An Introduction to Gauge Theory and Its Applications
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Geometric Analysis of Shapes and Its Application to Medical Image Analysis
Geometric Analysis of Shapes and Its application to Medical Image Analysis by Anirban Mukhopadhyay (Under the DIRECTION of Suchendra M. Bhandarkar) Abstract Geometric analysis of shapes plays an important role in the way the visual world is per- ceived by modern computers. To this end, low-level geometric features provide most obvious and important cues towards understanding the visual scene. A novel intrinsic geometric sur- face descriptor, termed as the Geodesic Field Estimate (GFE) is proposed. Also proposed is a parallel algorithm, well suited for implementation on Graphics Processing Units, for efficient computation of the shortest geodesic paths. Another low level geometric descriptor, termed as the Biharmonic Density Estimate, is proposed to provide an intrinsic geometric scale space signature for multiscale surface feature-based representation of deformable 3D shapes. The computer vision and graphics communities rely on mid-level geometric understanding as well to analyze a scene. Symmetry detection and partial shape matching play an important role as mid-level cues. A comprehensive framework for detection and characterization of partial intrinsic symmetry over 3D shapes is proposed. To identify prominent overlapping symmetric regions, the proposed framework is decoupled into Correspondence Space Voting followed by Transformation Space Mapping procedure. Moreover, a novel multi-criteria optimization framework for matching of partially visible shapes in multiple images using joint geometric embedding is also proposed. The ultimate goal of geometric shape analysis is to resolve high level applications of modern world. This dissertation has focused on three different application scenarios. In the first scenario, a novel approach for the analysis of the non-rigid Left Ventricular (LV) endocardial surface from Multi-Detector CT images, using a generalized isometry-invariant Bag-of-Features (BoF) descriptor, is proposed and implemented. -
Introduction to Gauge Theory Arxiv:1910.10436V1 [Math.DG] 23
Introduction to Gauge Theory Andriy Haydys 23rd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles . .4 2.1.1 Basic notions . .4 2.1.2 Operations on vector bundles . .5 2.1.3 Sections . .6 2.1.4 Covariant derivatives . .6 2.1.5 The curvature . .8 2.1.6 The gauge group . 10 2.2 Principal bundles . 11 2.2.1 The frame bundle and the structure group . 11 2.2.2 The associated vector bundle . 14 2.2.3 Connections on principal bundles . 16 2.2.4 The curvature of a connection on a principal bundle . 19 arXiv:1910.10436v1 [math.DG] 23 Oct 2019 2.2.5 The gauge group . 21 2.3 The Levi–Civita connection . 22 2.4 Classification of U(1) and SU(2) bundles . 23 2.4.1 Complex line bundles . 24 2.4.2 Quaternionic line bundles . 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory . 26 3.1.1 The Chern classes . 28 3.2 The Chern–Simons functional . 30 3.3 The modui space of flat connections . 32 3.3.1 Parallel transport and holonomy . -
HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
2. Chern Connections and Chern Curvatures1
1 2. Chern connections and Chern curvatures1 Let V be a complex vector space with dimC V = n. A hermitian metric h on V is h : V £ V ¡¡! C such that h(av; bu) = abh(v; u) h(a1v1 + a2v2; u) = a1h(v1; u) + a2h(v2; u) h(v; u) = h(u; v) h(u; u) > 0; u 6= 0 where v; v1; v2; u 2 V and a; b; a1; a2 2 C. If we ¯x a basis feig of V , and set hij = h(ei; ej) then ¤ ¤ ¤ ¤ h = hijei ej 2 V V ¤ ¤ ¤ ¤ where ei 2 V is the dual of ei and ei 2 V is the conjugate dual of ei, i.e. X ¤ ei ( ajej) = ai It is obvious that (hij) is a hermitian positive matrix. De¯nition 0.1. A complex vector bundle E is said to be hermitian if there is a positive de¯nite hermitian tensor h on E. r Let ' : EjU ¡¡! U £ C be a trivilization and e = (e1; ¢ ¢ ¢ ; er) be the corresponding frame. The r hermitian metric h is represented by a positive hermitian matrix (hij) 2 ¡(; EndC ) such that hei(x); ej(x)i = hij(x); x 2 U Then hermitian metric on the chart (U; ') could be written as X ¤ ¤ h = hijei ej For example, there are two charts (U; ') and (V; Ã). We set g = à ± '¡1 :(U \ V ) £ Cr ¡¡! (U \ V ) £ Cr and g is represented by matrix (gij). On U \ V , we have X X X ¡1 ¡1 ¡1 ¡1 ¡1 ei(x) = ' (x; "i) = à ± à ± ' (x; "i) = à (x; gij"j) = gijà (x; "j) = gije~j(x) j j For the metric X ~ hij = hei(x); ej(x)i = hgike~k(x); gjle~l(x)i = gikhklgjl k;l that is h = g ¢ h~ ¢ g¤ 12008.04.30 If there are some errors, please contact to: [email protected] 2 Example 0.2 (Fubini-Study metric on holomorphic tangent bundle T 1;0Pn). -
An Introduction to Complex Algebraic Geometry with Emphasis on The
AN INTRODUCTION TO COMPLEX ALGEBRAIC GEOMETRY WITH EMPHASIS ON THE THEORY OF SURFACES By Chris Peters Mathematisch Instituut der Rijksuniversiteit Leiden and Institut Fourier Grenoble i Preface These notes are based on courses given in the fall of 1992 at the University of Leiden and in the spring of 1993 at the University of Grenoble. These courses were meant to elucidate the Mori point of view on classification theory of algebraic surfaces as briefly alluded to in [P]. The material presented here consists of a more or less self-contained advanced course in complex algebraic geometry presupposing only some familiarity with the theory of algebraic curves or Riemann surfaces. But the goal, as in the lectures, is to understand the Enriques classification of surfaces from the point of view of Mori-theory. In my opininion any serious student in algebraic geometry should be acquainted as soon as possible with the yoga of coherent sheaves and so, after recalling the basic concepts in algebraic geometry, I have treated sheaves and their cohomology theory. This part culminated in Serre’s theorems about coherent sheaves on projective space. Having mastered these tools, the student can really start with surface theory, in particular with intersection theory of divisors on surfaces. The treatment given is algebraic, but the relation with the topological intersection theory is commented on briefly. A fuller discussion can be found in Appendix 2. Intersection theory then is applied immediately to rational surfaces. A basic tool from the modern point of view is Mori’s rationality theorem. The treatment for surfaces is elementary and I borrowed it from [Wi]. -
Karen Keskulla Uhlenbeck
2019 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck University of Texas at Austin “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Karen Keskulla Uhlenbeck is a founder of modern by earlier work of Morse, guarantees existence of Geometric Analysis. Her perspective has permeated minimisers of geometric functionals and is successful the field and led to some of the most dramatic in the case of 1-dimensional domains, such as advances in mathematics in the last 40 years. closed geodesics. Geometric analysis is a field of mathematics where Uhlenbeck realised that the condition of Palais— techniques of analysis and differential equations are Smale fails in the case of surfaces due to topological interwoven with the study of geometrical and reasons. The papers of Uhlenbeck, co-authored with topological problems. Specifically, one studies Sacks, on the energy functional for maps of surfaces objects such as curves, surfaces, connections and into a Riemannian manifold, have been extremely fields which are critical points of functionals influential and describe in detail what happens when representing geometric quantities such as energy the Palais-Smale condition is violated. A minimising and volume. For example, minimal surfaces are sequence of mappings converges outside a finite set critical points of the area and harmonic maps are of singular points and by using rescaling arguments, critical points of the Dirichlet energy. Uhlenbeck’s they describe the behaviour near the singularities major contributions include foundational results on as bubbles or instantons, which are the standard minimal surfaces and harmonic maps, Yang-Mills solutions of the minimising map from the 2-sphere to theory, and integrable systems. -
Characteristic Classes and K-Theory Oscar Randal-Williams
Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. -
A Geodesic Connection in Fréchet Geometry
A geodesic connection in Fr´echet Geometry Ali Suri Abstract. In this paper first we propose a formula to lift a connection on M to its higher order tangent bundles T rM, r 2 N. More precisely, for a given connection r on T rM, r 2 N [ f0g, we construct the connection rc on T r+1M. Setting rci = rci−1 c, we show that rc1 = lim rci exists − and it is a connection on the Fr´echet manifold T 1M = lim T iM and the − geodesics with respect to rc1 exist. In the next step, we will consider a Riemannian manifold (M; g) with its Levi-Civita connection r. Under suitable conditions this procedure i gives a sequence of Riemannian manifolds f(T M, gi)gi2N equipped with ci c1 a sequence of Riemannian connections fr gi2N. Then we show that r produces the curves which are the (local) length minimizer of T 1M. M.S.C. 2010: 58A05, 58B20. Key words: Complete lift; spray; geodesic; Fr´echet manifolds; Banach manifold; connection. 1 Introduction In the first section we remind the bijective correspondence between linear connections and homogeneous sprays. Then using the results of [6] for complete lift of sprays, we propose a formula to lift a connection on M to its higher order tangent bundles T rM, r 2 N. More precisely, for a given connection r on T rM, r 2 N [ f0g, we construct its associated spray S and then we lift it to a homogeneous spray Sc on T r+1M [6]. Then, using the bijective correspondence between connections and sprays, we derive the connection rc on T r+1M from Sc. -
Circles in a Complex Projective Space
Adachi, T., Maeda, S. and Udagawa, S. Osaka J. Math. 32 (1995), 709-719 CIRCLES IN A COMPLEX PROJECTIVE SPACE TOSHIAKI ADACHI, SADAHIRO MAEDA AND SEIICHI UDAGAWA (Received February 15, 1994) 0. Introduction The study of circles is one of the interesting objects in differential geometry. A curve γ(s) on a Riemannian manifold M parametrized by its arc length 5 is called a circle, if there exists a field of unit vectors Ys along the curve which satisfies, together with the unit tangent vectors Xs — Ϋ(s)9 the differential equations : FSXS = kYs and FsYs= — kXs, where k is a positive constant, which is called the curvature of the circle γ(s) and Vs denotes the covariant differentiation along γ(s) with respect to the Aiemannian connection V of M. For given a point lEJIί, orthonormal pair of vectors u, v^ TXM and for any given positive constant k, we have a unique circle γ(s) such that γ(0)=x, γ(0) = u and (Psγ(s))s=o=kv. It is known that in a complete Riemannian manifold every circle can be defined for -oo<5<oo (Cf. [6]). The study of global behaviours of circles is very interesting. However there are few results in this direction except for the global existence theorem. In general, a circle in a Riemannian manifold is not closed. Here we call a circle γ(s) closed if = there exists So with 7(so) = /(0), XSo Xo and YSo— Yo. Of course, any circles in Euclidean m-space Em are closed. And also any circles in Euclidean m-sphere Sm(c) are closed. -
Math 600 Day 6: Abstract Smooth Manifolds
Math 600 Day 6: Abstract Smooth Manifolds Ryan Blair University of Pennsylvania Tuesday September 28, 2010 Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 1 / 21 Outline 1 Transition to abstract smooth manifolds Partitions of unity on differentiable manifolds. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 2 / 21 A Word About Last Time Theorem (Sard’s Theorem) The set of critical values of a smooth map always has measure zero in the receiving space. Theorem Let A ⊂ Rn be open and let f : A → Rp be a smooth function whose derivative f ′(x) has maximal rank p whenever f (x) = 0. Then f −1(0) is a (n − p)-dimensional manifold in Rn. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 3 / 21 Transition to abstract smooth manifolds Transition to abstract smooth manifolds Up to this point, we have viewed smooth manifolds as subsets of Euclidean spaces, and in that setting have defined: 1 coordinate systems 2 manifolds-with-boundary 3 tangent spaces 4 differentiable maps and stated the Chain Rule and Inverse Function Theorem. Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 4 / 21 Transition to abstract smooth manifolds Now we want to define differentiable (= smooth) manifolds in an abstract setting, without assuming they are subsets of some Euclidean space. Many differentiable manifolds come to our attention this way. Here are just a few examples: 1 tangent and cotangent bundles over smooth manifolds 2 Grassmann manifolds 3 manifolds built by ”surgery” from other manifolds Ryan Blair (U Penn) Math 600 Day 6: Abstract Smooth Manifolds Tuesday September 28, 2010 5 / 21 Transition to abstract smooth manifolds Our starting point is the definition of a topological manifold of dimension n as a topological space Mn in which each point has an open neighborhood homeomorphic to Rn (locally Euclidean property), and which, in addition, is Hausdorff and second countable. -
Twistor Theory at Fifty: from Rspa.Royalsocietypublishing.Org Contour Integrals to Twistor Strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J
Downloaded from http://rspa.royalsocietypublishing.org/ on November 10, 2017 Twistor theory at fifty: from rspa.royalsocietypublishing.org contour integrals to twistor strings Michael Atiyah1,2, Maciej Dunajski3 and Lionel Review J. Mason4 Cite this article: Atiyah M, Dunajski M, Mason LJ. 2017 Twistor theory at fifty: from 1School of Mathematics, University of Edinburgh, King’s Buildings, contour integrals to twistor strings. Proc. R. Edinburgh EH9 3JZ, UK Soc. A 473: 20170530. 2Trinity College Cambridge, University of Cambridge, Cambridge http://dx.doi.org/10.1098/rspa.2017.0530 CB21TQ,UK 3Department of Applied Mathematics and Theoretical Physics, Received: 1 August 2017 University of Cambridge, Cambridge CB3 0WA, UK Accepted: 8 September 2017 4The Mathematical Institute, Andrew Wiles Building, University of Oxford, Oxford OX2 6GG, UK Subject Areas: MD, 0000-0002-6477-8319 mathematical physics, high-energy physics, geometry We review aspects of twistor theory, its aims and achievements spanning the last five decades. In Keywords: the twistor approach, space–time is secondary twistor theory, instantons, self-duality, with events being derived objects that correspond to integrable systems, twistor strings compact holomorphic curves in a complex threefold— the twistor space. After giving an elementary construction of this space, we demonstrate how Author for correspondence: solutions to linear and nonlinear equations of Maciej Dunajski mathematical physics—anti-self-duality equations e-mail: [email protected] on Yang–Mills or conformal curvature—can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang– Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations. -
Iasinstitute for Advanced Study
IAInsti tSute for Advanced Study Faculty and Members 2012–2013 Contents Mission and History . 2 School of Historical Studies . 4 School of Mathematics . 21 School of Natural Sciences . 45 School of Social Science . 62 Program in Interdisciplinary Studies . 72 Director’s Visitors . 74 Artist-in-Residence Program . 75 Trustees and Officers of the Board and of the Corporation . 76 Administration . 78 Past Directors and Faculty . 80 Inde x . 81 Information contained herein is current as of September 24, 2012. Mission and History The Institute for Advanced Study is one of the world’s leading centers for theoretical research and intellectual inquiry. The Institute exists to encourage and support fundamental research in the sciences and human - ities—the original, often speculative thinking that produces advances in knowledge that change the way we understand the world. It provides for the mentoring of scholars by Faculty, and it offers all who work there the freedom to undertake research that will make significant contributions in any of the broad range of fields in the sciences and humanities studied at the Institute. Y R Founded in 1930 by Louis Bamberger and his sister Caroline Bamberger O Fuld, the Institute was established through the vision of founding T S Director Abraham Flexner. Past Faculty have included Albert Einstein, I H who arrived in 1933 and remained at the Institute until his death in 1955, and other distinguished scientists and scholars such as Kurt Gödel, George F. D N Kennan, Erwin Panofsky, Homer A. Thompson, John von Neumann, and A Hermann Weyl. N O Abraham Flexner was succeeded as Director in 1939 by Frank Aydelotte, I S followed by J.