2020/2021

Научно-исследовательский семинар "Гладкие структуры на многообразиях"

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»

Лучший по критерию «Новизна полученных знаний»

Статус: Дисциплина общефакультетского пула

Кто читает: Факультет математики

Где читается: Факультет математики

Когда читается: 1, 2 модуль

Преподаватели: Тихомиров Александр Сергеевич

Язык: английский

Кредиты: 6

Контактные часы: 60

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

• 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

• 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
• 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

• 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

• Solution of problems from home task sheets.
  Written solutions
• Final exam
  Oral exam
• Solution of problems from midterm home task sheet
  Written solutions

Interim Assessment

• Interim assessment (2 module)
  0.5 * Final exam + 0.3 * Solution of problems from home task sheets. + 0.2 * Solution of problems from midterm home task sheet

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