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Jean Gallier

University of Pennsylvania, Computer Science, Faculty Member

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I am a computer scientist and mathematician.I am interested in all aspects of geometry,in particular differential geometry, also in harmonic analysis, and their applications in computer vision, machine learning, and robotics.
Phone: 215-898-4405

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Armando Marques-Guedes

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Oxford Brookes University

Roshan Chitrakar

Nepal College of Information Technology

Graduate Center of the City University of New York

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Universidade de Coimbra

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Books by Jean Gallier

Geometric Methods and Applications for Computer Science and Engineering, Second Edition

Geometric Methods and Applications for Computer Science and Engineetring, 2012

This book grew out of lectures notes that I have written as I have been teaching CIS610, Advanced... more This book grew out of lectures notes that I have written as I have been teaching CIS610, Advanced Geometric Methods in Computer Science, for the past two years. Many thanks to the copyeditor, David Kramer, who did a superb job. I also wish to thank some students and colleagues for their comments, including Koji Ashida,

Curves and surfaces in geometric modelling: Theory and algorithms, 1999

We will be dealing primarily with the following kinds of problems: • Approximating a shape (curve... more We will be dealing primarily with the following kinds of problems: • Approximating a shape (curve or surface). We will see how this can be done using polynomial curves or surfaces (also called Bézier curves or surfaces), spline curves or surfaces. • Interpolating a set of points, by a curve or a surface. Again, we will see how this can be done using spline curves or spline surfaces. • Drawing a curve or a surface. The tools and techniques developed for solving the approximation problem will be very useful for solving the other two problems. The material presented in this book is related to the classical differential geometry of curves and surfaces, and to numerical methods in matrix analysis. In fact, it is often possible to reduce problems involving certain splines to solving systems of linear equations. Thus, it is very helpful to be aware of efficient methods for numerical matrix analysis. For further information on these topics, readers are referred to the excellent texts by Gray [39], Strang [81], and Ciarlet [19]. Strang's beautiful book on applied mathematics is also highly recommended as a general reference [80]. There are other interesting applications of geometry to computer vision, computer graphics, and solid modeling. Some great references are Koenderink [46] and Faugeras [33] for computer vision, Hoffman [43] for solid modeling, and Metaxas [53] for physics-based deformable models. Novelties As far as we know, there is no fully developed modern exposition integrating the basic concepts of affine geometry as well as a presentation of curves and surfaces from the algorithmic point of view in terms of control points (in the polynomial case). There is also no reasonably thorough textbook presentation of the main surface subdivision schemes (Doo-Sabin, Catmull-Clark, Loop), and a technical discussion of convergence and smoothness.

Discmath root b

Mathematical Foundations and Aspects of Discrete Mathematics, 2024

This is a book about discrete mathematics which also discusses mathematical reasoning and logic. ... more This is a book about discrete mathematics which also discusses mathematical reasoning and logic. Since the publication of the first edition of this book a few years ago, we came to realize that for a significant number of readers, it is their first exposure to the rules of mathematical reasoning and to logic. As a consequence, the version of Chapter 1 from the first edition may be a bit too abstract and too formal for those readers, and they may find this discouraging. To remedy this problem, we have written a new version of the first edition of Chapter 1. This new chapter is more elementary, more intuitive, and less formal. It also contains less material, but as in the first edition, it is still possible to skip Chapter 1 without causing any problem or gap, because the other chapters of this book do not depend on the material of Chapter 1. It appears that enough readers are interested in the first edition of Chapter 1, so in this second edition, we reproduce it (slightly updated) as Chapter 11. Again, this chapter can be omitted without causing any problem or gap. My suggestion to readers who have not been exposed to mathematical reasoning and logic is to read, or at least skim, Chapter 1. On the other hand, my suggestion to more sophisticatd readers is to skip Chapter 1 and proceed directly to Chapter 2. From my point of view, they will miss some interesting considerations on the constructive nature of logic, but that's because we are very fond of foundational issues, and we realize that not everybody has the same level of interest in foundations! In this second edition, we tried to make the exposition simpler and clearer. We added some figures, some examples, clarified certain definitions, and simplified some proofs. A few changes and additions were also made. In Chapter 2 we added a section (Section 2.12) which describes the Haar transform on sequences in an elementary fashion as a certain bijection. We also show how the Haar transform can be used to compress audio signals (see Section 2.13). This is a spectacular and concrete illustration of the abstract notion of a bijection. We created a separate chapter (Chapter 3) dealing with the set-theoretical notions of equinumerosity, finite, countable, and infinite sets. In this new chapter we discuss the pigeonhole principle more extensively. In particular, we discuss the Frobenius vii viii Preface Preface ix The last three sections of this chapter contain more advanced material and are optional. The topics of these optional sections are generating functions (including the moment generating function and the characteristic function), the limit theorems (weak law of large numbers, central limit theorem, and strong law of large numbers), and Chernoff bounds. A beautiful exposition of discrete probability can be found in Chapter 8 of Concrete Mathematics, by Graham, Knuth, and Patashnik [1]. Comprehensive presentations can be found in Mitzenmacher and Upfal [3], Ross [4, 5], and Grimmett and Stirzaker [2]. Ross [4] contains an enormous amount of examples and exercises and is very easy to read.

Proofs, Computability, Undecidability, Complexity, And the Lambda Calculus An Introduction

Preface The main goal of this book is to present a mix of material dealing with 1. Proof systems.... more Preface The main goal of this book is to present a mix of material dealing with 1. Proof systems. 2. Computability and undecidability. 3. The Lambda Calculus. 4. Some aspects of complexity theory. Historically, the theory of computability and undecidability arose from Hilbert's efforts to completely formalize mathematics and from Gödel's first incompleteness theorem that showed that such a program was doomed to fail. People realized that to carry out both Hilbert's program and Gödel's work it was necessary to define precisely what is the notion of a computable function and the notion of a mechanically checkable proof. The first definition given around 1934 was that of the class of computable function in the sense of Herbrand-Gödel-Kleene. The second definition given by Church in 1935-1936 was the notion of a function definable in the λ-calculus. The equivalence of these two definitions was shown by Kleene in 1936. Shortly after in 1936, Turing introduced a third definition, that of a Turingcomputable function. Turing proved the equivalence of his definition with the Herbrand-Gödel-Kleene definition in 1937 (his proofs are rather sketchy compared to Kleene's proofs). All these historical papers can be found in a fascinating book edited by Martin Davis [27]. Negative results pointing out limitations of the notion of computability started to appear: Gödel's first (and second) incompleteness result, but also Church's theorem on the undecidability of validity in first-order logic, and Turing's result on the undecidability of the halting problem for Turing machines. Although originally the main focus was on the notion of function, these undecidability results triggered the study of computable and noncomputable sets of natural numbers. Other definitions of the computable functions were given later. From our point of view, the most important ones are 1. RAM programs and RAM-computable functions by Shepherdson and Sturgis (1963), and anticipated by Post (1944); see Machtey and Young [41].

Aspects of Convex Geometry Polyhedra, Linear Programming, Shellings, Voronoi Diagrams, Delaunay Triangulations

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used... more Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincaré formula (Poincaré, 1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a "correct" account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space. It turns out that in order to achieve (even partially) the above goals, I found that it was necessary to include quite a bit of background material on convex sets, polytopes, polyhedra and projective spaces. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also had to include some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of "projective polyhedron" based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new. Since Chapters 2, 3, 4, and 5 contain all the background (and more) needed to discuss linear programming (including the simplex algorithm and duality), we have included some chapters on linear programming.

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, 2022

for all p ≥ 0. Furthermore, this assignment is functorial (more precisely, it is a functor from t... more for all p ≥ 0. Furthermore, this assignment is functorial (more precisely, it is a functor from the category of cochain complexes and chain maps to the category of abelian groups and their homomorphisms). At first glance cohomology appears to be very abstract so it is natural to look for explicit examples. A way to obtain a cochain complex is to apply the operator (functor) Hom Z (-, G) to a chain complex C, where G is any abelian group. Given a fixed abelian group A, for any abelian group B we denote by Hom Z (B, A) the abelian group of all homomorphisms from B to A. Given any two abelian groups B and C, for any homomorphism f : The map Hom Z (f, A) is also denoted by Hom Z (f, id A ) or even Hom Z (f, id). Observe that the effect of Hom Z (f, id) on ϕ is to precompose ϕ with f . If f : B → C and g : C → D are homomorphisms of abelian groups, a simple computation shows that Hom R (g 1.1. EXACT SEQUENCES, CHAIN COMPLEXES, HOMOLOGY, COHOMOLOGY 21 Observe that Hom Z (f, id) and Hom Z (g, id) are composed in the reverse order of the composition of f and g. It is also immediately verified that Hom Z (id A , id) = id Hom Z (A,G) . We say that Hom Z (-, id) is a contravariant functor (from the category of abelian groups and group homomorphisms to itself). Then given a chain complex we can form the cochain complex obtained by applying Hom Z (-, G), and denoted Hom Z (C, G). The coboundary map d p is given by which means that for any f ∈ Hom Z (C p , G), we have Thus, for any (p + 1)-chain c ∈ C p+1 we have We obtain the cohomology groups H p (Hom Z (C, G)) associated with the cochain complex Hom Z (C, G). The cohomology groups H p (Hom Z (C, G)) are also denoted H p (C; G). This process was applied to the simplicial chain complex C * (K) associated with a simplicial complex K by Alexander and Kolmogoroff to obtain the simplicial cochain complex Hom Z (C * (K); G) denoted C * (K; G) and the simplicial cohomology groups H p (K; G) of the simplicial complex K; see Section 5.6. Soon after, this process was applied to the singular chain complex S * (X; Z) of a space X to obtain the singular cochain complex Hom Z (S * (X; Z); G) denoted S * (X; G) and the singular cohomology groups H p (X; G) of the space X; see Section 4.8. Given a continuous map f : X → Y , there is an induced chain map f : S * (Y ; G) → S * (X; G) between the singular cochain complexes S * (Y ; G) and S * (X; G), and thus homomorphisms of cohomology f * : H p (Y ; G) → H p (X; G). Observe the reversal: f is a map from X to Y , but f * maps H p (Y ; G) to H p (X; G). We say that the map X → (H p (X; G)) p≥0 is a contravariant functor from the category of topological spaces and continuous maps to the category of abelian groups and their homomorphisms. So far our homology groups have coefficients in Z, but the process of forming a cochain complex Hom Z (C, G) from a chain complex C allows the use of coefficients in any abelian obtained by applying Hom R (-, G), and denoted Hom R (C, G).

Differential Geometry and Lie Groups A Second Course

Differential Geometry and Lie Groups A Second Course, 2020

However, we often omit the subscript K unless confusion may arise.

Differential Geometry and Lie Groups A Computational Perspective

Differential Geometry and Lie Groups A Computational Perspective, 2020

To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie. To my parents Howard a... more To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie. To my parents Howard and Jane. discuss three results, one of which being the Hadamard and Cartan theorem about complete manifolds of non-positive curvature. The goal of Chapter 18 is to understand the behavior of isometries and local isometries, in particular their action on geodesics. We also intoduce Riemannian covering maps and Riemannian submersions. If π : M → B is a submersion between two Riemannian manifolds, then for every b ∈ B and every p ∈ π -1 (b), the tangent space T p M to M at p splits into two orthogonal components, its vertical component V p = Ker dπ p , and its horizontal component H p (the orthogonal complement of V p ). If the map dπ p is an isometry between H p and T b B, then most of the differential geometry of B can be studied by lifting B to M , and then projecting down to B again. We also introduce Killing vector fields, which play a technical role in the study of reductive homogeneous spaces. In Chapter 19, we return to Lie groups. Not every Lie group is a matrix group, so in order to study general Lie groups it is necessary to introduce left-invariant (and rightinvariant) vector fields on Lie groups. It turns out that the space of left-invariant vector fields is isomorphic to the tangent space g = T I G to G at the identity, which is a Lie algebra. By considering integral curves of left-invariant vector fields, we define the generalization of the exponential map exp : g → G to an arbitrary Lie group. The notion of immersed Lie subgroup is introduced, and the correspondence between Lie groups and Lie algebra is explored. We also consider the special classes of semidirect products of Lie algebras and Lie groups, the universal covering of a Lie group, and the Lie algebra of Killing vector fields on a Riemannian manifold. Chapter 20 deals with two topics: 1. A formula for the derivative of the exponential map for a general Lie group (not necessarily a matrix group). 2. A formula for the Taylor expansion of µ(X, Y ) = log(exp(X) exp(Y )) near the origin. The second problem is solved by a formula known as the Campbell-Baker-Hausdorff formula. An explicit formula was derived by Dynkin (1947), and we present this formula. Chapter 21 is devoted to the study of metrics, connections, geodesics, and curvature, on Lie groups. Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. As a consequence, it is possible to find explicit formulae for the Levi-Civita connection and the various curvatures, especially in the case of metrics which are both left and right-invariant. In Section 21.2 we give four characterizations of bi-invariant metrics. The first one refines the criterion of the existence of a left-invariant metric and states that every bi-invariant metric on a Lie group G arises from some Ad-invariant inner product on the Lie algebra g. In Section 21.3 we show that if G is a Lie group equipped with a left-invariant metric, then it is possible to express the Levi-Civita connection and the sectional curvature in terms Recall that a real symmetric matrix is called positive (or positive semidefinite) if its eigenvalues are all positive or null, and positive definite if its eigenvalues are all strictly positive. We denote the vector space of real symmetric n × n matrices by S(n), the set of symmetric positive matrices by SP(n), and the set of symmetric positive definite matrices by SPD(n). The next proposition shows that every symmetric positive definite matrix A is of the form e B for some unique symmetric matrix B. The set of symmetric matrices is a vector space, but it is not a Lie algebra because the Lie bracket [A, B] is not symmetric unless A and B commute, and the set of symmetric (positive) definite matrices is not a multiplicative group, so this result is of a different flavor as Theorem 2.6. Proposition 2.8. For every symmetric matrix B, the matrix e B is symmetric positive definite. For every symmetric positive definite matrix A, there is a unique symmetric matrix B such that A = e B . Proof. We showed earlier that e B = e B . If B is a symmetric matrix, then since B = B, we get e B = e B = e B , and e B is also symmetric. Since the eigenvalues λ 1 , . . . , λ n of the symmetric matrix B are real and the eigenvalues of e B are e λ 1 , . . . , e λn , and since e λ > 0 if λ ∈ R, e B is positive definite. To show the surjectivity of the exponential map, note that if A is symmetric positive definite, then by Theorem 12.3 from Chapter 12 of Gallier [50], there is an orthogonal matrix P such that A = P D P , where D is a diagonal matrix CHAPTER 2. THE MATRIX EXPONENTIAL; SOME MATRIX LIE GROUPS

Fundamentals of Optimization Theory With Applications to Machine Learning

Fundamentals of Optimization Theory With Applications to Machine Learning, 2020

for reporting typos and for helpful comments. Mary Pugh and William Yu (at the University of Toro... more for reporting typos and for helpful comments. Mary Pugh and William Yu (at the University of Toronto) taught a course using our book and reported a number of typos and errors. We warmly thank them as well as their students, not only for finding errors, but also for very hepful comments and suggestions for simplifying some proofs. Thanks to Gilbert Strang. We learned much from his books which have been a major source of inspiration. Special thanks to Steven Boyd. We learned a lot from his remarkable book on convex optimization and his papers, and Part III of our book is significantly inspired by his writings. The first author also wishes to express his deepest gratitute to Philippe G. Ciarlet who was his teacher and mentor in 1970-1972 while he was a student at ENPC in Paris. Professor Ciarlet was by far his best teacher. He also knew how to instill in his students the importance of intellectual rigor, honesty, and modesty. He still has his typewritten notes on measure theory and integration, and on numerical linear algebra. The latter became his wonderful book Ciarlet , from which we have borrowed heavily.

Linear Algebra for Computer Vision, Robotics, and Machine Learning

Linear algebra for computer vision, robotics and machine learning, 2022

In recent years, computer vision, robotics, machine learning, and data science have been some of ... more In recent years, computer vision, robotics, machine learning, and data science have been some of the key areas that have contributed to major advances in technology. Anyone who looks at papers or books in the above areas will be baffled by a strange jargon involving exotic terms such as kernel PCA, ridge regression, lasso regression, support vector machines (SVM), Lagrange multipliers, KKT conditions, etc. Do support vector machines chase cattle to catch them with some kind of super lasso? No! But one will quickly discover that behind the jargon which always comes with a new field (perhaps to keep the outsiders out of the club), lies a lot of "classical" linear algebra and techniques from optimization theory. And there comes the main challenge: in order to understand and use tools from machine learning, computer vision, and so on, one needs to have a firm background in linear algebra and optimization theory. To be honest, some probablity theory and statistics should also be included, but we already have enough to contend with. Many books on machine learning struggle with the above problem. How can one understand what are the dual variables of a ridge regression problem if one doesn't know about the Lagrangian duality framework? Similarly, how is it possible to discuss the dual formulation of SVM without a firm understanding of the Lagrangian framework? The easy way out is to sweep these difficulties under the rug. If one is just a consumer of the techniques we mentioned above, the cookbook recipe approach is probably adequate. But this approach doesn't work for someone who really wants to do serious research and make significant contributions. To do so, we believe that one must have a solid background in linear algebra and optimization theory. This is a problem because it means investing a great deal of time and energy studying these fields, but we believe that perseverance will be amply rewarded. Our main goal is to present fundamentals of linear algebra and optimization theory, keeping in mind applications to machine learning, robotics, and computer vision. This work consists of two volumes, the first one being linear algebra, the second one optimization theory and applications, especially to machine learning. This first volume covers "classical" linear algebra, up to and including the primary decomposition and the Jordan form. Besides covering the standard topics, we discuss a few topics that are important for applications. These include: 1. Haar bases and the corresponding Haar wavelets. knew how to instill in his students the importance of intellectual rigor, honesty, and modesty. He still has his typewritten notes on measure theory and integration, and on numerical linear algebra. The latter became his wonderful book Ciarlet , from which we have borrowed heavily. CHAPTER 2. VECTOR SPACES, BASES, LINEAR MAPS for all (x 1 , . . . , x n ) ∈ R n and all λ ∈ R. Axioms (V0)-(V3) are all satisfied, but (V4) fails. Less trivial examples can be given using the notion of a basis, which has not been defined yet. The field R itself can be viewed as a vector space over itself, addition of vectors being addition in the field, and multiplication by a scalar being multiplication in the field. Example 2.3. 1. The fields R and C are vector spaces over R. 2. The groups R n and C n are vector spaces over R, with scalar multiplication given by λ(x 1 , . . . , x n ) = (λx 1 , . . . , λx n ), for any λ ∈ R and with (x 1 , . . . , x n ) ∈ R n or (x 1 , . . . , x n ) ∈ C n , and C n is a vector space over C with scalar multiplication as above, but with λ ∈ C. n of polynomials of degree at most n with real coefficients is a vector space over R, and the ring C[X] n of polynomials of degree at most n with complex coefficients is a vector space over C, with scalar multiplication λ•P (X) of a polynomial 4. The ring R[X] of all polynomials with real coefficients is a vector space over R, and the ring C[X] of all polynomials with complex coefficients is a vector space over C, with the same scalar multiplication as above. 5. The ring of n × n matrices M n (R) is a vector space over R. 6. The ring of m × n matrices M m,n (R) is a vector space over R. 7. The ring C(a, b) of continuous functions f : (a, b) → R is a vector space over R, with the scalar multiplication λf of a function f : (a, b) → R by a scalar λ ∈ R given by (λf )(x) = λf (x), for all x ∈ (a, b). 8. A very important example of vector space is the set of linear maps between two vector spaces to be defined in Section 2.7. Here is an example that will prepare us for the vector space of linear maps. Let X be any nonempty set and let E be a vector space. The set of all functions f : X → E can be made into a vector space as follows: Given any two functions f : X → E and g : X → E, let (f + g) : X → E be defined such that i∈I λ i u i can be viewed as an affine combination. For example, if k is an index not in I, if we let J = I ∪ {k}, u k = 0, and λ k = 1 -i∈I λ i , then j∈J λ j u j is an affine combination and i∈I λ i u i = j∈J λ j u j . 49 However, we get new spaces. For example, in R 3 , the set of all affine combinations of the three vectors e 1 = (1, 0, 0), e 2 = (0, 1, 0), and e 3 = (0, 0, 1), is the plane passing through these three points. Since it does not contain 0 = (0, 0, 0), it is not a linear subspace. (2) Consider combinations i∈I λ i u i for which λ i ≥ 0, for all i ∈ I. These are called positive (or conic) combinations. It turns out that positive combinations of families of vectors are cones. They show up naturally in convex optimization. (3) Consider combinations i∈I λ i u i for which we require (1) and (2), that is i∈I λ i = 1, and λ i ≥ 0 for all i ∈ I. These are called convex combinations. Given any finite family of vectors, the set of all convex combinations of these vectors is a convex polyhedron. Convex polyhedra play a very important role in convex optimization. The notion of linear combination can also be defined for infinite index sets I. To ensure that a sum i∈I λ i u i makes sense, we restrict our attention to families of finite support. Definition 2.8. Given any field K, a family of scalars (λ i ) i∈I has finite support if λ i = 0 for all i ∈ I -J, for some finite subset J of I.

On Some Quadratic Optimization Problems Arising in Computer Vision

The goal of this paper is to find methods for solving various quadratic optimization problems, mo... more

Roughly speaking, probability theory deals with experiments whose outcome are not predictable wit... more Roughly speaking, probability theory deals with experiments whose outcome are not predictable with certainty. We often call such experiments random experiments. They are subject to chance. Using a mathematical theory of probability, we may be able to calculate the likelihood of some event. In the introduction to his classical book [1] (first published in 1888), Joseph Bertrand (1822-1900) writes (translated from French to English): "How dare we talk about the laws of chance (in French: le hasard)? Isn't chance the antithesis of any law? In rejecting this definition, I will not propose any alternative. On a vaguely defined subject, one can reason with authority. ..." Of course, Bertrand's words are supposed to provoke the reader. But it does seem paradoxical that anyone could claim to have a precise theory about chance! It is not my intention to engage in a philosophical discussion about the nature of chance. Instead, I will try to explain how it is possible to build some mathematical tools that can be used to reason rigorously about phenomema that are subject to chance. These tools belong to probability theory. These days, many fields in computer science such as machine learning, cryptography, computational linguistics, computer vision, robotics, and of course algorithms, rely a lot on probability theory. These fields are also a great source of new problems that stimulate the discovery of new methods and new theories in probability theory. Although this is an oversimplification that ignores many important contributors, one might say that the development of probability theory has gone through four eras whose key figures are:

Notes on Primality Testing And Public Key Cryptography Part 1: Randomized Algorithms Miller-Rabin and Solovay-Strassen Tests

Ever since written communication was used, people have been interested in trying to conceal the c... more Ever since written communication was used, people have been interested in trying to conceal the content of their messages from their adversaries. This has led to the development of techniques of secret communication, a science known as cryptography. The basic situation is that one party, A, say Albert, wants to send a message to another party, J, say Julia. However, there is a danger that some ill-intentioned third party, Machiavelli, may intercept the message and learn things that he is not supposed to know about and as a result, do evil things. The original message, understandable to all parties, is known as the plain text. To protect the content of the message, Albert encrypts his message. When Julia receives the encrypted message, she must decrypt it in order to be able to read it. Both Albert and Julia share some information that Machiavelli does not have, a key. Without a key, Machiavelli, is incapable of decrypting the message and thus, to do harm. There are many schemes for generating keys to encrypt and decrypt messages. We are going to describe a method involving public and private keys known as the RSA Cryptosystem, named after its inventors, Ronald Rivest, Adi Shamir, and Leonard Adleman (1978), based on ideas by Diffie and Hellman (1976). We highly recommend reading the orginal paper by Rivest, Shamir, and Adleman [19]. It is beautifully written and easy to follow. A very clear, but concise exposition can also be found in Koblitz [10]. An encyclopedic coverage of cryptography can be found in Menezes, van Oorschot, and Vanstone's Handbook [14].

Motion Interpolation in SIM(3

In this paper, we explore motion interpolation in the group SIM(3), the group of a ne transformat... more In this paper, we explore motion interpolation in the group SIM(3), the group of a ne transformations which are the composition of a rotation, a translation, and a uniform rescaling. The group SIM(n) is a Lie group with Lie algebra sim(n), and we give a formula for the exponential map exp : sim(n) ! SIM(n). We prove the surjectivity of the exponential map for any n 1, and for n = 3, we give an explicit formula and show how to compute logarithms. We use these algorithms for computing logarithms and exponentials to perform motion interpolation in SIM(3). Given a sequence A 0 , A 1 ,. .. , A n of transformations in SIM(3), we compute a sequence of logs X 0 , X 1 ,. .. , X n in sim(3) ⇡ R 7 , then fit a cubic spline c(t) that interpolates the X i , and then compute the curve e c(t) in SIM(3). However, the fact that the logarithm is multivalued causes problems. Whenever a rotation "crosses through

Solving the Elastic Net and Lasso Regression Problems

In this paper we investigate methods for solving the elastic net regression problem. The lasso re... more In this paper we investigate methods for solving the elastic net regression problem. The lasso regression problem is the special case of the elastic net regression problem where only the 1-norm of the weight vector is penalized. The elastic net regression problem has a unique minimal solution. This solution can be found directly from the primal program using ADMM, using a simple extension of the method used to solve lasso. For all of these problems we seek a minimal solution (w, b) where w is the weight vector and b is the intercept (we deal with the intercept directly, rather than by "centering" of the data).

by Jean Gallier and Dianna Xu

In the words of Milnor himself, the classification theorem for compact surfaces is a formidable r... more In the words of Milnor himself, the classification theorem for compact surfaces is a formidable result. This result was first proved rigorously in 1921 by Brahana. It is the culmination of the work of many, including Möbius, Jordan, von Dyck, and Dehn and Heegaard. Indeed, a rigorous proof requires, among other things, a precise definition of a surface and of orientability, a precise notion of triangulation, and a precise way of determining whether two surfaces are homeomorphic or not. This requires some notions of algebraic topology such as, fundamental groups, homology groups, and the Euler-Poincaré characteristic. Most steps of the proof are rather involved and it is easy to loose track.

If we want to somehow classify surfaces, we have to deal with the issue of deciding when we consi... more If we want to somehow classify surfaces, we have to deal with the issue of deciding when we consider two surfaces to be equivalent. It seems reasonable to treat homeomorphic surfaces as equivalent, but this leads to the problem of deciding when two surfaces are not homeomorphic, which is a very difficult problem. One way to approach this problem is to forget some of the topological structure of a surface and look for more algebraic objects that can be associated with a surface. For example, we can consider closed curves on a surface, and see how they can be deformed. It is also fruitful to give an algebraic structure to appropriate sets of closed curves on a surface, for example, a group structure. Two important tools for studying surfaces were invented by Poincaré, the fundamental group, and the homology groups. In this section, we take a look at the fundamental group. Roughly speaking, given a topological space E and some chosen point a ∈ E, a group π(E, a) called the fundamental group of E based at a is associated with (E, a), and to every continuous map f : (X, x) → (Y, y) such that f (x) = y, is associated a group homomorphism f * : π(X, x) → π(Y, y). Thus, certain topological questions about the space E can translated into algebraic questions about the group π(E, a). This is the paradigm of algebraic topology. In this section, we will focus on the concepts rather than dwelve into technical details. For a thorough presentation of the fundamental group and related concepts, the reader is referred to Massey [28, 29], Munkres [33], Bredon [6], Hatcher [16], Dold [11], Fulton [14] and Rotman [37]. We also recommend Sato [38] for an informal and yet very clear presentation.

It is remarkable that the compact (two-dimensional) polyhedras can be characterized up to homeomo... more It is remarkable that the compact (two-dimensional) polyhedras can be characterized up to homeomorphism. This situation is exceptional, as such a result is known to be essentially impossible for compact m-manifolds for m ≥ 4, and still open for compact 3-manifolds. In fact, it is possible to characterize the compact (two-dimensional) polyhedras in terms of a simple extension of the notion of a complex, called cell complex by Ahlfors and Sario. What happens is that it is possible to define an equivalence relation on cell complexes and it can be shown that every cell complex is equivalent to some specific normal form. Furthermore, every cell complex has a geometric realization which is a surface, and equivalent cell complexes have homeomorphic geometric realizations. Also, every cell complex is equivalent to a triangulated 2-complex. Finally, we can show that the geometric realizations of distinct normal forms are not homeomorphic.

This Chapter provides a review of basic topological notions. For a comprehensive account, we high... more This Chapter provides a review of basic topological notions. For a comprehensive account, we highly recommend Munkres [33], Amstrong [3], Dixmier [9], Singer and Thorpe [41], Lang [25], or Schwartz [39]. Most spaces considered will have a topological structure given by a metric or a norm and we first review these notions. We begin with metric spaces.

Geometric Methods and Applications for Computer Science and Engineering, Second Edition

Geometric Methods and Applications for Computer Science and Engineetring, 2012

This book grew out of lectures notes that I have written as I have been teaching CIS610, Advanced... more This book grew out of lectures notes that I have written as I have been teaching CIS610, Advanced Geometric Methods in Computer Science, for the past two years. Many thanks to the copyeditor, David Kramer, who did a superb job. I also wish to thank some students and colleagues for their comments, including Koji Ashida,

Curves and surfaces in geometric modelling: Theory and algorithms, 1999

We will be dealing primarily with the following kinds of problems: • Approximating a shape (curve... more We will be dealing primarily with the following kinds of problems: • Approximating a shape (curve or surface). We will see how this can be done using polynomial curves or surfaces (also called Bézier curves or surfaces), spline curves or surfaces. • Interpolating a set of points, by a curve or a surface. Again, we will see how this can be done using spline curves or spline surfaces. • Drawing a curve or a surface. The tools and techniques developed for solving the approximation problem will be very useful for solving the other two problems. The material presented in this book is related to the classical differential geometry of curves and surfaces, and to numerical methods in matrix analysis. In fact, it is often possible to reduce problems involving certain splines to solving systems of linear equations. Thus, it is very helpful to be aware of efficient methods for numerical matrix analysis. For further information on these topics, readers are referred to the excellent texts by Gray [39], Strang [81], and Ciarlet [19]. Strang's beautiful book on applied mathematics is also highly recommended as a general reference [80]. There are other interesting applications of geometry to computer vision, computer graphics, and solid modeling. Some great references are Koenderink [46] and Faugeras [33] for computer vision, Hoffman [43] for solid modeling, and Metaxas [53] for physics-based deformable models. Novelties As far as we know, there is no fully developed modern exposition integrating the basic concepts of affine geometry as well as a presentation of curves and surfaces from the algorithmic point of view in terms of control points (in the polynomial case). There is also no reasonably thorough textbook presentation of the main surface subdivision schemes (Doo-Sabin, Catmull-Clark, Loop), and a technical discussion of convergence and smoothness.

Discmath root b

Mathematical Foundations and Aspects of Discrete Mathematics, 2024

This is a book about discrete mathematics which also discusses mathematical reasoning and logic. ... more This is a book about discrete mathematics which also discusses mathematical reasoning and logic. Since the publication of the first edition of this book a few years ago, we came to realize that for a significant number of readers, it is their first exposure to the rules of mathematical reasoning and to logic. As a consequence, the version of Chapter 1 from the first edition may be a bit too abstract and too formal for those readers, and they may find this discouraging. To remedy this problem, we have written a new version of the first edition of Chapter 1. This new chapter is more elementary, more intuitive, and less formal. It also contains less material, but as in the first edition, it is still possible to skip Chapter 1 without causing any problem or gap, because the other chapters of this book do not depend on the material of Chapter 1. It appears that enough readers are interested in the first edition of Chapter 1, so in this second edition, we reproduce it (slightly updated) as Chapter 11. Again, this chapter can be omitted without causing any problem or gap. My suggestion to readers who have not been exposed to mathematical reasoning and logic is to read, or at least skim, Chapter 1. On the other hand, my suggestion to more sophisticatd readers is to skip Chapter 1 and proceed directly to Chapter 2. From my point of view, they will miss some interesting considerations on the constructive nature of logic, but that's because we are very fond of foundational issues, and we realize that not everybody has the same level of interest in foundations! In this second edition, we tried to make the exposition simpler and clearer. We added some figures, some examples, clarified certain definitions, and simplified some proofs. A few changes and additions were also made. In Chapter 2 we added a section (Section 2.12) which describes the Haar transform on sequences in an elementary fashion as a certain bijection. We also show how the Haar transform can be used to compress audio signals (see Section 2.13). This is a spectacular and concrete illustration of the abstract notion of a bijection. We created a separate chapter (Chapter 3) dealing with the set-theoretical notions of equinumerosity, finite, countable, and infinite sets. In this new chapter we discuss the pigeonhole principle more extensively. In particular, we discuss the Frobenius vii viii Preface Preface ix The last three sections of this chapter contain more advanced material and are optional. The topics of these optional sections are generating functions (including the moment generating function and the characteristic function), the limit theorems (weak law of large numbers, central limit theorem, and strong law of large numbers), and Chernoff bounds. A beautiful exposition of discrete probability can be found in Chapter 8 of Concrete Mathematics, by Graham, Knuth, and Patashnik [1]. Comprehensive presentations can be found in Mitzenmacher and Upfal [3], Ross [4, 5], and Grimmett and Stirzaker [2]. Ross [4] contains an enormous amount of examples and exercises and is very easy to read.

Proofs, Computability, Undecidability, Complexity, And the Lambda Calculus An Introduction

Preface The main goal of this book is to present a mix of material dealing with 1. Proof systems.... more Preface The main goal of this book is to present a mix of material dealing with 1. Proof systems. 2. Computability and undecidability. 3. The Lambda Calculus. 4. Some aspects of complexity theory. Historically, the theory of computability and undecidability arose from Hilbert's efforts to completely formalize mathematics and from Gödel's first incompleteness theorem that showed that such a program was doomed to fail. People realized that to carry out both Hilbert's program and Gödel's work it was necessary to define precisely what is the notion of a computable function and the notion of a mechanically checkable proof. The first definition given around 1934 was that of the class of computable function in the sense of Herbrand-Gödel-Kleene. The second definition given by Church in 1935-1936 was the notion of a function definable in the λ-calculus. The equivalence of these two definitions was shown by Kleene in 1936. Shortly after in 1936, Turing introduced a third definition, that of a Turingcomputable function. Turing proved the equivalence of his definition with the Herbrand-Gödel-Kleene definition in 1937 (his proofs are rather sketchy compared to Kleene's proofs). All these historical papers can be found in a fascinating book edited by Martin Davis [27]. Negative results pointing out limitations of the notion of computability started to appear: Gödel's first (and second) incompleteness result, but also Church's theorem on the undecidability of validity in first-order logic, and Turing's result on the undecidability of the halting problem for Turing machines. Although originally the main focus was on the notion of function, these undecidability results triggered the study of computable and noncomputable sets of natural numbers. Other definitions of the computable functions were given later. From our point of view, the most important ones are 1. RAM programs and RAM-computable functions by Shepherdson and Sturgis (1963), and anticipated by Post (1944); see Machtey and Young [41].

Aspects of Convex Geometry Polyhedra, Linear Programming, Shellings, Voronoi Diagrams, Delaunay Triangulations

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used... more Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincaré formula (Poincaré, 1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a "correct" account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space. It turns out that in order to achieve (even partially) the above goals, I found that it was necessary to include quite a bit of background material on convex sets, polytopes, polyhedra and projective spaces. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also had to include some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of "projective polyhedron" based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new. Since Chapters 2, 3, 4, and 5 contain all the background (and more) needed to discuss linear programming (including the simplex algorithm and duality), we have included some chapters on linear programming.

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, 2022

for all p ≥ 0. Furthermore, this assignment is functorial (more precisely, it is a functor from t... more for all p ≥ 0. Furthermore, this assignment is functorial (more precisely, it is a functor from the category of cochain complexes and chain maps to the category of abelian groups and their homomorphisms). At first glance cohomology appears to be very abstract so it is natural to look for explicit examples. A way to obtain a cochain complex is to apply the operator (functor) Hom Z (-, G) to a chain complex C, where G is any abelian group. Given a fixed abelian group A, for any abelian group B we denote by Hom Z (B, A) the abelian group of all homomorphisms from B to A. Given any two abelian groups B and C, for any homomorphism f : The map Hom Z (f, A) is also denoted by Hom Z (f, id A ) or even Hom Z (f, id). Observe that the effect of Hom Z (f, id) on ϕ is to precompose ϕ with f . If f : B → C and g : C → D are homomorphisms of abelian groups, a simple computation shows that Hom R (g 1.1. EXACT SEQUENCES, CHAIN COMPLEXES, HOMOLOGY, COHOMOLOGY 21 Observe that Hom Z (f, id) and Hom Z (g, id) are composed in the reverse order of the composition of f and g. It is also immediately verified that Hom Z (id A , id) = id Hom Z (A,G) . We say that Hom Z (-, id) is a contravariant functor (from the category of abelian groups and group homomorphisms to itself). Then given a chain complex we can form the cochain complex obtained by applying Hom Z (-, G), and denoted Hom Z (C, G). The coboundary map d p is given by which means that for any f ∈ Hom Z (C p , G), we have Thus, for any (p + 1)-chain c ∈ C p+1 we have We obtain the cohomology groups H p (Hom Z (C, G)) associated with the cochain complex Hom Z (C, G). The cohomology groups H p (Hom Z (C, G)) are also denoted H p (C; G). This process was applied to the simplicial chain complex C * (K) associated with a simplicial complex K by Alexander and Kolmogoroff to obtain the simplicial cochain complex Hom Z (C * (K); G) denoted C * (K; G) and the simplicial cohomology groups H p (K; G) of the simplicial complex K; see Section 5.6. Soon after, this process was applied to the singular chain complex S * (X; Z) of a space X to obtain the singular cochain complex Hom Z (S * (X; Z); G) denoted S * (X; G) and the singular cohomology groups H p (X; G) of the space X; see Section 4.8. Given a continuous map f : X → Y , there is an induced chain map f : S * (Y ; G) → S * (X; G) between the singular cochain complexes S * (Y ; G) and S * (X; G), and thus homomorphisms of cohomology f * : H p (Y ; G) → H p (X; G). Observe the reversal: f is a map from X to Y , but f * maps H p (Y ; G) to H p (X; G). We say that the map X → (H p (X; G)) p≥0 is a contravariant functor from the category of topological spaces and continuous maps to the category of abelian groups and their homomorphisms. So far our homology groups have coefficients in Z, but the process of forming a cochain complex Hom Z (C, G) from a chain complex C allows the use of coefficients in any abelian obtained by applying Hom R (-, G), and denoted Hom R (C, G).

Differential Geometry and Lie Groups A Second Course

Differential Geometry and Lie Groups A Second Course, 2020

However, we often omit the subscript K unless confusion may arise.

Differential Geometry and Lie Groups A Computational Perspective

Differential Geometry and Lie Groups A Computational Perspective, 2020

To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie. To my parents Howard a... more To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie. To my parents Howard and Jane. discuss three results, one of which being the Hadamard and Cartan theorem about complete manifolds of non-positive curvature. The goal of Chapter 18 is to understand the behavior of isometries and local isometries, in particular their action on geodesics. We also intoduce Riemannian covering maps and Riemannian submersions. If π : M → B is a submersion between two Riemannian manifolds, then for every b ∈ B and every p ∈ π -1 (b), the tangent space T p M to M at p splits into two orthogonal components, its vertical component V p = Ker dπ p , and its horizontal component H p (the orthogonal complement of V p ). If the map dπ p is an isometry between H p and T b B, then most of the differential geometry of B can be studied by lifting B to M , and then projecting down to B again. We also introduce Killing vector fields, which play a technical role in the study of reductive homogeneous spaces. In Chapter 19, we return to Lie groups. Not every Lie group is a matrix group, so in order to study general Lie groups it is necessary to introduce left-invariant (and rightinvariant) vector fields on Lie groups. It turns out that the space of left-invariant vector fields is isomorphic to the tangent space g = T I G to G at the identity, which is a Lie algebra. By considering integral curves of left-invariant vector fields, we define the generalization of the exponential map exp : g → G to an arbitrary Lie group. The notion of immersed Lie subgroup is introduced, and the correspondence between Lie groups and Lie algebra is explored. We also consider the special classes of semidirect products of Lie algebras and Lie groups, the universal covering of a Lie group, and the Lie algebra of Killing vector fields on a Riemannian manifold. Chapter 20 deals with two topics: 1. A formula for the derivative of the exponential map for a general Lie group (not necessarily a matrix group). 2. A formula for the Taylor expansion of µ(X, Y ) = log(exp(X) exp(Y )) near the origin. The second problem is solved by a formula known as the Campbell-Baker-Hausdorff formula. An explicit formula was derived by Dynkin (1947), and we present this formula. Chapter 21 is devoted to the study of metrics, connections, geodesics, and curvature, on Lie groups. Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. As a consequence, it is possible to find explicit formulae for the Levi-Civita connection and the various curvatures, especially in the case of metrics which are both left and right-invariant. In Section 21.2 we give four characterizations of bi-invariant metrics. The first one refines the criterion of the existence of a left-invariant metric and states that every bi-invariant metric on a Lie group G arises from some Ad-invariant inner product on the Lie algebra g. In Section 21.3 we show that if G is a Lie group equipped with a left-invariant metric, then it is possible to express the Levi-Civita connection and the sectional curvature in terms Recall that a real symmetric matrix is called positive (or positive semidefinite) if its eigenvalues are all positive or null, and positive definite if its eigenvalues are all strictly positive. We denote the vector space of real symmetric n × n matrices by S(n), the set of symmetric positive matrices by SP(n), and the set of symmetric positive definite matrices by SPD(n). The next proposition shows that every symmetric positive definite matrix A is of the form e B for some unique symmetric matrix B. The set of symmetric matrices is a vector space, but it is not a Lie algebra because the Lie bracket [A, B] is not symmetric unless A and B commute, and the set of symmetric (positive) definite matrices is not a multiplicative group, so this result is of a different flavor as Theorem 2.6. Proposition 2.8. For every symmetric matrix B, the matrix e B is symmetric positive definite. For every symmetric positive definite matrix A, there is a unique symmetric matrix B such that A = e B . Proof. We showed earlier that e B = e B . If B is a symmetric matrix, then since B = B, we get e B = e B = e B , and e B is also symmetric. Since the eigenvalues λ 1 , . . . , λ n of the symmetric matrix B are real and the eigenvalues of e B are e λ 1 , . . . , e λn , and since e λ > 0 if λ ∈ R, e B is positive definite. To show the surjectivity of the exponential map, note that if A is symmetric positive definite, then by Theorem 12.3 from Chapter 12 of Gallier [50], there is an orthogonal matrix P such that A = P D P , where D is a diagonal matrix CHAPTER 2. THE MATRIX EXPONENTIAL; SOME MATRIX LIE GROUPS

Fundamentals of Optimization Theory With Applications to Machine Learning

Fundamentals of Optimization Theory With Applications to Machine Learning, 2020

for reporting typos and for helpful comments. Mary Pugh and William Yu (at the University of Toro... more for reporting typos and for helpful comments. Mary Pugh and William Yu (at the University of Toronto) taught a course using our book and reported a number of typos and errors. We warmly thank them as well as their students, not only for finding errors, but also for very hepful comments and suggestions for simplifying some proofs. Thanks to Gilbert Strang. We learned much from his books which have been a major source of inspiration. Special thanks to Steven Boyd. We learned a lot from his remarkable book on convex optimization and his papers, and Part III of our book is significantly inspired by his writings. The first author also wishes to express his deepest gratitute to Philippe G. Ciarlet who was his teacher and mentor in 1970-1972 while he was a student at ENPC in Paris. Professor Ciarlet was by far his best teacher. He also knew how to instill in his students the importance of intellectual rigor, honesty, and modesty. He still has his typewritten notes on measure theory and integration, and on numerical linear algebra. The latter became his wonderful book Ciarlet , from which we have borrowed heavily.

Linear Algebra for Computer Vision, Robotics, and Machine Learning

Linear algebra for computer vision, robotics and machine learning, 2022

In recent years, computer vision, robotics, machine learning, and data science have been some of ... more In recent years, computer vision, robotics, machine learning, and data science have been some of the key areas that have contributed to major advances in technology. Anyone who looks at papers or books in the above areas will be baffled by a strange jargon involving exotic terms such as kernel PCA, ridge regression, lasso regression, support vector machines (SVM), Lagrange multipliers, KKT conditions, etc. Do support vector machines chase cattle to catch them with some kind of super lasso? No! But one will quickly discover that behind the jargon which always comes with a new field (perhaps to keep the outsiders out of the club), lies a lot of "classical" linear algebra and techniques from optimization theory. And there comes the main challenge: in order to understand and use tools from machine learning, computer vision, and so on, one needs to have a firm background in linear algebra and optimization theory. To be honest, some probablity theory and statistics should also be included, but we already have enough to contend with. Many books on machine learning struggle with the above problem. How can one understand what are the dual variables of a ridge regression problem if one doesn't know about the Lagrangian duality framework? Similarly, how is it possible to discuss the dual formulation of SVM without a firm understanding of the Lagrangian framework? The easy way out is to sweep these difficulties under the rug. If one is just a consumer of the techniques we mentioned above, the cookbook recipe approach is probably adequate. But this approach doesn't work for someone who really wants to do serious research and make significant contributions. To do so, we believe that one must have a solid background in linear algebra and optimization theory. This is a problem because it means investing a great deal of time and energy studying these fields, but we believe that perseverance will be amply rewarded. Our main goal is to present fundamentals of linear algebra and optimization theory, keeping in mind applications to machine learning, robotics, and computer vision. This work consists of two volumes, the first one being linear algebra, the second one optimization theory and applications, especially to machine learning. This first volume covers "classical" linear algebra, up to and including the primary decomposition and the Jordan form. Besides covering the standard topics, we discuss a few topics that are important for applications. These include: 1. Haar bases and the corresponding Haar wavelets. knew how to instill in his students the importance of intellectual rigor, honesty, and modesty. He still has his typewritten notes on measure theory and integration, and on numerical linear algebra. The latter became his wonderful book Ciarlet , from which we have borrowed heavily. CHAPTER 2. VECTOR SPACES, BASES, LINEAR MAPS for all (x 1 , . . . , x n ) ∈ R n and all λ ∈ R. Axioms (V0)-(V3) are all satisfied, but (V4) fails. Less trivial examples can be given using the notion of a basis, which has not been defined yet. The field R itself can be viewed as a vector space over itself, addition of vectors being addition in the field, and multiplication by a scalar being multiplication in the field. Example 2.3. 1. The fields R and C are vector spaces over R. 2. The groups R n and C n are vector spaces over R, with scalar multiplication given by λ(x 1 , . . . , x n ) = (λx 1 , . . . , λx n ), for any λ ∈ R and with (x 1 , . . . , x n ) ∈ R n or (x 1 , . . . , x n ) ∈ C n , and C n is a vector space over C with scalar multiplication as above, but with λ ∈ C. n of polynomials of degree at most n with real coefficients is a vector space over R, and the ring C[X] n of polynomials of degree at most n with complex coefficients is a vector space over C, with scalar multiplication λ•P (X) of a polynomial 4. The ring R[X] of all polynomials with real coefficients is a vector space over R, and the ring C[X] of all polynomials with complex coefficients is a vector space over C, with the same scalar multiplication as above. 5. The ring of n × n matrices M n (R) is a vector space over R. 6. The ring of m × n matrices M m,n (R) is a vector space over R. 7. The ring C(a, b) of continuous functions f : (a, b) → R is a vector space over R, with the scalar multiplication λf of a function f : (a, b) → R by a scalar λ ∈ R given by (λf )(x) = λf (x), for all x ∈ (a, b). 8. A very important example of vector space is the set of linear maps between two vector spaces to be defined in Section 2.7. Here is an example that will prepare us for the vector space of linear maps. Let X be any nonempty set and let E be a vector space. The set of all functions f : X → E can be made into a vector space as follows: Given any two functions f : X → E and g : X → E, let (f + g) : X → E be defined such that i∈I λ i u i can be viewed as an affine combination. For example, if k is an index not in I, if we let J = I ∪ {k}, u k = 0, and λ k = 1 -i∈I λ i , then j∈J λ j u j is an affine combination and i∈I λ i u i = j∈J λ j u j . 49 However, we get new spaces. For example, in R 3 , the set of all affine combinations of the three vectors e 1 = (1, 0, 0), e 2 = (0, 1, 0), and e 3 = (0, 0, 1), is the plane passing through these three points. Since it does not contain 0 = (0, 0, 0), it is not a linear subspace. (2) Consider combinations i∈I λ i u i for which λ i ≥ 0, for all i ∈ I. These are called positive (or conic) combinations. It turns out that positive combinations of families of vectors are cones. They show up naturally in convex optimization. (3) Consider combinations i∈I λ i u i for which we require (1) and (2), that is i∈I λ i = 1, and λ i ≥ 0 for all i ∈ I. These are called convex combinations. Given any finite family of vectors, the set of all convex combinations of these vectors is a convex polyhedron. Convex polyhedra play a very important role in convex optimization. The notion of linear combination can also be defined for infinite index sets I. To ensure that a sum i∈I λ i u i makes sense, we restrict our attention to families of finite support. Definition 2.8. Given any field K, a family of scalars (λ i ) i∈I has finite support if λ i = 0 for all i ∈ I -J, for some finite subset J of I.

On Some Quadratic Optimization Problems Arising in Computer Vision

The goal of this paper is to find methods for solving various quadratic optimization problems, mo... more

Roughly speaking, probability theory deals with experiments whose outcome are not predictable wit... more Roughly speaking, probability theory deals with experiments whose outcome are not predictable with certainty. We often call such experiments random experiments. They are subject to chance. Using a mathematical theory of probability, we may be able to calculate the likelihood of some event. In the introduction to his classical book [1] (first published in 1888), Joseph Bertrand (1822-1900) writes (translated from French to English): "How dare we talk about the laws of chance (in French: le hasard)? Isn't chance the antithesis of any law? In rejecting this definition, I will not propose any alternative. On a vaguely defined subject, one can reason with authority. ..." Of course, Bertrand's words are supposed to provoke the reader. But it does seem paradoxical that anyone could claim to have a precise theory about chance! It is not my intention to engage in a philosophical discussion about the nature of chance. Instead, I will try to explain how it is possible to build some mathematical tools that can be used to reason rigorously about phenomema that are subject to chance. These tools belong to probability theory. These days, many fields in computer science such as machine learning, cryptography, computational linguistics, computer vision, robotics, and of course algorithms, rely a lot on probability theory. These fields are also a great source of new problems that stimulate the discovery of new methods and new theories in probability theory. Although this is an oversimplification that ignores many important contributors, one might say that the development of probability theory has gone through four eras whose key figures are:

Notes on Primality Testing And Public Key Cryptography Part 1: Randomized Algorithms Miller-Rabin and Solovay-Strassen Tests

Ever since written communication was used, people have been interested in trying to conceal the c... more Ever since written communication was used, people have been interested in trying to conceal the content of their messages from their adversaries. This has led to the development of techniques of secret communication, a science known as cryptography. The basic situation is that one party, A, say Albert, wants to send a message to another party, J, say Julia. However, there is a danger that some ill-intentioned third party, Machiavelli, may intercept the message and learn things that he is not supposed to know about and as a result, do evil things. The original message, understandable to all parties, is known as the plain text. To protect the content of the message, Albert encrypts his message. When Julia receives the encrypted message, she must decrypt it in order to be able to read it. Both Albert and Julia share some information that Machiavelli does not have, a key. Without a key, Machiavelli, is incapable of decrypting the message and thus, to do harm. There are many schemes for generating keys to encrypt and decrypt messages. We are going to describe a method involving public and private keys known as the RSA Cryptosystem, named after its inventors, Ronald Rivest, Adi Shamir, and Leonard Adleman (1978), based on ideas by Diffie and Hellman (1976). We highly recommend reading the orginal paper by Rivest, Shamir, and Adleman [19]. It is beautifully written and easy to follow. A very clear, but concise exposition can also be found in Koblitz [10]. An encyclopedic coverage of cryptography can be found in Menezes, van Oorschot, and Vanstone's Handbook [14].

Motion Interpolation in SIM(3

In this paper, we explore motion interpolation in the group SIM(3), the group of a ne transformat... more In this paper, we explore motion interpolation in the group SIM(3), the group of a ne transformations which are the composition of a rotation, a translation, and a uniform rescaling. The group SIM(n) is a Lie group with Lie algebra sim(n), and we give a formula for the exponential map exp : sim(n) ! SIM(n). We prove the surjectivity of the exponential map for any n 1, and for n = 3, we give an explicit formula and show how to compute logarithms. We use these algorithms for computing logarithms and exponentials to perform motion interpolation in SIM(3). Given a sequence A 0 , A 1 ,. .. , A n of transformations in SIM(3), we compute a sequence of logs X 0 , X 1 ,. .. , X n in sim(3) ⇡ R 7 , then fit a cubic spline c(t) that interpolates the X i , and then compute the curve e c(t) in SIM(3). However, the fact that the logarithm is multivalued causes problems. Whenever a rotation "crosses through

Solving the Elastic Net and Lasso Regression Problems

In this paper we investigate methods for solving the elastic net regression problem. The lasso re... more In this paper we investigate methods for solving the elastic net regression problem. The lasso regression problem is the special case of the elastic net regression problem where only the 1-norm of the weight vector is penalized. The elastic net regression problem has a unique minimal solution. This solution can be found directly from the primal program using ADMM, using a simple extension of the method used to solve lasso. For all of these problems we seek a minimal solution (w, b) where w is the weight vector and b is the intercept (we deal with the intercept directly, rather than by "centering" of the data).

by Jean Gallier and Dianna Xu

In the words of Milnor himself, the classification theorem for compact surfaces is a formidable r... more In the words of Milnor himself, the classification theorem for compact surfaces is a formidable result. This result was first proved rigorously in 1921 by Brahana. It is the culmination of the work of many, including Möbius, Jordan, von Dyck, and Dehn and Heegaard. Indeed, a rigorous proof requires, among other things, a precise definition of a surface and of orientability, a precise notion of triangulation, and a precise way of determining whether two surfaces are homeomorphic or not. This requires some notions of algebraic topology such as, fundamental groups, homology groups, and the Euler-Poincaré characteristic. Most steps of the proof are rather involved and it is easy to loose track.

If we want to somehow classify surfaces, we have to deal with the issue of deciding when we consi... more If we want to somehow classify surfaces, we have to deal with the issue of deciding when we consider two surfaces to be equivalent. It seems reasonable to treat homeomorphic surfaces as equivalent, but this leads to the problem of deciding when two surfaces are not homeomorphic, which is a very difficult problem. One way to approach this problem is to forget some of the topological structure of a surface and look for more algebraic objects that can be associated with a surface. For example, we can consider closed curves on a surface, and see how they can be deformed. It is also fruitful to give an algebraic structure to appropriate sets of closed curves on a surface, for example, a group structure. Two important tools for studying surfaces were invented by Poincaré, the fundamental group, and the homology groups. In this section, we take a look at the fundamental group. Roughly speaking, given a topological space E and some chosen point a ∈ E, a group π(E, a) called the fundamental group of E based at a is associated with (E, a), and to every continuous map f : (X, x) → (Y, y) such that f (x) = y, is associated a group homomorphism f * : π(X, x) → π(Y, y). Thus, certain topological questions about the space E can translated into algebraic questions about the group π(E, a). This is the paradigm of algebraic topology. In this section, we will focus on the concepts rather than dwelve into technical details. For a thorough presentation of the fundamental group and related concepts, the reader is referred to Massey [28, 29], Munkres [33], Bredon [6], Hatcher [16], Dold [11], Fulton [14] and Rotman [37]. We also recommend Sato [38] for an informal and yet very clear presentation.

It is remarkable that the compact (two-dimensional) polyhedras can be characterized up to homeomo... more It is remarkable that the compact (two-dimensional) polyhedras can be characterized up to homeomorphism. This situation is exceptional, as such a result is known to be essentially impossible for compact m-manifolds for m ≥ 4, and still open for compact 3-manifolds. In fact, it is possible to characterize the compact (two-dimensional) polyhedras in terms of a simple extension of the notion of a complex, called cell complex by Ahlfors and Sario. What happens is that it is possible to define an equivalence relation on cell complexes and it can be shown that every cell complex is equivalent to some specific normal form. Furthermore, every cell complex has a geometric realization which is a surface, and equivalent cell complexes have homeomorphic geometric realizations. Also, every cell complex is equivalent to a triangulated 2-complex. Finally, we can show that the geometric realizations of distinct normal forms are not homeomorphic.

This Chapter provides a review of basic topological notions. For a comprehensive account, we high... more This Chapter provides a review of basic topological notions. For a comprehensive account, we highly recommend Munkres [33], Amstrong [3], Dixmier [9], Singer and Thorpe [41], Lang [25], or Schwartz [39]. Most spaces considered will have a topological structure given by a metric or a norm and we first review these notions. We begin with metric spaces.

Correction to: Differential Geometry and Lie Groups

Geometry and computing, 2020

Operators on Riemannian Manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, the Bochner Laplacian, and Weitzenböck Formulae

Geometry and computing, 2020

The Laplacian is a very important operator because it shows up in many of the equations used in p... more The Laplacian is a very important operator because it shows up in many of the equations used in physics to describe natural phenomena such as heat diffusion or wave propagation. Therefore, it is highly desirable to generalize the Laplacian to functions defined on a manifold. Furthermore, in the late 1930s, Georges de Rham (inspired by Elie Cartan) realized that it was fruitful to define a version of the Laplacian operating on differential forms, because of a fundamental and almost miraculous relationship between harmonics forms (those in the kernel of the Laplacian) and the de Rham cohomology groups on a (compact, orientable) smooth manifold. Indeed, as we will see in Section 8.6, for every cohomology group \(H^k_{\mathrm {DR}}(M)\), every cohomology class \([\omega ]\in H^k_{\mathrm {DR}}(M)\) is represented by a unique harmonic k-form ω. The connection between analysis and topology lies deep and has many important consequences. For example, Poincare duality follows as an “easy” consequence of the Hodge theorem.

Geodesics on Riemannian Manifolds

Geometry and computing, 2020

If (M, g) is a Riemannian manifold, then the concept of length makes sense for any piecewise smoo... more If (M, g) is a Riemannian manifold, then the concept of length makes sense for any piecewise smooth (in fact, C1) curve on M. It is then possible to define the structure of a metric space on M, where d(p, q) is the greatest lower bound of the length of all curves joining p and q. Curves on M which locally yield the shortest distance between two points are of great interest. These curves, called geodesics, play an important role and the goal of this chapter is to study some of their properties.

The Structure of O(p, q) and SO(p, q)

Geometry and computing, 2020

In this chapter, we take a closer look at the structure of the groups O(p, q) and SO(p, q) (also ... more In this chapter, we take a closer look at the structure of the groups O(p, q) and SO(p, q) (also SO 0 (p, q)). We begin with the polar form of matrices in O(p, q), and then we describe the topological structure of the groups O(p, q), SO(p, q), and SO 0 (p, q). For this, we briefly investigate a class of groups called pseudo-algebraic groups. Recall from Section 6.1 that the group O(p, q) is the set of all n × n-matrices We deduce immediately that | det(A)| = 1, and we also know that AI p,q A = I p,q holds. Unfortunately, when p = 0, 1 and q = 0, 1, it does not seem possible to obtain a formula as nice as that given in Proposition 6.2. Nevertheless, we can obtain a formula for a polar form factorization of matrices in O(p, q). Recall (for example, see Gallier [48], Chapter 12) that if S is a symmetric positive definite matrix, then there is a unique symmetric positive definite matrix, T , so that We denote T by S 1 2 or √ S. By S -1 2 , we mean the inverse of S 1 2 . In order to obtain the polar form of a matrix in O(p, q), we begin with the following proposition: Proposition 7.1. Every matrix X ∈ O(p, q) can be written as

Isometries, Local Isometries, Riemannian Coverings and Submersions, and Killing Vector Fields

Geometry and computing, 2020

The goal of this chapter is to understand the behavior of isometries and local isometries, in par... more The goal of this chapter is to understand the behavior of isometries and local isometries, in particular their action on geodesics. In Section 18.1 we show that isometries preserve the Levi-Civita connection. Local isometries preserve all concepts that are local in nature, such as geodesics, the exponential map, sectional, Ricci, and scalar curvature. In Section 18.2 we define Riemannian covering maps. These are smooth covering maps π: M → N that are also local isometries. There is a nice correspondence between the geodesics in M and the geodesics in N. We prove that if M is complete, N is connected, and π: M → N is a local isometry, then π is a Riemannian covering. In Section 18.3 we introduce Riemannian submersions. Given a submersion π: M → B between two Riemannian manifolds (M, g) and (B, h), for every b ∈ B in the image of π, the fiber π−1(b) is a Riemannian submanifold of M, and for every p ∈ π−1(b), the tangent space TpM to M at p splits into the two components $$\displaystyle T_p M = {\mathrm {Ker}\,} d\pi _p \oplus ({\mathrm {Ker}\,} d\pi _p)^{\perp }, $$ where \(\mathcal {V}_p = {\mathrm {Ker}\,} d\pi _p\) is the vertical subspace of TpM and \(\mathcal {H}_p = ({\mathrm {Ker}\,} d\pi _p)^{\perp }\) (the orthogonal complement of \(\mathcal {V}_p\) with respect to the metric gp on TpM) is the horizontal subspace of TpM. If the map dπp is an isometry between the horizontal subspace \(\mathcal {H}_p\) of TpM and Tπ(p)B for every p, then π is a Riemannian submersion. In this case most of the differential geometry of B can be studied by “lifting” from B to M, and then projecting down to B again. In Section 18.4 we define Killing vector fields. A Killing vector field X satisfies the condition $$\displaystyle X(\langle Y, Z{\rangle }) = \langle [X, Y], Z {\rangle } + \langle Y, [X, Z]{\rangle }, $$ for all \(Y, Z\in {\mathfrak {X}}(M)\). A vector field X is a Killing vector field iff the diffeomorphisms Φt induced by the flow Φ of X are isometries (on their domain). Killing vector fields play an important role in the study of reductive homogeneous spaces; see Section 23.4.

The Log-Euclidean Framework Applied to SPD Matrices

Geometry and computing, 2020

WHAT'S SO SPECIAL ABOUT KRUSKAL'S THEOREM AND THE ORDINAL 0? A SURVEY OF SOME RESULTS IN PROOF THEORY This paper has appeared in Annals of Pure and Applied Logic, 53 (1991), 199-260

This paper consists primarily of a survey of results of Harvey Friedman about some proof theoreti... more This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal Γ 0 . We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the "tree theorem", as well as a "finite miniaturization" of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called "natural independence phenomena", which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ 0 and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence. i WHAT'S SO SPECIAL ABOUT KRUSKAL'S THEOREM? of Kruskal's tree theorem are presented in section 4. Section 5 is devoted to several versions of the finite miniaturization of Kruskal's theorem due to Harvey Friedman. Section 6 is a fairly lengthy presentation of basic facts about the countable ordinals, normal functions, and Γ 0 . Most of this material is taken from Schütte [46], and we can only claim to have presented it our own way, and hopefully made it more accessible. Section 7 gives a glimpse at Veblen hierarchies. A constructive system of notations for Γ 0 is presented in section 8. The connection between Kruskal's tree theorem and Γ 0 due to Friedman is presented in section 9. A brief discussion of some relevant subsystems of second-order arithmetic occurs in section 10. An introduction to the theory of term orderings is presented in section 11, including the recursive path ordering and the lexicographic path ordering. A glimpse at slow-growing and fast-growing hierarchies is given in section 12. Finally, constructive proofs of Higman's lemma are briefly discussed in section 13. 2 Well Quasi-Orders (WQO's) We let N denote the set {0, 1, 2, . . .} of natural numbers, and N + denote the set {1, 2, . . .} of positive natural numbers. Given any n ∈ N + , we let [n] denote the finite set {1, 2, . . . , n}, and we let [0] = ∅. Given a set S, a finite sequence u over S, or string over S, is a function u : [n] → S, for some n ∈ N. The integer n is called the length of u and is denoted by |u|. The special sequence with domain ∅ is called the empty sequence, or empty string, and will be denoted by e. Strings can be concatenated in the usual way: Given two strings u : [m] → S and v : [n] → S, their concatenation denoted by u.v or uv, is the string uv : Clearly, concatenation is associative and e is an identity element. Occasionally, a finite sequence u of length n will be denoted as u 1 , . . . , u n (denoting u(i) as u i ), or as u 1 . . . u n . Strings of length 1 are identified with elements of S. The set of all strings over S is denoted as S * . An infinite sequence is a function s : N + → S. An infinite sequence s is also denoted by (s i ) i≥1 , or by s 1 , s 2 , . . . , s i , . . . . Given an infinite sequence s = (s i ) i≥1 , an infinite subsequence of s is any infinite sequence s = (s j ) j≥1 such that there is a strictly monotonic function 1 f : N + → N + , and s i = s f (i) for all i > 0. An infinite subsequence s of s associated with the function f is also denoted as s = (s f (i) ) i≥1 . We now review preorders and well-foundedness. Definition 2.1 Given a set A, a binary relation ⊆ A × A on the set A is a preorder 1 A function f : N + → N + is strictly monotonic (or increasing) iff for all i, j > 0, i < j implies that f (i) < f (j).

$Research paper thumbnail of Construction of Manifolds from Gluing Data $$\circledast $$$

Construction of Manifolds from Gluing Data $$\circledast $$

Geometry and computing, 2020

The definition of a manifold given in Chapter 8 assumes that the underlying set M is already know... more The definition of a manifold given in Chapter 8 assumes that the underlying set M is already known. However, there are situations where we only have some indirect information about the overlap of the domains Ui of the local charts defining our manifold M in terms of the transition functions $$\displaystyle \varphi _i^j = \varphi _{j i}{\colon } \varphi _i(U_i\cap U_j)\rightarrow \varphi _j(U_i\cap U_j), $$ but where M itself is not known. For example, this situation happens when trying to construct a surface approximating a 3D-mesh. If we let Ωij = φi(Ui ∩ Uj) and Ωji = φj(Ui ∩ Uj), then φji can be viewed as a “gluing map” $$\displaystyle \varphi _{j i}{\colon } \Omega _{i j}\rightarrow \Omega _{j i} $$ between two open subsets of Ωi and Ωj, respectively.

Bundles, Metrics on Bundles, and Homogeneous Spaces

Geometry and computing, 2020

A transitive action ⋅: G × X → X of a group G on a set X yields a description of X as a quotient ... more A transitive action ⋅: G × X → X of a group G on a set X yields a description of X as a quotient G∕Gx, where Gx is the stabilizer of any element, x ∈ X. The points of X are identified with the left cosets gGx (g ∈ G). If X is a “well-behaved” topological space (say a locally compact Hausdorff space), G is a “well-behaved” topological group (say a locally compact topological group which is countable at infinity), and the action is continuous, then G∕Gx is homeomorphic to X. In particular these conditions are satisfied if G is a Lie group and X is a manifold. Intuitively, the above theorem says that G can be viewed as a family of “fibers” gGx, all isomorphic to Gx, these fibers being parametrized by the “base space” X, and varying smoothly when the point corresponding to the coset gGx moves in X. We have an example of what is called a fiber bundle, in fact, a principal fiber bundle. This view of G as a family of fibers gGx, with x ∈ X, is a special case of the notion of a fiber bundle.

Clifford Algebras, Clifford Groups, and the Groups Pin(n) and Spin(n)

Geometry and computing, 2020

The main goal of this chapter is to explain how rotations in R n are induced by the action of a c... more The main goal of this chapter is to explain how rotations in R n are induced by the action of a certain group Spin(n) on R n , in a way that generalizes the action of the unit complex numbers U(1) on R 2 , and the action of the unit quaternions SU(2) on R 3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and R n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra Cl n , the Clifford algebra associated with R n. Furthermore, for n ≥ 3, we are lucky, because the group Spin(n) is topologically simpler than the group SO(n). Indeed, for n ≥ 3, the group Spin(n) is simply connected (a fact that it is not so easy to prove without some machinery), whereas SO(n) is not simply connected. Intuitively speaking, SO(n) is more twisted than Spin(n). In fact, we will see that Spin(n) is a double cover of SO(n). Since the spinor groups are certain well-chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. This chapter provides a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl p,q associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful "8-periodicity theorem" of Élie Cartan and Raoul Bott (with proofs). We also explain when Spin(p, q) is a double-cover of SO(p, q). The reader should be warned that a certain amount of algebraic (and topological) background is expected. This being said, perseverant readers will be rewarded by being exposed to some beautiful and nontrivial concepts and results, including Élie Cartan and Raoul Bott "8-periodicity theorem." Going back to rotations as transformations induced by group actions, recall that if V is a vector space, a linear action (on the left) of a group G on V is a map α : G× V → V satisfying the following conditions, where, for simplicity of notation, we denote α(g, v) by g • v:

Exterior Tensor Powers and Exterior Algebras

Geometry and computing, 2020

In this chapter we consider alternating (also called skew-symmetric) multilinear maps and exterio... more In this chapter we consider alternating (also called skew-symmetric) multilinear maps and exterior tensor powers (also called alternating tensor powers), denoted \(\bigwedge ^n(E)\). In many respects alternating multilinear maps and exterior tensor powers can be treated much like symmetric tensor powers, except that sgn(σ) needs to be inserted in front of the formulae valid for symmetric powers.

Groups and Group Actions

Geometry and computing, 2020

This chapter provides the foundations for deriving a class of manifolds known as homogeneous spac... more This chapter provides the foundations for deriving a class of manifolds known as homogeneous spaces. It begins with a short review of group theory, introduces the concept of a group acting on a set, and defines the Grassmannians and Stiefel manifolds as homogenous manifolds arising from group actions of Lie groups. The last section provides an overview of topological groups, of which Lie groups are a special example, and contains more advanced material that may be skipped upon first reading.

A Review of Point Set Topology

Geometry and computing, 2020

This chapter contains a review of the topological concepts necessary for studying differential ge... more This chapter contains a review of the topological concepts necessary for studying differential geometry and includes the following material: 1. The definition of a topological space in terms of open sets; 2. The definition of a basis for a topology; 3. The definition of the subspace topology; 4. The definition of the product topology; 5. The definition of continuity and notion of a homeomorphism; 6. The definition of a limit of a sequence; 7. The definition of connectivity and path-wise connectivity; 8. The definition of compactness; 9. The definition of the quotient topology.

Distributions and the Frobenius Theorem

Geometry and computing, 2020

Given any smooth manifold M (of dimension n), for any smooth vector field X on M, it is known tha... more Given any smooth manifold M (of dimension n), for any smooth vector field X on M, it is known that for every point p ∈ M, there is a unique maximal integral curve through p; see Warner or Gallier and Quaintance. Furthermore, any two distinct integral curves do not intersect each other, and the union of all the integral curves is M itself. A nonvanishing vector field X can be viewed as the smooth assignment of a one-dimensional vector space to every point of M, namely \(p \mapsto \mathbb {R} X_p \subseteq T_p M\), where \(\mathbb {R} X_p\) denotes the line spanned by Xp. Thus, it is natural to consider the more general situation where we fix some integer r, with 1 ≤ r ≤ n, and we have an assignment p↦D(p) ⊆ TpM, where D(p) is some r-dimensional subspace of TpM such that D(p) “varies smoothly” with p ∈ M. Is there a notion of integral manifold for such assignments? Do they always exist?

Integration on Manifolds

Geometry and computing, 2020

The purpose of this chapter is to generalize the theory of integration known for functions define... more The purpose of this chapter is to generalize the theory of integration known for functions defined on open subsets of \(\mathbb {R}^n\) to manifolds. As a first step, we explain how differential forms defined on an open subset of \(\mathbb {R}^n\) are integrated. Then, if M is a smooth manifold of dimension n, and if ω is an n-form on M (with compact support), the integral ∫Mω is defined by patching together the integrals defined on small-enough open subsets covering M using a partition of unity. If (U, φ) is a chart such that the support of ω is contained in U, then the pullback (φ−1)∗ω of ω is an n-form on \(\mathbb {R}^n\), so we know how to compute its integral ∫φ(U)(φ−1)∗ω. To ensure that these integrals have a consistent value on overlapping charts, we need for M to be orientable. Actually, there is a more general notion of integration on a manifold that uses densities instead differential forms, but we do not need such generality.

$Research paper thumbnail of The Derivative of $$\exp $$ and Dynkin’s Formula $$\circledast $$$

The Derivative of $$\exp $$ and Dynkin’s Formula $$\circledast $$

Geometry and computing, 2020

By Proposition 2.5, we know that if [X, Y ] = 0, then \(\exp (X + Y) = \exp (X)\exp (Y)\), whenev... more By Proposition 2.5, we know that if [X, Y ] = 0, then \(\exp (X + Y) = \exp (X)\exp (Y)\), whenever \(X,Y\in {\mathfrak {g}}\), but this is generally false if X and Y do not commute. For X and Y in a small enough open subset \(U \subseteq {\mathfrak {g}}\) containing 0, we know by Proposition 19.6 that \(\exp \) is a diffeomorphism from U to its image, so the function μ: U × U → U given by $$\displaystyle \mu (X, Y) = \log (\exp (X)\exp (Y)) $$ is well defined and it turns out that for U small enough, it is analytic. Thus, it is natural to seek a formula for the Taylor expansion of μ near the origin.

Connections on Manifolds

Geometry and computing, 2020

Given a manifold M, in general, for any two points p, q ∈ M, there is no “natural” isomorphism be... more

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry

Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, 2022

Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups

arXiv (Cornell University), 2013

One of the main goals of these notes is to explain how rotations in R n are induced by the action... more One of the main goals of these notes is to explain how rotations in R n are induced by the action of a certain group Spin(n) on R n , in a way that generalizes the action of the unit complex numbers U(1) on R 2 , and the action of the unit quaternions SU(2) on R 3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and R n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra Cl n , the Clifford algebra associated with R n .

Spectral Theory of Unsigned and Signed Graphs Applications to Graph Clustering: a Survey

arXiv (Cornell University), Jan 18, 2016