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Some fundamental issues concerning degrees of unsolvability
 In [6], 2005. Preprint
, 2007
"... Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to f ..."
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Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, corecursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.
A biased view of symplectic cohomology
"... 2. Weinstein domains 2 3. Symplectic cohomology 5 4. Growth measures and affine varieties 14 ..."
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2. Weinstein domains 2 3. Symplectic cohomology 5 4. Growth measures and affine varieties 14
How to make a triangulation of S 3 polytopal
 Trans. Am. Math. Soc
, 2004
"... We introduce a method of studying triangulations T of S 3 that connects topological and geometrical approaches. In its center is a new numerical invariant p(T), called polytopality. Although its definition is purely topological, it is sensitive for geometric properties of T, as follows. We consider ..."
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We introduce a method of studying triangulations T of S 3 that connects topological and geometrical approaches. In its center is a new numerical invariant p(T), called polytopality. Although its definition is purely topological, it is sensitive for geometric properties of T, as follows. We consider local moves called expansions, that generalize stellar subdivisions of simplicial complexes. Let d(T) be the length of a shortest sequence of expansions relating T with the boundary complex of a convex 4–polytope. In this paper we obtain both lower and upper bounds for d(T) in terms of p(T). Using previous results [9] based on the Rubinstein–Thompson algorithm, we obtain an upper bound for d(T) in terms of the number n of tetrahedra of T. The bound is exponential in n 2, and we prove here that in general one can not replace it by a subexponential bound. Our results yield another recognition algorithm for S 3 that is conceptionally much simpler, though slower, as the Rubinstein–Thompson algorithm.
ATINT: A POLYMAKE EXTENSION FOR ALGORITHMIC TROPICAL INTERSECTION THEORY
, 2012
"... In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study of moduli spaces, where the underlying combinatorics of the ..."
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In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.
© 2010 Springer Basel AG GAFA Geometric And Functional Analysis SINGULARITIES, EXPANDERS AND TOPOLOGY OF MAPS. PART 2: FROM COMBINATORICS TO TOPOLOGY VIA ALGEBRAIC ISOPERIMETRY
, 2010
"... Abstract. We find lower bounds on the topology of the fibers F −1 (y) ⊂ X of continuous maps F: X → Y in terms of combinatorial invariants of certain polyhedra and/or of the cohomology algebras H ∗ (X). Our exposition is conceptually related to but essentially independent of Part 1 of the paper. Co ..."
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Abstract. We find lower bounds on the topology of the fibers F −1 (y) ⊂ X of continuous maps F: X → Y in terms of combinatorial invariants of certain polyhedra and/or of the cohomology algebras H ∗ (X). Our exposition is conceptually related to but essentially independent of Part 1 of the paper. Contents 1 Definitions, Problems and Selected Inequalities 417 1.1 ΔInequalities for the multiplicities of maps of the nskeleton of the Nsimplex to R n 1.2 (n − k)Planes crossing many ksimplices in R n..................... 419 1.3 Riemannian and subRiemannian waist inequalities................... 420
Topics in Logic and . . .
, 2005
"... This is a set of lecture notes from a 15week graduate course at the Pennsylvania State University taught as Math 574 by Stephen G. Simpson in Spring 2005. The course was intended for students already familiar with the basics of mathematical logic. The course covered some topics which are important ..."
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This is a set of lecture notes from a 15week graduate course at the Pennsylvania State University taught as Math 574 by Stephen G. Simpson in Spring 2005. The course was intended for students already familiar with the basics of mathematical logic. The course covered some topics which are important in contemporary mathematical logic and foundations but usually omitted from introductory courses. These notes were typeset by the students in the course: John Ethier, Esteban
Hardness of embedding simplicial complexes in R^d
, 2009
"... Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for a ..."
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Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5sphere implies that EMBEDd→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is NPhardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d −2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spie˙z, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.