Fields institute exterior differential systems and lie. Between topology and quantum field theory a conference in celebration of dan freeds 60th birthday january 1418, 2019 university of texas at austin register at. I would like to thank marco gualtieri, joel kamnitzer, yael karshon. Generalized complex geometry marco gualtieri oxford university d. Marco gualtieri search results annals of mathematics. To marco gualtieri for a number of very stimulating courses and seminars over the last few years, from which i have learned immensely. Thesis abstract generalized complex geometry is a new kind of geometrical structure which contains complex and symplectic geometry as its extremal special cases. We state the result for r 5 without proof, although the method of computation is described in x9. Morse theory has provided the inspiration for exciting developments.
Combinatorial geometry with application to field theory by linfan. This is in contrast to micro theories, which focus in detail on more specific elements of the. The lojasiewicz inequality is a key estimate in the process of proving that the gradient. Classical morse theory centered around simple statements like morse inequalities, concerning just the betti number s. Critical point of smooth functions on a smooth manifold briefly, morse theory 1 is divided into two parts. Symposium on the geometry and topology of manifolds speaker. This thesis would not be possible with their guidance, patience, advice, encouragement and help over the years. For further information on morse theory, the following sources are extremely useful. Hence, under certain conditions, a chernsimons path integral in three dimensions is equivalent to an n4 path integral in four dimensions. Speakers a celebration of raoul botts legacy in mathematics. Wood, banach algebras and bott periodicity, topology, 4 196566, pp. A function f is a morse function if all of its critical points are nondegenerate, where a nondegenerate critical points xis one where h xfis nondegenerate. Wells differential analysis on complex manifolds isbn 0387904190. These phenomena have an analog for chernsimons gauge theory in three dimensions.
A d ddimensional sigmamodel is a quantum field theory that is induced from certain differential geometric and differential cohomological data, to be thought of as encoding the background geometry on which quantum objects of dimension d d propagate the operation of tduality is a map that interchanges pairs of such geometric data for 2dimensional conformal field theory sigmamodels. This theorem provides a natural complement to taubes fredholm theory for general endperiodic. Panel session raoul bott as teacher, mentor, and colleague with nancy hingston chair, paul baum, jim stasheff and loring tu. We also define and study generalized complex branes, which interpolate between flat bundles on lagrangian submanifolds and holomorphic bundles on complex submanifolds. On the other hand, the work of kitchloo and cohen build on morse theory in the context of quantum topology, and refine the use of morse theory in low dimensional topology. Lectures on morse homology pennsylvania state university. As we will see in chapter 4, however, most smooth functions are morse. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a b. University of toronto title and abstract tba march 31, 2015 3 9. A short introduction to morse theory alessandro fasse email. An important technique in the study of symplectic and hyperk. The aim of this focused research workshop is to bring together mathematicians working in two different, but related, fields.
For r 3 and r 4 with odd degree, we use morse theory to calculate the betti numbers, verifying conjectural betti numbers due to chuang, diaconescu, and pan in 4, which was adapted to genus 0 by mozgovoy in 18. Songhao li doctor of philosophy graduate department of mathematics university of toronto. Gabor pete, section 2 of morse theory, lecture notes 19992001 pdf. Marco gualtieri toronto, spiro karigiannis waterloo. Existence of morse functions surgery, handle attachment, and cobordisms morse inequalities bott morse functions morse floer homology morse smale dynamics wittens approach to morse theory morse theory on the loop space of a riemannian manifold floer homology and pseudoholomorphic curves the. We apply the theory of differential spaces to subcartesian spaces equipped with families.
Exterior differential systems and cartans equivalence method on the one hand and lie theory especially lie pseudogroups and lie groupoidsalgebroids on the other. On the other hand, the work of kitchloo and cohen build on morse theory in the context of quantum topology, and refine the. Recall from linalg that for any nondegenerate symmetric binlinear form b. Volume 7, number 2, september 1982 lectures on morse theory, old and new by raoul bott1 morse theory is a beautiful and natural extension of the minimum principle for a continuous function on a compact space. Morse boundaries boundaries play an important role in the study of hyperbolic spaces and hyperbolic groups. Pims symposium on the geometry and topology of manifolds program june 29 july 9, 2015 the university of british columbia earth sciences building esb 2207 main mall, vancouver organizers. Lie theory and mathematical physics in higher structure. Morse, the calculus of variations in the large, american mathematical society, new york, 1934. Morse theory has previously been adapted to discrete and applied settings. Geometry of cohiggs bundles steven rayan, balliol college submitted for d. Generalized complex geometry annals of mathematics. Gualtieri has informed that he has presented this idea as early as 2009 17 see also section 4. We develop a theory of reduction for generalized kahler and hyperkahler structures which uses the generalized riemannian metric in an essential way, and which is not described with reference solely to a single generalized.
I would also like to thank two other professors here at uoft, marco gualtieri and ragnar olafbuchweitz, both of whom i have learned a lot of mathematics from over the past year. Symposium on the geometry and topology of manifolds. They arise as generalised holomorphic bundles on ordinary complex manifolds. Normsquare localization for hamiltonian lgspaces yiannis loizides doctor of philosophy. Tromba, morse lemma in banach spaces, the euler characteristic of vector fields on banach manifolds and a globalization of lerayschauder degree, to appear.
Morse theory with j j2 was developed by kirwan kir84 by direct analogy with morse theory with the yangmills functional in two dimensions ab82. This book describes morse theory for finite dimensions. Tromba, morse lemma in banach spaces, eulerpoincare index theory on banach manifolds, ann. We obtain equivalent assertions among the integrability conditions of. Aaron fenyes university of toronto marco gualtieri university of toronto nathan harman mit. Bredon, topologyand geometry, springerverlag, new york. The more complicated the topology of m, the greater the number of. The authors would like to thank marco gualtieri, yael karshon, eckhard meinrenken, paul selick, and ping xu for useful discussions. I would like to thank the former for sacri cing lots of personal time to guide me towards interesting new research direc. R, there exists a basis such that, in this basis, bis diagonal with 1. Copenhagen university week 1 automorphisms of manifolds and graph homology the lecture will describe the rational cohomology of the classifying space of the groups of homotopy automorphisms and block.
Milnors morse theory, isbn 0691080089, wells differential analysis on complex manifolds, isbn 0387904190. Marco gualtieri, course page, lecture notes and links. Geometry and topology of lie transformation groups tom. Generalized complex geometry, as developed by hitchin. In chapter 2, we study morse theory with moment maps. In these lectures i would like to discuss it in the context of two problems in analysis which have selfevident. Generalized complex geometry by marco gualtieri abstract generalized complex geometry encompasses complex and symplectic geometry as its extremal special cases. In this short introduction we will follow the excellent book of yukio matsumoto1. July 2628, 2017, 2449 simons building building 2 mit, cambridge, ma 029, usa speakers. I will define the notion of dbrane on a holomorphic poisson manifold and give some methods as well as consequences of their construction. Forman 2002 studies discrete morse functions that assign a single value to eac h cell in a complex. In particular, she proved that the kirwan map, a natural map from the kequivariant cohomology of m to the ordinary cohomology of the reduced space mk, is surjective. Here, on one hand, the work of hingston and goresky recast string topology in terms of morse theory as applied to loop spaces by bott in the 1950s.
Dg nov 2007 the equivariant cohomology theory of twisted generalized complex manifolds yi lin abstract. Lees introduction to smooth manifolds isbn 0387954481. Twisted stable homotopy theory by christopher lee douglas bachelor of science, massachusetts institute of technology, june 1999 master of science, oxford university, june 2001. Boris botvinnik oregon, ian hambleton mcmaster, alexei kovalev cambridge and. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. Warners foundations of differentiable manifolds and lie groups isbn 0387908943. Researchers from groups underrepresented in mathematics.
Here is a more detailed presentation of the contents. Moduli space techniques in algebraic geometry and symplectic. The main technical tool we employ is morse theory with moment maps. Morse theory deals with both finitedimensional and infinitedimensional spaces. New york journal of mathematics cohiggs bundles on p.
In particular, it is believed that morse theory on infinitedimensional spaces will become more and more important in the future as mathematics advances. I would like the assignments to be handed back as a pdf document typeset by latex. A marco gualtieri marco gualtieri courses 1 mat0f geometry and topology core course 3. A new look at the path integral of quantum mechanics inspire. The stokes groupoids marco gualtieri university of toronto fields institute workshop on eds and lie theory, december 11, 20 based onarxiv. Thus in the hypothesis of the previous theorem, we could have said that fis a c. Cmi supported conferences clay mathematics institute. Bott, on the periodicity theorem for complex vector bundles, acts, mathematica, vol. In academics, macro theories attempt to explain the entirety of a subject in general or broad terms.
Morse theory, loop spaces, and manifolds with corners 15. In another context, a morse function can also mean an anharmonic oscillator see morse potential in mathematics, specifically in differential topology, morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. The second part consists of applications of morse theory over the reals, while the last part describes the basics and some applications of complex morse theory, a. Following ideas of atiyah and bott, kirwan studied symplectic reductions mk using kequivariant morse theory on m. The proof is based on a version of infinite dimensional morse theory which is due to floer. Threlfall, varlationsrechnung im grossen, published in the united states by chelsea, new york, 1951. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of morse functions. We introduce a lie algebroid which allows us to view such structures as symplectic forms. Presentation greatly improved, one subsection added, errors corrected, references added. The kinds of theorems we would like to prove in morse theory will typically only apply to morse functions. Special thanks go to marco gualtieri for ever engaging causeries too often by email and for making me laugh more and with less restraint than anyone. Morse theory is the method of studying the topology of a smooth manifold m m by the study of morse functions m. Yukio matsumoto gakushuin university tokyo may 2007 proposed schedule friday 11.
In the talks given by these speakers, as well as those of. Poster for connections in geometry and physics 2009. I would also like to thank two other professors here at uoft, marco gualtieri and ragnar olaf. The local part is related to the idea of a critical point of a smooth function, the hessian of a function at its critical point, the morse index of a critical point, etc. Existence of perfect morse functions of spaces with semifree circle action kogan, mikhail, journal of symplectic geometry, 2003 a new morse theory and strong resonance problems li, shujie, topological methods in nonlinear analysis, 2003. Generalized complex geometry, as developed by hitchin, contains complex and symplectic geometry as its extremal special cases. In chapter 1 we introduce the basic notions of the theory and we describe the main properties of. Lie groupoids and generalized almost subtangent manifolds. Introduction to morse theory let mand nbe smooth manifolds, and let f. The dynamics of a nice enough f 2 c1m reconstructs msmoothly.
Geometry and topology of lie transformation groups tom baird. It is a pleasure to thank jemima merisca and ida bulat, who welcomed me warmly when i arrived at the department, and provided help and encouragement over the years. Morse theory, loop spaces, and manifolds with corners. This panel session will also feature the film by vanessa scott about her grandfather, raoul bott, a peek into the book. Morse theory is the method of studying the topology of a smooth manifold m by the study of. The topology and geometry of hyperk\ ahler quotients.
658 1303 945 1501 1149 366 187 441 1413 1261 336 941 209 953 1515 1064 439 1232 816 1208 408 1427 517 1340 599 279 1456 350 700 132 427 283 495 1056