support the motion of the body. With the advancements ... More

A Representation Theorem for Smooth Brownian MartingalesMay 02 2012Nov 05 2015We show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times the Malliavin ... More

A parallel repetition theorem for all entangled gamesApr 15 2016The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. ... More

A quantum lower bound for distinguishing random functions from random permutationsOct 10 2013Dec 20 2013The problem of distinguishing between a random function and a random permutation on a domain of size $N$ is important in theoretical cryptography, where the security of many primitives depend on the problem's hardness. We study the quantum query complexity ... More

Frobenius-Schur indicators for near group and Haagerup-Izumi fusion categoriesOct 19 2015Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion ... More

UV photolysis, organic molecules in young disks, and the origin of meteoritic amino acidsDec 14 2011The origin of complex organic molecules such as amino acids and their precursors found in meteorites and comets is unknown. Previous studies have accounted for the complex organic inventory of the Solar System by aqueous chemistry on warm meteoritic parent ... More

Projective geometry over F_1 and the Gaussian binomial coefficientsJul 07 2004There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is ... More

Symmetrization of monoïds as hypergroupsSep 08 2013We adapt the construction of the Grothendieck group associated to a commutative mono\"id to handle idempotent mono\"ids. Our construction works for a restricted class of commutative mono\"ids, it agrees with the Grothendieck group construction in many ... More

Second Yamabe Constant on Riemannian ProductsMay 05 2015Dec 01 2016Let $(M^m,g)$ be a closed Riemannian manifold $(m\geq 2)$ of positive scalar curvature and $(N^n,h)$ any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second $N-$Yamabe constant of $(M\times N,g+th)$ as $t$ goes ... More

Alternating quotients of free groupsApr 30 2010Dec 09 2011We strengthen Marshall Hall's Theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let {g_1,...,g_n} be a finite subset ... More

Virtual retractions, conjugacy separability and omnipotenceSep 13 2008We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe ... More

Measure theory over boolean toposesNov 06 2014In this paper we develop a notion of measure theory over boolean toposes which is analogous to noncommutative measure theory, i.e. to the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative ... More

2-adic behavior of numbers of domino tilingsAug 30 2000We study the 2-adic behavior of the number of domino tilings of a 2n-by-2n square as nvaries. It was previously known that this number was of the form 2^n f(n)^2, where f(n) is an odd, positive integer. We show that the function f is uniformly continuous ... More

On the semi-classical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domainMay 23 2014In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2\Delta+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\Omega\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is assumed to ... More

Spectral projections of the complex cubic oscillatorOct 17 2013We prove the spectral instability of the complex cubic oscillator $-\frac{d^2}{dx^2}+ix^3+i\alpha x$ for non-negative values of the parameter $\alpha$, by getting the exponential growth rate of $\|\Pi_n(\alpha)\|$, where $\Pi_n(\alpha)$ is the spectral ... More

The Effects Of Computerizing Banking OperationsAug 01 2013Computerizing banking operation has a far-reaching consequences,both positively and negatively.But here,i have been able to deal with both effects and proffer solutions on the best way to go about computerizing banking operations.

Sensitivity to Serial Dependency of Input Processes: A Robust ApproachJun 21 2016Procedures in assessing the impact of serial dependency on performance analysis are usually built on parametrically specified models. In this paper, we propose a robust, nonparametric approach to carry out this assessment, by computing the worst-case ... More

A Geometric Bohr toposFeb 06 2015In this short note, we construct a variant of the Bohr topos of a C*-algebra which takes into account the topology of the algebra in a finer way and such that this construction is stable under pullback along geometric morphisms. This generalizes a construction ... More

Symmetry and specializability in continued fractionsAug 30 2000We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following theorem: Suppose f(x) ... More

Toposes, quantales and C* algebras in the atomic caseNov 14 2013We start by reviewing the relation between toposes and Grothendieck quantales. We improve results of previous work on this relation by giving both a characterisation of the map from the tensor product of two internal sup-lattices to another sup-lattice ... More

Complete C*-categories and a topos theoretic Green-Julg theoremDec 10 2015We investigate what would be a correct definition of categorical completeness for C*-categories and propose several variants of such a definition that make the category of Hilbert modules over a C*-algebra a free (co)completion. We extend results about ... More

An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo RotaApr 21 2009We sketch the outlines of Gian Carlo Rota's interaction with the ideas that Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota variously called 'Grassmann-Cayley ... More

One-ended subgroups of graphs of free groups with cyclic edge groupsFeb 14 2011Feb 26 2011Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of infinite index. ... More

A formalism for the study of Natural Tensors Fields of type (0,2) on Manifolds and FibrationsDec 11 2008Dec 15 2009In order to study tensor fields of type (0,2) on manifolds and fibrations we introduce the notion of s-spaces. With the help of these objects we generalized the concept of natural tensor without making use of the theory of natural operators and differential ... More

Regular polygraphs and the Simpson conjectureJul 07 2018We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and the "regular" conjecture, involving two different notions ... More

Algebraic models of homotopy types and the homotopy hypothesisSep 15 2016We introduce and study a notion of cylinder coherator similar to the notion of Grothendieck coherator which define more flexible notion of weak infinity groupoids. We show that each such cylinder coherator produces a combinatorial semi-model category ... More

Symmetries in LDDMM with higher order momentum distributionsJun 14 2013Jul 17 2013In some implementations of the Large Deformation Diffeomorphic Metric Mapping formulation for image registration we consider the motion of particles which locally translate image data. We then lift the motion of the particles to obtain a motion on the ... More

Applying Bayesian Inference to Galileon Solutions of the Muon ProblemSep 23 2016Nov 14 2016We derive corrections to atomic energy levels from disformal couplings in Galileon theories. Through Bayesian inference, we constrain the cut-off radii and Galileon scale via these corrections. To connect different atomic systems, we assume the various ... More

Second Yamabe Constant on Riemannian ProductsMay 05 2015Let $(M^m,g)$ be a closed Riemannian manifold $(m\geq 2)$ of positive scalar curvature and $(N^n,h)$ any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second $N-$Yamabe constant of $(M\times N,g+th)$ as $t$ goes ... More

Spectral instability of some non-selfadjoint anharmonic oscillatorsJan 22 2013The purpose of this Note is to highlight the spectral instability of some non-selfadjoint differential operators, by studying the growth rate of the norms of the spectral projections $\Pi_n$ associated with their eigenvalues. More precisely, we are concerned ... More

Hadronic Vacuum Polarization in True MuoniumNov 14 2016The leading-order hadronic vacuum polarization contribution to the hyperfine splitting of true muonium is reevaluated in two ways. The first considers a more complex pionic form factor and better estimates of the perturbative QCD contributions. The second, ... More

Cohomology of Commuting Varieties of Connected Compact Reductive Lie GroupsAug 05 2016We calculate the rational cohomology of the commuting variety $X_{G, n}$ consisting of $n$-tuples of commuting elements of a compact reductive group $G$. This is done by studying a map from a related variety $Y_{G, n}$, which has easily calculated cohomology. ... More

Order and disorder in energy minimizationMar 16 2010Jun 20 2012How can we understand the origins of highly symmetrical objects? One way is to characterize them as the solutions of natural optimization problems from discrete geometry or physics. In this paper, we explore how to prove that exceptional objects, such ... More

Residually free 3-manifoldsFeb 06 2008Sep 12 2008We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M \cong \Sigma\times ... More

Isoparametric functions and nodal solutions of the Yamabe equationJul 02 2018We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.

A modification of the Chang-Wilson-Wolff Inequality via the Bellman FunctionMay 26 2018We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we present how it was originally used by Donald Burkholder to prove $L^p$ boundedness of the $\pm 1$ martingale ... More

The localic Istropy group of a toposJun 15 2017It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T. We show that this group is in fact the group of points ... More

What do ultraproducts remember about the original structures?Apr 28 2018We describe a syntactic method for taking proofs which use ultraproducts and translating them into direct, constructive proofs.

On generating a diminimal set of polyhedral maps on the torusJul 17 2001We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal ... More

Elementarily free groups are subgroup separableNov 16 2005Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela.

Weak model categories in classical and constructive mathematicsJul 07 2018We introduce a notion of "weak model category". It is a weakening of the classical notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunction, Quillen equivalence and most of the usual categorical homotopy theory ... More

Copula-Based Univariate Time Series Structural Shift Identification TestJul 05 2016An approach is proposed to determine structural shift in time-series assuming non-linear dependence of lagged values of dependent variable. Copulas are used to model non-linear dependence of time series components.

Localic Metric spaces and the localic Gelfand dualityNov 04 2014In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localic C*-algebras. In order to do so we develop a constructive ... More

Priority Arguments and Epsilon SubstitutionsDec 18 2008Kreisel has observed that the termination proof for Hilbert's epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments ... More

Dynamic instabilities of fracture under biaxial strain using a phase field modelFeb 23 2004Jun 11 2004We present a phase field model of the propagation of fracture under plane strain. This model, based on simple physical considerations, is able to accurately reproduce the different behavior of cracks (the principle of local symmetry, the Griffith and ... More

Relatively exchangeable structuresSep 22 2015Oct 01 2015We study random relational structures that are \emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure $\mathfrak{M}$. When $\mathfrak{M}$ has {\em trivial definable closure}, every ... More