2020/2021

Research Seminar "Smooth Structures on Manifolds"

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 1, 2 module

Instructors: Alexander S. Tikhomirov

Language: English

ECTS credits: 6

Contact hours: 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