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The geometry of dynamical triangulations
 Lecture Notes in Physics m50
, 1997
"... The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct ..."
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The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct mathematical fields ranging from global riemannian geometry, moduli theory, number theory, and topology. Our exposition will concentrate on these points so that graduate students may find in these notes a useful exposition of some of the rigorous results one can establish in this field and hopefully a source of inspiration for new exciting problems. We also illustrate the deep and beautiful interplay between the analytical aspects of dynamical triangulations and the results of MonteCarlo simulations. The techniques described here are rather novel and allow us to address successfully many high points of great current interest in the subject of simplicial quantum gravity while requiring very lit1 tle in the way of fancy field theoretical arguments. As a consequence, these
NIELSEN ROOT THEORY AND HOPF DEGREE THEORY
 PACIFIC JOURNAL OF MATHEMATICS VOL. 198, NO. 1
, 2001
"... ... a point c ∈ N is a homotopy invariant lower bound for the number of roots at c, that is, for the cardinality of f −1 (c). There is a formula for calculating N(f; c) if M and N are closed oriented manifolds of the same dimension. We extend the calculation of N(f; c) to manifolds that are not orie ..."
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... a point c ∈ N is a homotopy invariant lower bound for the number of roots at c, that is, for the cardinality of f −1 (c). There is a formula for calculating N(f; c) if M and N are closed oriented manifolds of the same dimension. We extend the calculation of N(f; c) to manifolds that are not orientable, and also to manifolds that have nonempty boundaries and are not compact, provided that the map f is boundarypreserving and proper. Because of its connection with degree theory, we introduce the transverse Nielsen root number for maps transverse to c, obtain computational results for it in the same setting, and prove that the two Nielsen root numbers are sharp lower bounds in dimensions other than 2. We apply these extended root theory results to the degree theory for maps of not necessarily orientable manifolds introduced by Hopf in 1930. Thus we reestablish, in a new and modern treatment,
Spin^cStructures and Homotopy Equivalences
"... We show that a homotopy equivalence between manifolds induces a correspondence between their spin c structures, even in the presence of 2torsion. This is proved by generalizing spin c structures to Poincar'e complexes. A procedure is given for explicitly computing the correspondence under rea ..."
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We show that a homotopy equivalence between manifolds induces a correspondence between their spin c structures, even in the presence of 2torsion. This is proved by generalizing spin c structures to Poincar'e complexes. A procedure is given for explicitly computing the correspondence under reasonable hypotheses. AMS Classification numbers Primary: 57N13, 57R15 Secondary: 57P10, 57R19 Keywords: 4manifold, SeibergWitten invariant, Poincar'e complex 0 Partially supported by NSF grant DMS9625654. Research at MSRI is supported in part by NSF grant DMS9022140. 1 1. Introduction The theory of spin c structures has attained new importance through its recent application to the topology of smooth 4manifolds. Among smooth, closed, oriented 4manifolds (with b 1 + b + odd) a typical homeomorphism type contains many diffeomorphism types. The only invariants known to distinguish such diffeomorphism types are those arising from gauge theory, as pioneered by Donaldson (e.g., [1])....
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental