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The A-truncated K-moment problemOct 25 2012Aug 28 2014Let A be a finite subset of N^n, and K be a compact semialgebraic set in R^n. An A-tms is a vector y indexed by elements in A. The A-truncated K-moment problem (A-TKMP) studies whether a given A-tms y admits a K-measure or not. This paper proposes a numerical ... More

A Matrix Positivstellensatz with lifting polynomialsJan 15 2018Given the projections of two semialgebraic sets defined by polynomial matrix inequalities, it is in general difficult to determine whether one is contained in the other. To address this issue we propose a new matrix Positivstellensatz that uses lifting ... More

Optimality Conditions and Finite Convergence of Lasserre's HierarchyJun 01 2012Apr 15 2013Lasserre's hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally. This paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of Lasserre's ... More

Symmetric Tensor Nuclear NormsMay 28 2016This paper studies nuclear norms of symmetric tensors. As recently shown by Friedland and Lim, the nuclear norm of a symmetric tensor can be achieved at a symmetric decomposition. We discuss how to compute symmetric tensor nuclear norms, depending on ... More

Nearly Low Rank Tensors and Their ApproximationsDec 23 2014The low rank tensor approximation problem (LRTAP) is to find a tensor whose rank is small and that is close to a given one. This paper studies the LRTAP when the tensor to be approximated is close to a low rank one. Both symmetric and nonsymmetric tensors ... More

Discriminants and Nonnegative PolynomialsFeb 10 2010Apr 23 2010For a semialgebraic set K in R^n, let P_d(K) be the cone of polynomials in R^n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary. When K=R^n and d is even, we show that its boundary lies on the irreducible ... More

Polynomial Matrix Inequality and Semidefinite RepresentationAug 03 2009Mar 28 2011Consider a convex set S defined by a matrix inequality of polynomials or rational functions over a domain. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a higher dimensional ... More

Linear Optimization with Cones of Moments and Nonnegative PolynomialsMay 13 2013Jul 17 2014Let A be a finite subset of N^n and R[x]_A be the space of real polynomials whose monomial powers are from A. Let K be a compact basic semialgebraic set of R^n such that R[x]_A contains a polynomial that is positive on K. Denote by P_A(K) the cone of ... More

Polynomial Optimization with Real VarietiesNov 08 2012Jun 04 2013We consider the optimization problem of minimizing a polynomial f(x) subject to polynomial constraints h(x)=0, g(x)>=0. Lasserre's hierarchy is a sequence of sum of squares relaxations for finding the global minimum. Let K be the feasible set. We prove ... More

Convex Hulls of Quadratically Parameterized Sets With Quadratic ConstraintsOct 11 2011Let V be a semialgebraic set parameterized by quadratic polynomials over a quadratic set T. This paper studies semidefinite representation of its convex hull by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by ... More

Tight Relaxations for Polynomial Optimization and Lagrange Multiplier ExpressionsJan 06 2017Apr 06 2018This paper proposes tight semidefinite relaxations for polynomial optimization. The optimality conditions are investigated. We show that generally Lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical ... More

Local Versus Global Conditions in Polynomial OptimizationMay 01 2015This paper reviews local and global optimality conditions in polynomial optimization. We summarize the relationship between them.

Sum of Squares Method for Sensor Network LocalizationMay 24 2006Sep 18 2007This paper has been withdrawn by the author due to its publication

Free Semidefinite Representation of Matrix Power FunctionsMay 18 2013Oct 09 2014Consider the matrix power function X^p defined over the cone of positive definite matrices S^{n}_{++}. It is known that X^p is convex over S^{n}_{++} if p is in [-1,0] or [1,2] and X^p is concave over S^{n}_{++} if p is in [0,1]. We show that the hypograph ... More

Low Rank Symmetric Tensor ApproximationsSep 06 2017For a given symmetric tensor, we aim at finding a new one whose symmetric rank is small and that is close to the given one. There exist linear relations among the entries of low rank symmetric tensors. Such linear relations can be expressed by polynomials, ... More

The Hierarchy of Local Minimums in Polynomial OptimizationNov 17 2013Nov 25 2014This paper studies the hierarchy of local minimums of a polynomial in the space. For this purpose, we first compute H-minimums, for which the first and second order optimality conditions are satisfied. To compute each H-minimum, we construct a sequence ... More

Certifying Convergence of Lasserre's Hierarchy via Flat TruncationJun 13 2011Aug 06 2012This paper studies how to certify the convergence of Lasserre's hierarchy of semidefinite programming relaxations for solving multivariate polynomial optimization. We propose flat truncation as a general certificate for this purpose. Assume the set of ... More

An Exact Jacobian SDP Relaxation for Polynomial OptimizationJun 11 2010Given polynomials f(x), g_i(x), h_j(x), we study how to minimize f on the semialgebraic set S = { x \in R^n: h_1(x)=...=h_{m_1}(x) =0, g_1(x) >= 0, ..., g_{m_2}(x) >= 0}. Let f_{min} be the minimum of f on S. Suppose S is nonsingular and f_{min} is achievable ... More

First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular PolynomialsJun 28 2008A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite ... More

Generating Polynomials and Symmetric Tensor DecompositionsAug 25 2014Oct 02 2015This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization ... More

Algebraic Degree of Polynomial OptimizationFeb 09 2008Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on optimizers, ... More

Real Eigenvalues of nonsymmetric tensorsMar 24 2015This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the contrast, every nonsymmetric ... More

Matrix Cubes Parametrized by EigenvaluesApr 28 2008An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix cube problems whose constraints are the minimum and maximum eigenvalue function on an affine space of symmetric matrices. An LMI representation ... More

The Split Feasibility Problem with PolynomialsJul 28 2017This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations for representing ... More

Semidefinite Relaxations for Best Rank-1 Tensor ApproximationsAug 29 2013May 28 2014This paper studies the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors, this is equivalent ... More

Sparse SOS Relaxations for Minimizing Functions that are Summations of Small PolynomialsJun 20 2006Oct 05 2007This paper discusses how to find the global minimum of functions that are summations of small polynomials (``small'' means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational ... More

Positive Maps and Separable MatricesApr 24 2015Mar 26 2016A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite ... More

Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment ProblemsAug 22 2009Sep 16 2009We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence $y$, we show that $y$ lies in the closure of truncated moment ... More

On the complexity of Putinar's PositivstellensatzOct 14 2005Sep 18 2007This paper has been withdrawn by the author due to its publication

Hankel tensor decompositions and ranksJun 12 2017Jan 28 2019Hankel tensors are generalizations of Hankel matrices. This article studies the relations among various ranks of Hankel tensors. We give an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic $n$-dimensional ... More

Regularization Methods for SDP Relaxations in Large Scale Polynomial OptimizationSep 19 2009Dec 05 2011We study how to solve semidefinite programming relaxations for large scale polynomial optimization. When interior-point methods are used, typically only small or moderately large problems could be solved. This paper studies regularization methods for ... More

On the complexity of Putinar's PositivstellensatzDec 14 2008We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schm\"udgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information ... More

A Semidefinite Approach for Truncated K-Moment ProblemsMay 02 2011Sep 06 2012A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported? This paper proposes ... More

Shape Optimization of Transfer FunctionsNov 09 2004Sep 18 2007This paper has been withdrawn due to its publication

Minimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of SquaresNov 05 2004Sep 18 2007This paper has been withdrawn by the authors due to its publication

An elementary and constructive solution to Hilbert's 17th Problem for matricesOct 12 2006Oct 27 2006We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let $A$ be an $n \times n$ symmetric matrix with entries in the polynomial ring $\mathbb R[x_1,...,x_m]$. ... More

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and SetsSep 25 2007Dec 07 2008A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For ... More

Semidefinite Representation of Convex SetsMay 28 2007Jul 21 2008Let $S =\{x\in \re^n: g_1(x)\geq 0, ..., g_m(x)\geq 0\}$ be a semialgebraic set defined by multivariate polynomials $g_i(x)$. Assume $S$ is convex, compact and has nonempty interior. Let $S_i =\{x\in \re^n: g_i(x)\geq 0\}$, and $\bdS$ (resp. $\bdS_i$) ... More

Structured Semidefinite Representation of Some Convex SetsFeb 13 2008Linear matrix Inequalities (LMIs) have had a major impact on control but formulating a problem as an LMI is an art. Recently there is the beginnings of a theory of which problems are in fact expressible as LMIs. For optimization purposes it can also be ... More

The Saddle Point Problem of PolynomialsSep 04 2018This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: ... More

Tensor Eigenvalue Complementarity ProblemsJan 20 2016This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one ... More

Bilevel Polynomial Programs and Semidefinite Relaxation MethodsAug 27 2015May 24 2016A bilevel program is an optimization problem whose constraints involve another optimization problem. This paper studies bilevel polynomial programs (BPPs), i.e., all the functions are polynomials. We reformulate BPPs equivalently as semi-infinite polynomial ... More

A Complete Semidefinite Algorithm for Detecting Copositive Matrices and TensorsNov 10 2017A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard. This paper proposes ... More

Minimizing Polynomials via Sum of Squares over the Gradient IdealNov 15 2004Sep 18 2007This paper has been withdrawn by the authors due to its publication

Tensor Eigenvalue Complementarity ProblemsJan 20 2016May 29 2017This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one ... More

Global Minimization of Rational Functions and the Nearest GCDsJan 05 2006Sep 18 2007This paper has been withdrawn by the authors due to its publication

Minimizing Polynomials Over Semialgebraic SetsFeb 17 2005This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in $\re^m$ defined by finitely many polynomial equations and inequalities, using the Karush-Kuhn-Tucker (KKT) system and sum of squares (SOS) relaxations. ... More

The Algebraic Degree of Semidefinite ProgrammingNov 19 2006Sep 08 2008Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts ... More

Semidefinite Representation of the $k$-EllipseJan 31 2007The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\binom{k}{k/2}$ if ... More

Completely Positive Binary TensorsAug 07 2018A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it satisfies two linear ... More

Bilevel Polynomial Programs and Semidefinite Relaxation MethodsAug 27 2015Nov 03 2016A bilevel program is an optimization problem whose constraints involve another optimization problem. This paper studies bilevel polynomial programs (BPPs), i.e., all the functions are polynomials. We reformulate BPPs equivalently as semi-infinite polynomial ... More

Nonlinear compressed sensing based on composite mappings and its pointwise linearizationJun 07 2015Classical compressed sensing (CS) allows us to recover structured signals from far few linear measurements than traditionally prescribed, thereby efficiently decreasing sampling rates. However, if there exist nonlinearities in the measurements, is it ... More

First-principles theory of frozen-ion flexoelectricityAug 25 2011We demonstrate that the frozen-ion contribution to the flexoelectric coefficient is given solely in terms of the sum of third moments of the charge density distortions induced by atomic displacements, even for ferroelectric or piezoelectric materials. ... More

Electrically driven octahedral rotations in SrTiO3 and PbTiO3Dec 04 2012Dec 10 2012We investigate the oxygen octahedral rotations that occur in two perovskites, SrTiO3 and PbTiO3, as a function of applied three-dimensional electric displacement field, allowing us to map out the phase diagram of rotations in both the paraelectric and ... More

All Real Eigenvalues of Symmetric TensorsMar 14 2014Dec 13 2014This paper studies how to compute all real eigenvalues of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle eigenvalues can not. We propose a new ... More

The noncommutative Waring problemMar 14 2019This paper poses and treats a noncommutative version of the classical Waring problem for polynomials. That is, for a homogeneous \nc \ polynomial $p$, we find a condition equivalent to $p$ being expressible as sums of powers of homogeneous \nc \ polynomials. ... More

Invariance of the spark, NSP order and RIP order under elementary transformations of matricesFeb 10 2015The theory of compressed sensing tells us that recovering all k-sparse signals requires a sensing matrix to satisfy that its spark is greater than 2k, or its order of the null space property (NSP) or the restricted isometry property (RIP) is 2k or above. ... More

Semidefnite Relaxation Bounds for Indefinite Homogeneous Quadratic OptimizationJan 02 2007In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{x^* C x \mid x^* A_k ... More

Adaptive Channel Allocation Spectrum Etiquette for Cognitive Radio NetworksFeb 07 2006In this work, we propose a game theoretic framework to analyze the behavior of cognitive radios for distributed adaptive channel allocation. We define two different objective functions for the spectrum sharing games, which capture the utility of selfish ... More

Salient Object Detection via High-to-Low Hierarchical Context AggregationDec 28 2018Apr 01 2019Recent progress on salient object detection mainly aims at exploiting how to effectively integrate convolutional side-output features in convolutional neural networks (CNN). Based on this, most of the existing state-of-the-art saliency detectors design ... More

Salient Object Detection via High-to-Low Hierarchical Context AggregationDec 28 2018Recent progress on salient object detection mainly aims at exploiting how to effectively integrate convolutional side-output features in convolutional neural networks (CNN). Based on this, most of the existing state-of-the-art saliency detectors design ... More

Logic Blog 2015fFeb 14 2016The 2015 Logic Blog contains a large variety of results connected to logic, some of them unlikely to be submitted to a journal. For the first time there is a group theory part. There are results in higher randomness, and in computable ergodic theory.

Lowness, randomness, and computable analysisJul 24 2016Analytic concepts contribute to our understanding of randomness of reals via algorithmic tests. They also influence the interplay between randomness and lowness notions. We provide a survey, written on the occasion of Rod Downey's 60th birthday.

Rescaling Limits in Non-Archimedean DynamicsDec 03 2016Jan 20 2018Suppose $\{f_t\}$ is an analytic one-parameter family of rational maps defined over a non-Archimedean field $K$. We prove a finiteness theorem for the set of rescalings for $\{f_t\}$. This complements results of J. Kiwi.

Entropy degeneration of convex projective surfacesMar 15 2015Nov 17 2015We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and ... More

The Space-like Surfaces with Vanishing Conformal Form in the Conformal SpaceAug 15 2011The conformal geometry of surfaces in the conformal space $\mathbf Q^n_1$ is studied. We classify the space-like surfaces in $\mathbf Q^n_1$ with vanishing conformal form up to conformal equivalence.

Classification of solutions to Toda systems of types $C$ and $B$ with singular sourcesAug 25 2015In this paper, the classification in [Lin,Wei,Ye] of solutions to Toda systems of type $A$ with singular sources is generalized to Toda systems of types $C$ and $B$. Like in the $A$ case, the solution space is shown to be parametrized by the abelian subgroup ... More

Meromorphic cubic differentials and convex projective structuresMar 09 2015Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed ... More

Meromorphic cubic differentials and convex projective structuresMar 09 2015Jan 06 2017Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed ... More

The quasi-Poisson Goldman formulaJan 22 2013Feb 10 2013We prove a quasi-Poisson bracket formula for the space of representations of the fundamental groupoid of a surface with boundary, which generalizes Goldman's Poisson bracket formula. We also deduce a similar formula for quasi-Poisson cross-sections.

Calculus of Cost FunctionsMar 05 2017Cost functions provide a framework for constructions of sets Turing below the halting problem that are close to computable. We carry out a systematic study of cost functions. We relate their algebraic properties to their expressive strength. We show that ... More

On the Hilbert geometry of simplicial Tits setsNov 05 2011Jul 11 2014The moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli space, the entropy ... More

Left-orderablity for surgeries on $(-2,3,2s+1)$-pretzel knotsFeb 28 2018In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a $(-2,3,2s+1)$-pretzel knot ($s\ge 3$) with slope $\frac{p}{q}$ is not left orderable if $\frac{p}{q}\ge 2s+3$, and that it is left orderable if $\frac{p}{q}$ ... More

On characteristic integrals of Toda field theoriesMar 05 2013Apr 06 2014Characteristic integrals of Toda field theories associated to simple Lie algebras are presented in the most explicit forms, both in terms of the formulas and in terms of the proofs.

Logic Blog 2014Apr 30 2015The 2014 Logic Blog starts with open questions from the May IMS program in Singapore. It contains results on randomness, including answers to some open questions in higher randomness. There are structural results on equivalence relations, and metric spaces. ... More

Fundamental elements of an affine Weyl groupOct 08 2013Jun 03 2014Fundamental elements are certain special elements of affine Weyl groups introduced by Gortz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne-Lusztig varieties. In this paper, we obtain characterizations of the ... More

Compactifications of the moduli spaces of Newton mapsMar 22 2018We study various compactifications of moduli space of Newton maps. Mainly, we focus on GIT compactifiaction and Deligne-Mumford compactification. Then we explore the relations among these compactifications.

Indeterminacy Loci of Iterate MapsSep 05 2017We consider the indeterminacy locus $I(\Phi_n)$ of the iterate map $\Phi_n:\overline{M}_d-rightarrow\overline{M}_{d^n}$, where $\overline{M}_d$ is the GIT compactification of the moduli space $M_d$ of degree $d$ complex rational maps. We give natural ... More

Weak Solutions of the Chern-Ricci flow on compact complex surfacesJan 18 2017In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the ... More

Logic Blog 2017Apr 15 2018The blog is somewhat shorter than in previous years, It contains new insights in a variety of areas, including computability, quantum algorithmic version of the SMB theorem, descriptions of groups (both discrete and profinite), metric spaces. There are ... More

An Application of Maximum Principle to space-like Hypersurfaces with Constant Mean Curvature in Anti-de Sitter SpaceAug 16 2011In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space $H^{n+1}_1(-1)$. we prove that if a complete space-like hypersurface with constant mean curvature $x:\mathbf M\rightarrow H^{n+1}_1(-1) $ has two distinct ... More

An Alternative Explanation on the Two Relaxation Rates in Cuprate SuperconductorsNov 22 2000Dec 05 2000Transport properties of high transition temperature (high-Tc) superconductors have been shown to have two distinct relaxation rates. We argue that this apparent inconsistence can be resolved with an effective carrier density n linear in temperature T. ... More

Variational bounds on the ground-state energy of three electrons and one hole in two-dimensionDec 31 2000Jan 09 2001We consider a model of three electrons and one hole confined in a two-dimensional (2D) plane, interacting with one another through Coulomb forces. Using a Ritz variational method we find an upper bound of \approx -0.0112me^4/8\pi^2 \epsilon ^2 \hbar ^2 ... More

Intrinsic construction of invariant functions on simple Lie algebrasMar 05 2013Apr 06 2014An algorithm for constructing primitive adjoint-invariant functions on a complex simple Lie algebra is presented. The construction is intrinsic in the sense that it does not resort to any representation. A primitive invariant function on the whole Lie ... More

On the Minimum Area of Null Homotopies of Curves Traced TwiceNov 29 2014Dec 31 2014We provide an efficient algorithm to compute the minimum area of a homotopy between two closed plane curves, given that they divide the plane into finite number of regions. For any positive real number $\varepsilon>0$, we construct a closed plane curve ... More

On Sha's secondary Chern-Euler classJan 17 2009Feb 08 2010For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincar\'e-Hopf ... More

A Functor Converting Equivariant Homology to HomotopyMar 18 2006Aug 01 2007In this paper, we prove an equivariant version of the classical Dold-Thom theorem. Associated to a finite group, a CW-complex on which this group acts and a covariant coefficient system in the sense of Bredon, we functorially construct a topological abelian ... More

Logic Blog 2012Feb 15 2013The 2012 logic blog has focussed on the following: Randomness and computable analysis/ergodic theory; Systematizing algorithmic randomness notions; Traceability; Higher randomness; Calibrating the complexity of equivalence relations from computability ... More

Logic Blog 2011Mar 23 2014This year's logic blog has focussed on: 1. Demuth randomness 2. traceability 3. The connection of computable analysis and randomness 4. $K$-triviality in metric spaces.

Logic Blog 2013Mar 23 2014Jun 19 2014The 2013 logic blog has focussed on the following: 1. Higher randomness. Among others, the Borel complexity of $\Pi^1_1$ randomness and higher weak 2 randomness is determined. 2. Reverse mathematics and its relationship to randomness. For instance, what ... More

Calibrating the complexity of Delta 2 sets via their changesFeb 03 2013The computational complexity of a Delta 2 set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Loef random sets, where we quantify the changes of initial segments. Secondly, we look ... More

The complexity of isomorphism between countably based profinite groupsApr 03 2016A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based profinite group as ... More

Rescaling Limits in Non-Archimedean DynamicsDec 03 2016Suppose $\{f_t\}$ is an analytic one-parameter family of rational maps defined over a non-Archimedean field $K$. We prove a finiteness theorem for the set of rescalings for $\{f_t\}$. This complements results of J. Kiwi.

The convolution algebra structure on $K^G(\mathcal{B} \times \mathcal{B})$Nov 08 2011We show that the convolution algebra $K^G(\mathcal{B} \times \mathcal{B})$ is isomorphic to the Based ring of the lowest two-sided cell of the extended affine Weyl group associated to $G$, where $G$ is a connected reductive algebraic group over the field ... More

On isomorphism numbers of "$F$-crystals"Mar 09 2014In this note, we show that for an ``$F$-crystal" (the equal characteristic analogue of $F$-crystals), its {\it isomorphism number} and its {\it level torsion} coincide. This confirms a conjure of Vasiu \cite{Va} in the equal characteristic case.

Iteration at the Boundary of Newton MapsMar 21 2018Let $\{N_t\}$ be a holomorphic family of degree $d\ge 3$ Newton maps. By studying the related Berkovich dynamics, we obtain an estimate of the weak limit of the maximal measures of $N_t$. Moreover, we give a complete description of the rescaling limits ... More

Logic Blog 2016Mar 05 2017Mar 09 2017This year's logic blog contains a variety of results, some of them available only here. Highlights include the resolution of the Gamma question by Monin, and a number of entries on topological group theory and its connection to logic. There's also a new ... More

Secondary Chern-Euler forms and the Law of Vector FieldsSep 25 2009Aug 14 2010The Law of Vector Fields is a term coined by Gottlieb for a relative Poincar\'e-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner ... More

Secondary Chern-Euler class for general submanifoldJun 22 2009Jul 13 2009We define and study the secondary Chern-Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study index for a vector field with non-isolated singularities on a submanifold. As an application, our studies give ... More

Coding Methods in Computability Theory and Complexity TheoryAug 29 2013A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees, it has been ... More