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29
"Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics
 BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
, 1993
"... Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and de ..."
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Cited by 24 (1 self)
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Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.
Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 21 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Higher fundamental functors for simplicial sets, Cahiers Topologie Géom
 Diff. Catég
"... Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how ..."
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Cited by 11 (8 self)
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Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (nonreversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy ncategory functor ↑Πn: Smp = nCat, left adjoint to a nerve Nn = nCat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
A Brouwer fixed point theorem for graph endomorphisms
, 2012
"... We prove a Lefschetz formula L(T) = ∑ x∈F iT (x) for graph endomorphisms T: G → G, where G is a general finite simple graph and F is the set of simplices fixed by T. The degree iT (x) of T at the simplex x is defined as (−1) dim(x) sign(T x), a graded sign of the permutation of T restricted to the ..."
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Cited by 8 (6 self)
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We prove a Lefschetz formula L(T) = ∑ x∈F iT (x) for graph endomorphisms T: G → G, where G is a general finite simple graph and F is the set of simplices fixed by T. The degree iT (x) of T at the simplex x is defined as (−1) dim(x) sign(T x), a graded sign of the permutation of T restricted to the simplex. The Lefschetz number L(T) is defined similarly as in the continuum as L(T) = ∑ k (−1)ktr(Tk), where Tk is the map induced on the k’th cohomology group Hk (G) of G. The theorem can be seen as a generalization of the NowakowskiRival fixed edge theorem [26]. A special case is the identity map T, where the formula reduces to the EulerPoincaré formula equating the Euler characteristic with the cohomological Euler characteristic. The theorem assures that if L(T) is nonzero, then T has a fixed clique. A special case is the discrete Brouwer fixed point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is starshaped in the sense that only the zero’th cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. If A is the automorphism group of a graph, we look at the average Lefschetz number L(G). We prove that this is the Euler characteristic of the chain G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function ζT (z) = exp ( ∑∞ n=1 L(T n) zn) is a product n of two dynamical zeta functions and therefore has an analytic continuation as a rational function. This explicitly computable product formula involves the dimension and the signature of prime orbits.
Dynamic coverage verification in mobile sensor networks via switched higher order Laplacians
 in Robotics: Science & Systems
, 2007
"... Abstract — In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on timeevolving simplicial complexes. The simplicial complexes model the connectivity ..."
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Cited by 7 (1 self)
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Abstract — In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on timeevolving simplicial complexes. The simplicial complexes model the connectivity of agents in the network, and the homology groups of the simplicial complexes provides information about the coverage properties of the network. Our main result states that the asymptotic stability the switched linear system implies that every point of the domain covered by the mobile sensor nodes is visited infinitely often, hence verifying dynamic coverage. The enabling mathematical technique for this result is the theory of higher order Laplacian operators, which is a generalization of the graph Laplacian used in spectral graph theory and continuoustime consensus problems.
An index formula for simple graphs
, 2012
"... Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is ..."
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Cited by 7 (6 self)
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Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is a discrete analogue of the index of the gradient vector field ∇f and where Bf (x) is a graph defined by G and f. The PoincaréHopf formula ∑ x jf (x) = χ(G) allows so to express the Euler characteristic χ(G) of G in terms of smaller dimensional graphs defined by the unit sphere S(x) and the ”hypersurface graphs ” Bf (x). For odd dimensional geometric graphs, Bf (x) is a geometric graph of dimension dim(G)−2 and jf (x) = −χ(Bf (x))/2 = 0 implying χ(G) = 0 and zero curvature K(x) = 0 for all x. For even dimensional geometric graphs, the formula becomes jf (x) = 1−χ(Bf (x))/2 and allows with PoincaréHopf to write the Euler characteristic of G as a sum of the Euler characteristic of smaller dimensional graphs. The same integral geometric index formula also is valid for compact Riemannian manifolds M if f is a Morse function, S(x) is a sufficiently small geodesic sphere around x and Bf (x) = S(x) ∩ {y  f(y) = f(x)}. 1.
Computing Cocycles on Simplicial Complexes
"... In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we ..."
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Cited by 3 (1 self)
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In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cupi products over integers on a simplicial complex at chain level. 1
A UNIVERSAL PROPERTY FOR THE JIANGSU ALGEBRA
, 707
"... Abstract. We prove that the infinite tensor power of a unital separable C ∗algebra absorbs the JiangSu algebraZ tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no un ..."
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Cited by 3 (1 self)
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Abstract. We prove that the infinite tensor power of a unital separable C ∗algebra absorbs the JiangSu algebraZ tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C ∗algebras without characters known to lie outside it. This category moreover contains the vast majority of our stockintrade separable amenable C ∗algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and crossed products by countable discrete groups. One consequence of our main result is that strongly selfabsorbing ASH algebras areZstable, and therefore satisfy the hypotheses of a recent classification theorem of W. Winter. One concludes thatZ is the only projectionless strongly selfabsorbing ASH algebra, completing the classification of strongly selfabsorbing ASH algebras. 1.
POINCARÉ DUALITY SPACES
"... At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topological nmanifold X n satisfy the relation bi(X): = dimRHi(X; R) ..."
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At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topological nmanifold X n satisfy the relation bi(X): = dimRHi(X; R)