On the Topology of Discrete Strategies∗
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Combinatorial Topology
Chapter 6 Basics of Combinatorial Topology 6.1 Simplicial and Polyhedral Complexes In order to study and manipulate complex shapes it is convenient to discretize these shapes and to view them as the union of simple building blocks glued together in a “clean fashion”. The building blocks should be simple geometric objects, for example, points, lines segments, triangles, tehrahedra and more generally simplices, or even convex polytopes. We will begin by using simplices as building blocks. The material presented in this chapter consists of the most basic notions of combinatorial topology, going back roughly to the 1900-1930 period and it is covered in nearly every algebraic topology book (certainly the “classics”). A classic text (slightly old fashion especially for the notation and terminology) is Alexandrov [1], Volume 1 and another more “modern” source is Munkres [30]. An excellent treatment from the point of view of computational geometry can be found is Boissonnat and Yvinec [8], especially Chapters 7 and 10. Another fascinating book covering a lot of the basics but devoted mostly to three-dimensional topology and geometry is Thurston [41]. Recall that a simplex is just the convex hull of a finite number of affinely independent points. We also need to define faces, the boundary, and the interior of a simplex. Definition 6.1 Let be any normed affine space, say = Em with its usual Euclidean norm. Given any n+1E affinely independentpoints a ,...,aE in , the n-simplex (or simplex) 0 n E σ defined by a0,...,an is the convex hull of the points a0,...,an,thatis,thesetofallconvex combinations λ a + + λ a ,whereλ + + λ =1andλ 0foralli,0 i n. -
WEIGHT FILTRATIONS in ALGEBRAIC K-THEORY Daniel R
November 13, 1992 WEIGHT FILTRATIONS IN ALGEBRAIC K-THEORY Daniel R. Grayson University of Illinois at Urbana-Champaign Abstract. We survey briefly some of the K-theoretic background related to the theory of mixed motives and motivic cohomology. 1. Introduction. The recent search for a motivic cohomology theory for varieties, described elsewhere in this volume, has been largely guided by certain aspects of the higher algebraic K-theory developed by Quillen in 1972. It is the purpose of this article to explain the sense in which the previous statement is true, and to explain how it is thought that the motivic cohomology groups with rational coefficients arise from K-theory through the intervention of the Adams operations. We give a basic description of algebraic K-theory and explain how Quillen’s idea [42] that the Atiyah-Hirzebruch spectral sequence of topology may have an algebraic analogue guides the search for motivic cohomology. There are other useful survey articles about algebraic K-theory: [50], [46], [23], [56], and [40]. I thank W. Dwyer, E. Friedlander, S. Lichtenbaum, R. McCarthy, S. Mitchell, C. Soul´e and R. Thomason for frequent and useful consultations. 2. Constructing topological spaces. In this section we explain the considerations from combinatorial topology which give rise to the higher algebraic K-groups. The first principle is simple enough to state, but hard to implement: when given an interesting group (such as the Grothendieck group of a ring) arising from some algebraic situation, try to realize it as a low-dimensional homotopy group (especially π0 or π1) of a space X constructed in some combinatorial way from the same algebraic inputs, and study the homotopy type of the resulting space. -
Combinatorial Topology and Applications to Quantum Field Theory
Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Chair Professor Ian Agol Professor Constantin Teleman Professor Joel Moore Fall 2018 Abstract Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vivek Shende, Chair Topology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and dis- crete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists. 1 To my loving parents, kind friends, and patient teachers. i Contents I Discrete Topology Toolbox1 1 Basics4 1.1 Discrete Spaces..........................4 1.1.1 Cellular Maps and Cellular Approximation.......6 1.1.2 Triangulations and Barycentric Subdivision......6 1.1.3 PL-Manifolds and Combinatorial Duality........8 1.1.4 Discrete Morse Flows...................9 1.2 Chains, Cycles, Cochains, Cocycles............... 13 1.2.1 Chains, Cycles, and Homology.............. 13 1.2.2 Pushforward of Chains.................. 15 1.2.3 Cochains, Cocycles, and Cohomology......... -
Homology Groups of Homeomorphic Topological Spaces
An Introduction to Homology Prerna Nadathur August 16, 2007 Abstract This paper explores the basic ideas of simplicial structures that lead to simplicial homology theory, and introduces singular homology in order to demonstrate the equivalence of homology groups of homeomorphic topological spaces. It concludes with a proof of the equivalence of simplicial and singular homology groups. Contents 1 Simplices and Simplicial Complexes 1 2 Homology Groups 2 3 Singular Homology 8 4 Chain Complexes, Exact Sequences, and Relative Homology Groups 9 ∆ 5 The Equivalence of H n and Hn 13 1 Simplices and Simplicial Complexes Definition 1.1. The n-simplex, ∆n, is the simplest geometric figure determined by a collection of n n + 1 points in Euclidean space R . Geometrically, it can be thought of as the complete graph on (n + 1) vertices, which is solid in n dimensions. Figure 1: Some simplices Extrapolating from Figure 1, we see that the 3-simplex is a tetrahedron. Note: The n-simplex is topologically equivalent to Dn, the n-ball. Definition 1.2. An n-face of a simplex is a subset of the set of vertices of the simplex with order n + 1. The faces of an n-simplex with dimension less than n are called its proper faces. 1 Two simplices are said to be properly situated if their intersection is either empty or a face of both simplices (i.e., a simplex itself). By \gluing" (identifying) simplices along entire faces, we get what are known as simplicial complexes. More formally: Definition 1.3. A simplicial complex K is a finite set of simplices satisfying the following condi- tions: 1 For all simplices A 2 K with α a face of A, we have α 2 K. -
Snake Lemma - Wikipedia, the Free Encyclopedia
Snake lemma - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Snake_lemma Snake lemma From Wikipedia, the free encyclopedia The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms. Contents 1 Statement 2 Explanation of the name 3 Construction of the maps 4 Naturality 5 In popular culture 6 See also 7 References 8 External links Statement In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider a commutative diagram: where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernels and cokernels of a, b, and c: Furthermore, if the morphism f is a monomorphism, then so is the morphism ker a → ker b, and if g' is an epimorphism, then so is coker b → coker c. Explanation of the name To see where the snake lemma gets its name, expand the diagram above as follows: 1 of 4 28/11/2012 01:58 Snake lemma - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Snake_lemma and then note that the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering snake. Construction of the maps The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. -
Central Schemes for the Modified Buckley-Leverett Equation
CENTRAL SCHEMES FOR THE MODIFIED BUCKLEY-LEVERETT EQUATION DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Ying Wang, B.Sc., M.Sc., M.A.S. Graduate Program in Mathematics The Ohio State University 2010 Dissertation Committee: Professor Chiu-Yen Kao, Advisor Professor Avner Friedman Professor Fei-Ran Tian c Copyright by Ying Wang July, 2010 ABSTRACT In fluid dynamics, the Buckley-Leverett (BL) equation is a transport equation used to model two-phase flow in porous media. One application is secondary recovery by water-drive in oil reservoir simulation. The modified Buckley-Leverett (MBL) equation differs from the classical BL equation by including a balanced diffusive- dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Rie- mann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this thesis, we first show that the solution of the finite interval [0,L] boundary value problem converges to that of the half-line [0, +∞) problem for the MBL equation as L → +∞. This result provides a justification for the use of the finite interval boundary value problem in numerical studies for the half line problem. Furthermore, we extend the classical central schemes for the hyperbolic conserva- tion laws to solve the MBL equation which is of pseudo-parabolic type. -
A Proof-Theoretic Approach to Hierarchical Math Library Organization
A Proof-Theoretic Approach to Hierarchical Math Library Organization Kamal Aboul-Hosn and Terese Damhøj Andersen Department of Computer Science, Cornell University, Ithaca, New York, USA [email protected] [email protected] Abstract. The relationship between theorems and lemmas in mathe- matical reasoning is often vague. No system exists that formalizes the structure of theorems in a mathematical library. Nevertheless, the deci- sions we make in creating lemmas provide an inherent hierarchical struc- ture to the statements we prove. In this paper, we develop a formal system that organizes theorems based on scope. Lemmas are simply the- orems with a local scope. We develop a representation of proofs that captures scope and present a set of proof rules to create and reorganize the scopes of theorems and lemmas. The representation and rules al- low systems for formalized mathematics to more accurately reflect the natural structure of mathematical knowledge. 1 Introduction The relationship between theorems and lemmas in mathematical reasoning is often vague. What makes a statement a lemma, but not a theorem? One might say that a theorem is “more important,” but what does it mean for one statement to be “more important” than another? When writing a proof for a theorem, we often create lemmas as a way to break down the complex proof, so perhaps we expect the proofs of lemmas to be shorter than the proofs of theorems. We also create lemmas when we have a statement that we do not expect to last in readers’ minds, i.e., it is not the primary result of our work. -
Lemmas in Olympiad Geometry
Lemmas In Olympiad Geometry Navneel Singhal July 14, 2016 Geometry is the art of correct reasoning from incorrectly drawn figures. - Henri Poincar´e 1 Introduction Here is a collection of some useful lemmas in geometry, some of them well known, some obscure and some by the author himself. This list of lemmas is also intended to be a list of some easier problems and also as some configurations that frequently appear on contests. Usually these lemmas will be intermediate results that will help you reach the solution in many cases, and maybe even trivialize the problem. These will help you write some really elegant solutions (and will also help you to simplify your bashes in cases of some problems that don't yield easily to synthetic solutions.) So have fun proving these lemmas and using them to the fullest advantage in your Olympiad journey! 2 Some Notations • By (XYZ) we denote the circumcircle of ∆XYZ, by (XY ) the circle with XY as diameter, and by (M; r) the circle with centre M and radius r, the radius being dropped when the context is clear. • We denote H as the orthocentre, O as the circumcentre, G as the centroid, N as the nine-point centre, I as the incentre, Na as the Nagel point and Ge as the Gergonne point of ∆ABC 3 Orthocenter related properties • O and H are isogonal conjugates. • Reflections of the orthocenter over the sides and the midpoints of the sides lie on (ABC). 1 • The O is the orthocenter of the medial triangle. a consequence is that AH=2OM. -
Coquelicot: a User-Friendly Library of Real Analysis for Coq
Coquelicot: A User-Friendly Library of Real Analysis for Coq Sylvie Boldo, Catherine Lelay and Guillaume Melquiond Abstract. Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems. As such, its support is warranted in proof assis- tants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems. The Coq system comes with an axiomatization of standard real numbers and a li- brary of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results. For instance, power series are not developed further than their definition. Moreover, the definitions of integrals and derivatives are based on dependent types, which make them especially cumbersome to use in practice. To palliate these inadequacies, we have designed a user-friendly library: Coquelicot. An easier way of writing formulas and theorem statements is achieved by re- lying on total functions in place of dependent types for limits, derivatives, integrals, power series, and so on. To help with the proof process, the library comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals, two- dimensional differentiability, asymptotic behaviors. It also offers some automation for performing differentiability proofs. Moreover, Coquelicot is a conservative extension of Coq’s standard library and we provide correspondence theorems between the two libraries. We have exercised the library on several use cases: in an exam at university entry level, for the definitions and properties of Bessel functions, and for the solution of the one-dimensional wave equation. -
Detailed Proofs of Lemmas, Theorems, and Corollaries
Detailed Proofs of Lemmas, Theorems, and Corollaries Dahua Lin John Fisher CSAIL, MIT CSAIL, MIT A List of Lemmas, Theorems, and Theorem 4. The hierarchically bridging Markov Corollaries chain with bk < 1 for k = 0;:::;K − 1, and fk < 1 for k = 1;:::;K is ergodic. If we write the equilibrium For being self-contained, we list here all the lemmas, distribution in form of (αµ0; β1µ1; : : : ; βK µK ), then theorems, and corollaries in the main paper. (S1) µ0 equals the target distribution µ; (S2) for each k ≥ 1, and y 2 Y , µ (y) is proportional to the total Lemma 1. The joint transition matrix P given by k k + probability of its descendant target states (the target Eq.(1) in the main paper has a stationary distribution states derived by filling all its placeholders); (S3) α, in form of (αµ ; βµ ), if and only if X Y the probability of being at the target level, is given by α−1 = 1 + PK (b ··· b )=(f ··· f ). µX QB = µY ; and µY QF = µX : (1) k=1 0 k−1 1 k Corollary 2. If bk=fk+1 ≤ κ < 1 for each k = Under this condition, we have αb = βf. Further, if 1;:::;K, then α > 1 − κ. both PX and PY are both reversible, then P+ is also reversible, if and only if B Proofs µX (x)QB(x; y) = µY (y)QF (y; x); (2) Here, we provide the proofs of the lemmas and theo- for all x 2 X and y 2 Y . rems presented in the paper. -
Motivic Homotopy Theory
Voevodsky’s Nordfjordeid Lectures: Motivic Homotopy Theory Vladimir Voevodsky1, Oliver R¨ondigs2, and Paul Arne Østvær3 1 School of Mathematics, Institute for Advanced Study, Princeton, USA [email protected] 2 Fakult¨at f¨ur Mathematik, Universit¨atBielefeld, Bielefeld, Germany [email protected] 3 Department of Mathematics, University of Oslo, Oslo, Norway [email protected] 148 V. Voevodsky et al. 1 Introduction Motivic homotopy theory is a new and in vogue blend of algebra and topology. Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology. This text is a report from Voevodsky’s summer school lectures on motivic homotopy in Nordfjordeid. Its first part consists of a leisurely introduction to motivic stable homotopy theory, cohomology theories for algebraic varieties, and some examples of current research problems. As background material, we recommend the lectures of Dundas [Dun] and Levine [Lev] in this volume. An introductory reference to motivic homotopy theory is Voevodsky’s ICM address [Voe98]. The appendix includes more in depth background material required in the main body of the text. Our discussion of model structures for motivic spectra follows Jardine’s paper [Jar00]. In the first part, we introduce the motivic stable homotopy category. The examples of motivic cohomology, algebraic K-theory, and algebraic cobordism illustrate the general theory of motivic spectra. In March 2000, Voevodsky [Voe02b] posted a list of open problems concerning motivic homotopy theory. There has been so much work done in the interim that our update of the status of these conjectures may be useful to practitioners of motivic homotopy theory. -
K-Theory, Bott Periodicity, and Elliptic Operators
K-THEORY, BOTT PERIODICITY, AND ELLIPTIC OPERATORS CAMERON KRULEWSKI Abstract. We give the background for and a proof of the Bott periodicity theorem. Our paper develops a foundation of topological K-theory and offers a summary of Michael Atiyah's 1967 proof of the complex case and the real equivariant case of Bott periodicity, found in the paper \Bott Periodicity and the Index of Elliptic Operators" [1]. Motivation is given for steps Atiyah's proof, but many details are skipped in favor of offering an understandable overview. Contents Introduction 1 1. Background 2 1.1. Useful Terms 2 1.2. K-Theory 3 1.3. Important Objects and Conventions 11 2. Overview of Atiyah's 1967 Proof with Elliptic Operators 14 2.1. The Template 14 2.2. The \Formal Trick" 15 2.3. Elliptic Operators on Vector Bundles 17 2.4. The Complex Case 23 2.5. The Spinor Case 24 2.6. Conclusion 31 Acknowledgments 31 References 32 Introduction Homotopy groups are very useful for classifying topological spaces, but are no- toriously difficult to calculate. Bott's 1959 result [9] massively simplified the calcu- lation for several classical groups by revealing a periodic structure, which in turn stimulated the development of K-theory and other fields. Written succinctly, Bott's result is that the unitary group is 2-periodic, while the orthogonal and symplectic groups are 8-periodic. That is, πk(U) = πk+2(U) πk(O) = πk+8(O) πk(Sp) = πk+8(Sp): Date: 7 January, 2018. 1 2 CAMERON KRULEWSKI Since its original discovery, Bott periodicity has inspired a slew of diverse proof approaches [18], as well as several different formulations of the theorem that extend its result.