Subjects

Your browser does not support JavaScript, or its support is disabled. Some features may not be available.

Introduction to Differential Topology - NMAG452

Title:	Úvod do diferenciální topologie
Guaranteed by:	Mathematical Institute of Charles University (32-MUUK)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2015 to 2015
Semester:	summer
E-Credits:	3
Hours per week, examination:	summer s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time

Guarantor:	prof. RNDr. Vladimír Souček, DrSc. Mgr. Martin Doubek, Ph.D.
Teacher(s):	Mgr. Martin Doubek, Ph.D.
Class:	M Mgr. MSTR M Mgr. MSTR > Povinně volitelné
Classification:	Mathematics > Geometry, Topology and Category
Incompatibility :	NMAT009
Interchangeability :	NMAT009

Opinion survey results Examination dates Schedule Noticeboard

Annotation -

Differential topology studies the relationship between analytic concepts (critical points of functions or functionals, solution spaces of systems of PDEs, zeroes of vector fields, diffeomorphism groups, etc) and topological concepts (Euler characteristics, CW structure, homotopy type, intersection forms, etc). We will focus on basic aspects of Sard's Theorem and Morse theory and their applications.

Last update: T_MUUK (02.03.2017)

Literature -

Lee, J. : Introduction to Smooth Manifolds, Springer 2012

Hirsch, M. W. : Differential Topology, Springer 1997

Kock, J. : Frobenius Algebras and 2D Topological Quantum Field Theories, Cambridge 2003

Last update: Somberg Petr, doc. RNDr., Ph.D. (02.03.2017)

Syllabus -

(Nondegenerate) critical point, critical value and regular value of a smooth map

Flow, generator of flow, existence of a flow with prescribed generator

First Morse theorem

Reeb theorem

Morse lemma

Second Morse theorem

CW structure on smooth compact manifolds

Smooth knot, ambient isotopy

Milnor theorem on detecting unknot via curvature

Morse inequalities

Index of a vector field

Hopf theorem on calculating Euler characteristics using a vector field

Smooth immersion and embedding, submanifold

Weak Whitney theorem

Existece of Morse functions

(Un)oriented cobordism

Monoidal category and functor

Topological quantum field theory (TQFT)

Frobenius algebra

Characterization of 2-dimensional TQFT's via Frobenius algebras

test

Last update: Somberg Petr, doc. RNDr., Ph.D. (02.03.2017)