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Reid's recipe and derived categoriesDec 24 2008May 15 2012We prove two existing conjectures which describe the geometrical McKay correspondence for a finite abelian G in SL3(C) such that C^3/G has a single isolated singularity. We do it by studying the relation between the derived category mechanics of computing ... More

Derived McKay correspondence via pure-sheaf transformsJun 30 2006Feb 04 2008In most cases where it had been shown to exist the derived McKay correspondence D(Y) --> D^G(C^n) can be written as a Fourier-Mukai transform which sends point sheaves of the crepant resolution Y to pure sheaves in D^G(C^n). We give a sufficient condition ... More

$\mathbb{P}^n$-functorsMay 14 2019We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, ... More

On adjunctions for Fourier-Mukai transformsApr 18 2010Aug 16 2012We show that the adjunction counits of a Fourier-Mukai transform $\Phi$ from $D(X_1)$ to $D(X_2)$ arise from maps of the kernels of the corresponding Fourier-Mukai transforms. In a very general setting of proper separable schemes of finite type over a ... More

On uniqueness of P-twistsNov 17 2017Jun 13 2019We prove that for any $\mathbb{P}^n$-functor $F$, split or non-split, all the convolutions (double cones) of the three-term complex $FHR \rightarrow FR \rightarrow$ Id defining its $\mathbb{P}$-twist are isomorphic.

A derived approach to geometric McKay correspondence in dimension threeMar 20 2008May 15 2012We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail when G is abelian and C^3/G has a single isolated singularity. More precisely, we ... More

Spherical DG-functorsSep 19 2013Oct 19 2015For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A -> B. We construct its associated autoequivalences: the twist T of D(B) and the co-twist F of D(A). We give powerful sufficiency criteria for a quasi-functor to be ... More

Orthogonally spherical objects and spherical fibrationsNov 02 2010Oct 19 2015We introduce a relative version of the spherical objects of Seidel and Thomas. Define an object E in the derived category D(Z x X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z) and D(X) in ... More

Bar category of modules and homotopy adjunction for tensor functorsDec 30 2016Jul 31 2018Given a DG-category A we introduce the bar category of modules Modbar(A). It is a DG-enhancement of the derived category D(A) of A which is isomorphic to the category of DG A-modules with A-infinity morphisms between them. However, it is defined intrinsically ... More

Derived Reid's recipe for abelian subgroups of SL3(C)May 14 2012Sep 26 2014For any finite subgroup G in SL3(C), work of Bridgeland-King-Reid constructs an equivalence between the G-equivariant derived category of C^3 and the derived category of the crepant resolution Y = G-Hilb(C^3) of C^3/G. When G is abelian we show that this ... More

Families of G-constellations over resolutions of quotient singularitiesMay 14 2003Let G be a finite subgroup of GL_n(C). A study is made of the ways in which resolutions of the quotient space C^n / G can parametrise G-constellations, that is, G-regular finite length sheaves. These generalise G-clusters, which are used in the McKay ... More

Natural G-Constellation FamiliesDec 31 2005Dec 25 2008Let G be a finite subgroup of GL_n(C). G-constellations are a scheme-theoretic generalization of orbits of G in C^n. We study flat families of G-constellations parametrised by an arbitrary resolution of the quotient space C^n/G. We develop a geometrical ... More

On uniqueness of P-twistsNov 17 2017Jun 23 2018We prove that for any $\mathbb{P}^n$-functor $F$, split or non-split, all the convolutions (double cones) of the three-term complex $FHR \rightarrow FR \rightarrow$ Id defining its $\mathbb{P}$-twist are isomorphic.

$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategoriesJan 21 2004The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Lagu\~{n}a, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract ... More

What `shape' is space-time?Oct 22 2002Some examples from the mathematics of shape are presented that question some of the almost hidden assumptions behind results on limiting behaviour of finitary approximations to space-time. These are presented so as to focus attention on the observational ... More

Bootstrapping Deep Lexical Resources: Resources for CoursesSep 15 2007We propose a range of deep lexical acquisition methods which make use of morphological, syntactic and ontological language resources to model word similarity and bootstrap from a seed lexicon. The different methods are deployed in learning lexical items ... More

Computing area in group presentationsJun 28 2016Jul 23 2016The width of a word $w$ representing an element in a free group $F(a,b)$ was defined by Jiang to be the minimal $N$ such that $w$ freely equals a product of $N$ conjugates of powers of $a$ and $b$. In 1991 Grigorchuk and Kurchanov gave an algorithm computing ... More

Nonparametric Stochastic Discount Factor DecompositionDec 15 2014Aug 19 2016Stochastic discount factor (SDF) processes in dynamic economies admit a permanent-transitory decomposition in which the permanent component characterizes pricing over long investment horizons. This paper introduces econometric methods to extract the permanent ... More

Quantum Solitons in Affine Toda Field TheoriesOct 03 1991The spectra of $A_r$ affine Toda field theories with imaginary coupling constant, are investigated. Soliton solutions are found, which, despite the non-unitary form of the Lagrangian, have real classical masses and are stable to small perturbations. The ... More

Wilson Loop Area Law for 2D Yang-Mills in Generalized Axial GaugeJan 18 2016Jul 25 2016We prove that Wilson loop expectation values for arbitrary simple closed contours obey an area law up to second order in perturbative two-dimensional Yang-Mills theory. Our analysis occurs within a general family of axial-like gauges, which include and ... More

Death and extended persistence in computational algebraic topologySep 04 2016The main aim of this paper is to explore the ideas of persistent homology and extended persistent homology, and their stability theorems, using ideas from [Bubenik and Scott, 2014; Cohen-Steiner, Edelsbrunner, and Harer, 2007; and Cohen-Steiner, Edelsbrunner, ... More

Topological States in the Kuramoto ModelApr 07 2017The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among the oscillators. In this paper we study steady state solutions of the Kuramoto model by assigning ... More

Simultaneous current graph constructions for minimum triangulations and complete graph embeddingsFeb 01 2019The problems of the genus of the complete graphs and minimum triangulations for each surface were both solved using the theory of current graphs, and each of them divided into twelve different cases, depending on the residue modulo 12 of the number of ... More

Some non-abelian covers of knots with non-trivial Alexander polynomialFeb 18 2019Let $K$ be a tame knot embedded in $\mathbf{S}^3$. We address the problem of finding the minimal degree non-cyclic cover $p:X \rightarrow \mathbf{S}^3 \smallsetminus K$. When $K$ has non-trivial Alexander polynomial we construct finite non-abelian representations ... More

The Seiberg-Witten Equations on Manifolds with Boundary II: Lagrangian Boundary Conditions for a Floer TheoryAug 11 2010Jun 01 2016In this paper, we study the Seiberg-Witten equations on the product R x Y, where Y is a compact 3-manifold with boundary. Following the approach of Salamon and Wehrheim in the instanton case, we impose Lagrangian boundary conditions for the Seiberg- Witten ... More

Bounds on the Norm of the Backward Shift and Related Operators in Hardy and Bergman SpacesFeb 10 2017We study bounds for the backward shift operator $f \mapsto (f(z)-f(0))/z$ and the related operator $f \mapsto f - f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound ... More

Regularity of Extremal Functions in Weighted Bergman and Fock Type SpacesNov 07 2014We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $\text{Re} \int_\Omega f(z) \overline{k(z)}\, ... More

A Class of Profinite Hopf-Galois Extensions Over QOct 14 2016Nov 18 2016For $p$ a prime and $a\in\mathbb{Q}$, where $a$ is not a $p^n$-th power of any rational number, the extension $\mathbb{Q}(w_n)/\mathbb{Q}$ where $w_n=\root p^n \of a$ is separable but non-normal. The Hopf-Galois theory for separable extensions was determined ... More

Ricci flow from some spaces with asymptotically cylindrical singularitiesMay 23 2018We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured ... More

Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph (TROPICAL EXCHANGE) is NP-CompleteOct 14 2016Given a directed graph, it is known that the problem of finnding a set of vertex-disjoint cycles with the maximum total number of vertices (MAX SIZE EXCHANGE) can be solved in polynomial time. Given a vertex-colored graph, if a set of vertices contains ... More

Improvements to Turing's MethodMar 11 2009Aug 05 2010This paper refines the argument of Lehman by reducing the size of the constants in Turing's method. This improvement is given in Theorem 1 and scope for further improvements is also given. Analogous improvements to Dirichlet L-functions and Dedekind zeta-functions ... More

An introduction to varieties in weighted projective spaceApr 08 2016Apr 17 2016Weighted projective space arises when we consider the usual geometric definition for projective space and allow for non-trivial weights. On its own, this extra freedom gives rise to more than enough interesting phenomena, but it is the fact that weighted ... More

On the sum of integers from some multiplicative sets and some powers of integersOct 17 2016We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2\pi i/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely many such integers. ... More

A Class of Profinite Hopf-Galois Extensions Over QOct 14 2016For $p$ a prime and $a\in\mathbb{Q}$, where $a$ is not a $p^n$-th power of any rational number, the extension $\mathbb{Q}(w_n)/\mathbb{Q}$ where $w_n=\root p^n \of a$ is separable but non-normal. The Hopf-Galois theory for separable extensions was determined ... More

A note on the lattice Dirac-Kaehler equationJul 18 1995A lattice version of the Dirac-Kaehler equation (DKE) describing fermions was discussed in articles by Becher and Joos. The decomposition of lattice Dirac-Kaehler fields (inhomogeneous cochains) to lattice Dirac fields remained as an open problem. I show ... More

On the sum of integers from some multiplicative sets and some powers of integersOct 17 2016Oct 18 2016We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2\pi i/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely many such integers. ... More

Adaptive testing on a regression function at a pointAug 15 2014Oct 14 2015We consider the problem of inference on a regression function at a point when the entire function satisfies a sign or shape restriction under the null. We propose a test that achieves the optimal minimax rate adaptively over a range of H\"{o}lder classes, ... More

Structural Properties of ${\cal R}_2$ Part IMar 07 2016This is the first of two papers establishing structural properties of ${\cal R}_2$.

Face distributions of embeddings of complete graphsAug 07 2017Aug 30 2018A longstanding open question of Archdeacon and Craft asks whether every complete graph has a minimum genus embedding with at most one nontriangular face. We exhibit such an embedding for each complete graph except $K_8$, the complete graph on 8 vertices, ... More

The Prouhet-Tarry-Escott problem for Gaussian integersNov 04 2010Feb 13 2011Given natural numbers $n$ and $k$, with $n>k$, the Prouhet-Tarry-Escott (PTE) problem asks for distinct subsets of $\Z$, say $X=\{x_1,...,x_n\}$ and $Y=\{y_1,...,y_n\}$, such that \[x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for $i=1,...,k$. Many partial solutions ... More

On the parity of the multiplicative order of certain products of integers related to Gauss factorialsMar 08 2016Mar 15 2016In this note, we prove that under some conditions, certain products of integers related to Gauss factorials are always quadratic residues.

Ramsey for $\mathcal{R}_{1}$ ultrafilter mappings and their Dedekind cutsJan 13 2014Associated to each ultrafilter $\mathcal{U}$ on $\omega$ and each map $p:\omega\rightarrow \omega$ is a Dedekind cut in the ultrapower $\omega^{\omega}/p( \mathcal{U})$. Blass has characterized, under CH, the cuts obtainable when $\mathcal{U}$ is taken ... More

Quantum Yang-Mills Theory in Two Dimensions: Exact versus PerturbativeAug 25 2015Feb 20 2018The standard Feynman diagrammatic approach to quantum field theories assumes that perturbation theory approximates the full quantum theory at small coupling even when a mathematically rigorous construction of the latter is absent. On the other hand, two-dimensional ... More

Bergman-Hölder Functions, Area Integral Means and Extremal ProblemsAug 02 2016We study certain weighted area integral means of analytic functions in the unit disc. We relate the growth of these means to the property of being mean H\"older continuous with respect to the Bergman space norm. In contrast with earlier work, we use the ... More

Stochastic Feynman Rules for Yang-Mills Theory on the PlaneJul 25 2016Feb 20 2018We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather ... More

A symplectic Gysin sequenceJul 11 2008We use the theory of pseudo-holomorphic quilts to establish a counterpart, in symplectic Floer homology, to the Gysin sequence for the homology of a sphere-bundle. In a motivating class of examples, this "symplectic Gysin sequence" is precisely analogous ... More

On the Discrepancy of the Roots of $x^2+1$ and $x^2+2$ to Prime ModuliMar 07 2011Oct 03 2012In this paper, we make a conjecture (conjecture 1) related to the Bateman-Horn conjecture and proceed to study the roots of $x^2+1$ and $x^2+2$ to prime moduli, assuming the truth of the Bateman-Horn conjecture and conjecture 1 and using the Erd\H{o}s-Turan-Koksma ... More

Extremal Problems in Bergman Spaces and an Extension of Ryabykh's $H^p$ Regularity Theorem For $1<p<\infty$Feb 05 2015We study linear extremal problems in the Bergman space $A^p$ of the unit disc, where $1 < p < \infty$. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if $q \le q_1 < \infty$ and ... More

Quantum Soliton Mass Corrections in SL(N) Affine Toda TheorySep 08 1992The first quantum mass corrections for the solitons of complex $sl(n)$ affine Toda field theory are calculated. The corrections are real and preserve the classical mass ratios. The formalism also proves that the solitons are classically stable.

Finiteness and Paradoxical Decompostions in C*-Dynamical SystemsFeb 21 2015We discuss the interplay between K-theoretical dynamics and the structure theory for certain C*-algebras arising from crossed products. For noncommutative C*-systems we present notions of minimality and topological transitivity in the K-theoretic framework ... More

From abstract alpha-Ramsey theory to abstract ultra-Ramsey theoryJan 15 2016We work within the framework of the Alpha-Theory introduced by Benci and Di Nasso. The Alpha-Theory postulates a few natural properties for an infinite "ideal" number $\alpha$. The formulation provides an elementary axiomatics for the methods of abstract ... More

Quantum Yang-Mills Theory in Two Dimensions: Exact versus PerturbativeAug 25 2015Sep 07 2016The standard Feynman diagrammatic approach to quantum field theories assumes that perturbation theory approximates the full quantum theory at small coupling even when a mathematically rigorous construction of the latter is absent. On the other hand, two-dimensional ... More

Dissecting Equilateral Triangles into Non-Congruent Equilateral Triangles: A Novel Proof of Tutte's ResultDec 15 2014In this paper, we show that an equilateral triangle cannot be dissected into finitely many smaller equilateral triangles, no two of which share two vertices. We do this without the use of Electrical Networks.

Exposing the Hidden Web: An Analysis of Third-Party HTTP Requests on 1 Million WebsitesNov 02 2015This article provides a quantitative analysis of privacy-compromising mechanisms on 1 million popular websites. Findings indicate that nearly 9 in 10 websites leak user data to parties of which the user is likely unaware; more than 6 in 10 websites spawn ... More

A further simplification of Tarski's axioms of geometryJun 01 2013A slight modification to one of Tarski's axioms of plane Euclidean geometry is proposed. This modification allows another of the axioms to be omitted from the set of axioms and proven as a theorem. This change to the system of axioms simplifies the system ... More

Irregular triangulations of complete graphs on 12s vertices in orientable surfacesMar 30 2018May 09 2018We present a family of index 1 abelian current graphs whose derived embeddings can be modified into triangulations of $K_{12s}$ for $s \geq 4$. Our construction is significantly simpler than previous methods for finding genus embeddings of $K_{12s}$, ... More

Revisiting Mayer: Symmetric solutions for sporadic cases of the Map Color TheoremMar 25 2018The original proof of the genus of the complete graphs $K_n$ depended on Mayer's \emph{ad hoc} solutions for $n = 18, 20, 23$. Recently, an improved solution for $K_{20}$ was found by the author. The purpose of this note is to use the theory of current ... More

On Milgram's construction and the Duke embedding conjecturesMar 10 2013Apr 18 2013Milgram constructed a 28-vertex cubic graph of genus 4 that disproved Duke's conjecture relating Betti number to minimum genus. We apply Milgram's method to construct to find graphs of higher genus violating Duke's conjecture, which gives a sharper bound ... More

Selective but not RamseyDec 19 2013We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space $\mathcal{R}$, are the notions of selective for $\mathcal{R}$ and Ramsey for $\mathcal{R}$ equivalent? Every topological Ramsey space $\mathcal{R}$ has ... More

Ricci Flow recovering from pinched discsApr 21 2017We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular point. We conjecture ... More

Uniform Approximation of Extremal Functions in Weighted Bergman SpacesMay 18 2017We discuss approximation of extremal functions by polynomials in the weighted Bergman spaces $A^p_\alpha$ where $-1 < \alpha < 0$ and $-1 < \alpha < p-2$. We obtain bounds on how close the approximation is to the true extremal function in the $A^p_\alpha$ ... More

Bounds on Integral Means of Bergman Projections and their DerivativesMar 13 2015Sep 29 2016We bound integral means of the Bergman projection of a function in terms of integral means of the original function. As an application of these results, we bound certain weighted Bergman space norms of derivatives of Bergman projections in terms of weighted ... More

Wilson Loop Area Law for 2D Yang-Mills in Generalized Axial GaugeJan 18 2016Feb 21 2018We prove that Wilson loop expectation values for arbitrary simple closed contours obey an area law up to second order in perturbative two-dimensional Yang-Mills theory. Our analysis occurs within a general family of axial-like gauges, which include and ... More

Lagrangian correspondences and Donaldson's TQFT construction of the Seiberg-Witten invariants of 3-manifoldsNov 15 2011Sep 08 2013Using Morse-Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3-manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold ... More

Nonparametric Stochastic Discount Factor DecompositionDec 15 2014May 19 2017Stochastic discount factor (SDF) processes in dynamic economies admit a permanent-transitory decomposition in which the permanent component characterizes pricing over long investment horizons. This paper introduces an empirical framework to analyze the ... More

On primes of the form $n_1^u + n_2^v + k$, on averageFeb 07 2012Oct 03 2012We use the Hardy-Littlewood circle method to study primes of the form $n_1^u + n_2^v + k$, on average.

Gram's Law Fails a Positive Proportion of the TimeNov 06 2008This paper extends the work done by Titchmarsh on Gram's Law (an attempt to locate the zeroes of the zeta-function on the critical line). Herewith it is shown that a positive proportion of Gram intervals violate Gram's Law; and also that a weaker notion ... More

The Bouniakowsky conjecture and the density of polynomial roots to prime moduliJun 06 2009We establish a result linking the Bouniakowsky conjecture and the density of polynomial roots to prime moduli.

Volume Bounds for the Phase-locking Region in the Kuramoto Model with Asymmetric CouplingAug 16 2018The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among them. It has been observed that if the natural frequencies of the oscillators are similar they ... More

The Prescribed Ricci Curvature Problem on Three-Dimensional Unimodular Lie GroupsJul 12 2016Let G be a three-dimensional unimodular Lie group, and let T be a left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g, c) consisting of a left-invariant Riemannian metric ... More

Tidal Tails of Interacting GalaxiesOct 25 2012Oct 30 2012This paper has been withdrawn by the author. A computer simulation of two galaxies, passing in parabolic orbits, was made to show their interaction and the tidal patterns formed. The galaxies were modelled as a point masses surrounded by 5 densely packed, ... More

Stochastic Feynman Rules for Yang-Mills Theory on the PlaneJul 25 2016Oct 13 2016We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather ... More

Geometric Aspects of Multiagent SystemsOct 25 2002Recent advances in Multiagent Systems (MAS) and Epistemic Logic within Distributed Systems Theory, have used various combinatorial structures that model both the geometry of the systems and the Kripke model structure of models for the logic. Examining ... More

The Prescribed Ricci Curvature Problem on Three-Dimensional Unimodular Lie GroupsJul 12 2016Oct 05 2016Let G be a three-dimensional unimodular Lie group, and let T be a left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g, c) consisting of a left-invariant Riemannian metric ... More

Computing area in presentations of the trivial groupJun 28 2016Nov 01 2016We give polynomial-time dynamic-programming algorithms finding the areas of words in the presentations $\langle a, b \mid a, b \rangle$ and $\langle a, b \mid a^k, b^k; \ k \in \mathbb{N} \rangle$ of the trivial group. In the first of these two cases, ... More

Quantization of the Nonlinear Sigma Model RevisitedAug 19 2014Sep 06 2016We revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous, mathematical point of view. Our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may ... More

The Prescribed Ricci Curvature Problem on Three-Dimensional Unimodular Lie GroupsJul 12 2016Oct 09 2017Let G be a three-dimensional unimodular Lie group, and let T be a left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g, c) consisting of a left-invariant Riemannian metric ... More

Hamiltonian handleslides for Heegaard Floer homologyJan 03 2008Feb 27 2008A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms ... More

Cohomogeneity-One Quasi-Einstein MetricsJul 28 2018Let $G/H$ be a connected, simply connected homogeneous space of a compact Lie group $G$. We study $G$-invariant quasi-Einstein metrics on the cohomogeneity one manifold $G/H\times (0,1)$ imposing the so-called monotypic condition on $G/H$. We obtain estimates ... More

Computing area in presentations of the trivial groupJun 28 2016Dec 16 2016We give polynomial-time dynamic-programming algorithms finding the areas of words in the presentations $\langle a, b \mid a, b \rangle$ and $\langle a, b \mid a^k, b^k; \ k \in \mathbb{N} \rangle$ of the trivial group. In the first of these two cases, ... More

MF Actions and K-theoretic DynamicsApr 16 2014We study the interplay of C*-dynamics and K-theory. Notions of chain recurrence for transformations groups (X,G) and MF actions for non-commutative C*-dynamical systems (A,G) are translated into K-theoretical language, where purely algebraic conditions ... More

Winding of simple walks on the square latticeSep 12 2017Jan 18 2018A method is described to count simple diagonal walks on $\mathbb{Z}^2$ with a fixed starting point and endpoint on one of the axes and a fixed winding angle around the origin. The method involves the decomposition of such walks into smaller pieces, the ... More

A new upper bound for $|ζ(1+ it)|$Oct 25 2012It is known that $|\zeta(1+ it)|\ll (\log t)^{2/3}$. This paper provides a new explicit estimate, viz.\ $|\zeta(1+ it)|\leq 3/4 \log t$, for $t\geq 3$. This gives the best upper bound on $|\zeta(1+ it)|$ for $t\leq 10^{2\cdot 10^{5}}$.

Extremal Problems in Bergman Spaces and an Extension of Ryabykh's TheoremJan 31 2013We study linear extremal problems in the Bergman space $A^p$ of the unit disc for $p$ an even integer. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if the Taylor coefficients of ... More

Stable Commutator Length in Amalgamated Free ProductsOct 08 2013May 12 2014We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups ... More

Stochastic Feynman Rules for Yang-Mills Theory on the PlaneJul 25 2016We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather ... More

The Analytic Structure of Trigonometric S MatricesMay 11 1993May 17 1993$S$-matrices associated to the vector representations of the quantum groups for the classical Lie algebras are constructed. For the $a_{m-1}$ and $c_m$ algebras the complete $S$-matrix is found by an application of the bootstrap equations. It is shown ... More

Towards in-situ cleaning of a trapped ion quantum computerJul 08 2012Sep 08 2012A plasma glow discharge system was created using a conventional microwave oven to ignite and maintain the plasma. The system was used for plasma cleaning and its properties were analysed to assess its viability for removing surface contaminants, which ... More

Categoricity for Patterns of Order 2Apr 09 2011A categoricity theorem is established for patterns of resemblance of order 2 showing that the order in which patterns arise in a wide range of hierarchies is the same.

Anisotropic Function Spaces and Elliptic Boundary Value ProblemsOct 19 2010Mar 01 2011In this paper, we study anisotropic Bessel potential and Besov spaces, where the anisotropy measures the extra amount of regularity in certain directions. Some basic properties of these spaces are established along with applications to elliptic boundary ... More

Structural Properties of ${\cal R}_2$ Part IIMar 07 2016This is the second of two papers establishing structural properties of ${\cal R}_2$.

A modest improvement on the function $S(T)$Oct 21 2010This paper contains a small improvement to the explicit bounds on the growth of the function $S(T)$. It is shown how more substantial improvements are possible if one has better explicit bounds on the growth of $|\zeta(\frac{1}{2}+it)|$.

As Scales Become Separated: Lectures on Effective Field TheoryMar 08 2019These lectures aim to provide a pedagogical introduction to the philosophical underpinnings and technical features of Effective Field Theory (EFT). Improving control of $S$-matrix elements in the presence of a large hierarchy of physical scales $m \ll ... More

Analysis of a Double Kruskal TheoremMar 07 2016The strength of an extension of Kruskal's Theorem to certain pairs of cohabitation trees is calibrated.

Continuity of the asymptotics of expected zeros of fewnomialsNov 27 2013In "Random complex fewnomials, I," B. Shiffman and S. Zelditch determine the limiting formula as N goes to infinity of the (normalized) expected distribution of complex zeros of a system of k random n-nomials in m variables where the coefficients are ... More

Solution of Extremal Problems in Bergman Spaces Using the Bergman ProjectionNov 21 2013In this paper we discuss the explicit solution of certain extremal problems in Bergman spaces. In order to do this, we develop methods to calculate the Bergman projections of various functions. As a special case, we deal with canonical divisors for certain ... More

Ricci Flow Emerging from Rotationally Symmetric Degenerate NeckpinchesNov 13 2014Feb 05 2015In previous work, Angenent, Isenberg, and Knopf created type-II Ricci flow neckpinch singularities. In this paper we construct solutions to Ricci flow whose initial data is the singular metric resulting from these singularities. We show in particular ... More

A Pohozaev Identity on Warped Product SolitonsFeb 02 2016Mar 10 2016Warped product metrics are a class of Riemannian metrics on cross products $B \times F$ which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove ... More

The Seiberg-Witten Equations on Manifolds with Boundary I: The Space of Monopoles and Their Boundary ValuesAug 11 2010Sep 08 2013In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a Banach manifold ... More

Characteristic Subgroup Lattices and Hopf-Galois StructuresJun 18 2018The Hopf-Galois structures on normal extensions $K/k$ with $G=Gal(K/k)$ are in one-to-one correspondence with the set of regular subgroups $N\leq B=Perm(G)$ that are normalized by the left regular representation $\lambda(G)\leq B$. Each such $N$ corresponds ... More