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Обычная версия сайта
2019/2020

Научно-исследовательский семинар "Гладкие структуры на многообразиях"

Статус: Дисциплина общефакультетского пула
Когда читается: 1, 2 модуль
Язык: английский
Кредиты: 6

Course Syllabus

Abstract

The smooth topology of four-dimensional manifolds is unique in the sense that it provides phenomena having no analogues neither in smaller, nor in higher dimensions. For instance, on many four-manifolds there were found an infinite, and on ℝ 4 even uncountable number of smooth structures. These phenomena were invented in 80-90-ies in the works of S.Donaldson, C. Taubes and many other geometers in connection with the application of methods of modern differential geometry to four-dimensional topology. This is a new area of mathematics lying at the junction of global analysis and gauge theory which is related to the Yang – Mills equations. Their solutions — the so-called instantons — lead to new invariants of smoothstructures on four- manifolds. In this course we give an introduction to the invariants of smooth structures related to instantons and show how they work in four-dimensional topology.
Learning Objectives

Learning Objectives

  • Students will gain understanding of the gauge-theoretic instanton moduli spaces of 4-manifolds and of their application to constructing a new type of invariants of smooth structure on simply-connected compact toplogical manifolds. These are Donaldson polynomial invariants which are then used to finding a large number of different smooth structures on a given topological manifold.
Expected Learning Outcomes

Expected Learning Outcomes

  • Students should be able to demonstrate understanding by giving a thirty minute presentation on one of the following topics: topological classification of simply connected compact manifolds by their intersection form, classification of principal SU(2)- and SO(3)-bundles over smooth compact manifolds via characteristic classes, general construction of instanton moduli spaces over smooth 4-manifolds
  • Also students should perform explicit description of instanton moduli spaces of charge one on the 4-sphere, compactness results of Uhlenbeck on instanton moduli, construction of Donaldson polynomial invariants of smooth 4-manifolds, examples of their application to nondecomposability of smooth manifolds and to distinguishing smooth structures on a given manifod, Kobayashi-Hitchin correspondence between instanton moduli spaces and algebraic-geometric moduli spaces for complex algebraic surfaces.
Course Contents

Course Contents

  • Smooth structures on topological manifolds
  • Vector and principal bundles. Connections
  • Curvature and characteristic classes
  • The space of connections
  • The Yang – Mills equations and the moduli space
  • Compactness and gluing theorems
  • Definite intersection forms.
  • The Donaldson polynomial invariants
  • The connected sum theorem
  • The Kobayashi – Hitchin correspondence
  • Smooth structures on complex algebraic surfaces
Assessment Elements

Assessment Elements

  • non-blocking Homemade problem sheets
  • non-blocking Midterm grade
  • non-blocking Final exam
    Written exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    0.5 * Final exam + 0.3 * Homemade problem sheets + 0.2 * Midterm grade
Bibliography

Bibliography

Recommended Core Bibliography

  • Akbulut, S. (2016). 4-Manifolds (Vol. First edition). Oxford: OUP Oxford. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1506249

Recommended Additional Bibliography

  • Robert E. Gompf, & András I. Stipsicz. (2015). 4-Manifolds and Kirby Calculus. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971013
  • Robert Friedman, & John W. Morgan. (2017). Gauge Theory and the Topology of Four-Manifolds. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1549640