Master Class 2016/2017 in Geometry, Topology and Physics

 

Description

Geometry and physics have been interconnected since ancient times, providing inspiration and intuition, as well language for each other. Geometric ideas lay in the foundations of electromagnetism, special and general relativity, while symplectic geometry is the modern language of mechanics, which, in turn, provides it with methods and motivation.

The 2016-2017 master class covers some of the most important and actively developed subjects of the research area in-between geometry, topology and physics providing an entry point into the forefront research for students starting to work in this field.

The program is aimed at Master students and beginning PhD students. The participants will enroll in a one-year program at the University of Geneva starting in September 2016, providing 60 ECTS credits. Participants will be offered the possibility of obtaining a master degree from the university of Geneva by completing a Master thesis for 30 additional ECTS credits.

Tuition fee for regular (master) students at the University of Geneva is 500 Swiss Francs per semester. There is no tuition fee for exchange students.

A number of fellowships covering accommodation and local expenses are available.

download poster: pdf, 751 KB

Registration

For full consideration, candidates should apply before December 31, 2015 by filling out the following form:

REGISTRATION FORM

If you have any questions concerning the program, please write to: swissmap email.

Program

First semester will start wil a 1-2 week introductory workshop designed to bring all students up to speed on the topics needed in the courses.

Schedule of the courses: fall semester (pdf), spring semster (pdf)

Video lectures

First Semester Second Semester

Symplectic geometry of moment maps (A. Alekseev)

Equivariant symplectic geometry or moment map theory is the field built around several extraordinary results: the Marsden-Weinstein symplectic reduction theory, moment map convexity theorems, Duistermaat-Heckman localization for oscillating integrals, and the "quantization commutes with reduction" principle. The goal of the course is to give an introduction into the field and to discuss (some of) these important results.

Program of the course:

  • Basics of symplectic geometry: definition, examples
  • Recall: group actions of compact groups, normal forms on a neighborhood of an orbit
  • Recall: Morse theory, Morse inequalities, perfect Morse functions
  • Hamiltonian G-spaces: definition, examples
  • Symplectic reduction, examples. Symplectic cuts
  • Atiyah-Guillemin-Sternberg convexity theorem
  • Duistermaat-Heckman localization I: Berline-Vergne approach
  • Equivariant cohomology: Cartan model, examples
  • Localization II: Atiyah-Bott approach
  • Kirwan surjectivity theorem, intersection pairings on reduced spaces
  • G non-abelian: what changes?
  • Kähler manifolds, geometric quantization, Guillemin-Sternberg [Q,R]=0.

prerequisites: differential geometry

Poisson geometry and quantization (P. Ševera)

Poisson geometry, or the geometry of Poisson brackets, is a natural generalization of symplectic geometry. It originates from classical mechanics. A celebrate theorem of Kontsevich and Tamarkin states that any Poisson structure can be "quantized", i.e. turned to a non-commutative deformation of the algebra of functions on the manifold. The first aim of the course is to give an introduction to Poisson geometry and some of the most interesting examples of Poisson manifolds, in particular Poisson-Lie groups. We shall then study Drinfeld associators, which provide a link between braids and the problem of quantization of Poisson manifolds. While Drinfeld associators can be used to quantize all Poisson manifolds, we shall content ourselves with a simple method of quantization of the most interesting examples, in particular of Poisson-Lie groups.

Program of the course:

  • Poisson structures and symplectic leaves
  • Deformation quantization, Moyal product
  • The Poisson structure on the dual of a Lie algebra, moment maps
  • Poisson-Lie groups and Lie bialgebras
  • Moduli spaces of flat connections and their Poisson structures
  • Knizhnik-Zamolodchikov equation and Drinfeld associator
  • Braided monoidal categories and construction of non-commutative algebras 
  • Quantization of Lie bialgebras
  • Quantization of moduli spaces
  • Drinfeld associators and the Grothendieck-Teichmueller group

prerequisites: differential geometry, Lie theory

Topological aspects of algebraic geometry I (I. Itenberg)

Algebraic geometry is geometry of varieties defined by polynomial equations. Coefficients of these equations may be of different nature, classically the main examples are integer, real and complex coefficients. The fields of real and complex numbers come with a natural Euclidean topology. This makes real and complex algebraic varieties especially nice and accessible to techniques going beyond standard commutative algebra considerations.

We start the course by topological theory of algebraic curves in the real plane (this subject goes back to Harnack, Klein, and Hilbert and is sometimes referred as Hilbert's 16th problem). Then we look at the so-called amoebas of complex algebraic varieties as well as a certain limiting procedure known as the tropical limit. The limiting objects are themselves subject to a kind of algebraic geometry known as tropical geometry.

prerequisites: basic geometry and topology courses

Topological aspects of algebraic geometry II (G. Mikhalkin)

see description in first semester

Symmetries and moduli spaces I (S.Galkin)

The purpose of the course is to give an introduction to the construction and geometric properties of moduli spaces from the point of view of algebraic geometry. These spaces appear in physics as well, as ground states of various gauge theories. A central tool in the construction of moduli spaces is an appropriate notion of symmetries, represented by groups acting on spaces, and the introduction of these objects into algebraic geometry will be the central theme of the course.

Program of the course:

  • Projective and affine algebraic varieties, smooth points and singularities.
  • Algebraic groups and representations. Vector bundles.
  • First examples: projective spaces and grassmannians.
  • Toric varieties: Polytopes and fans.
  • Toric varieties: Chambers and quotient constructions.
  • Intersection theory and vector bundles.

Symmetries and moduli spaces II (A. Szenes)

continuation:

  • More quotients: grassmannians and flag varieties.
  • Configuration of points on the line: the Hilbert-Mumford criterion.
  • Vector bundles on Riemann surfaces: classification.
  • Quot schemes and the construction of the moduli spaces of vector bundles.
  • Intersection theory on quotients.

Introduction to quantum topology I (A. Virelizier)

Quantum Topology (QT) is a branch of mathematics that studies various applications of methods and principles of quantum theory in low-dimensional topology. Historically, QT came to existence in 80’s with the discovery by V. Jones of his famous polynomial invariant of knots. Among the important mathematical notions related to QT is that of a Topological Quantum Field Theory (TQFT), i.e., a quantum field theory that has topological invariants as its observables. Important examples of TQFTs are related with quantum Chern-Simons gauge field theory in 2+1 dimensions and the theory of quantum groups. The problem of understanding quantum Chern-Simons theory with non-compact gauge groups is of special importance and interest because of its connections to geometric approach of Thurston to topology of 3-manifolds, as well as 2+1-dimensional quantum gravity.

The course  will start with basics on knot diagrams and Reidemeister moves, the Jones Polynomial, quantum groups and their finite dimensional representations, R-matrices,   monoidal (fusion, modular,…) categories, Kirby calculus, and then continue with more advanced constructions of TQFTs, in particular, constructions related to Chern-Simons theories with non-compact gauge groups in the case of Teichmüller spaces of punctured surfaces.

prerequisites: basic courses in algebra, geometry, topology, and analysis.
 

Introduction to quantum topology II (R. Kashaev)

see description in first semester

Quantum mechanics for mathematicians (M. Mariño)

The aim of this course is to give a self-contained introduction to the classical and quantum mechanics to students in mathematics. The program includes:

  • Lagrangian mechanics
  • Hamiltonan mechanics
  • Foundations of quantum mechanics
  • Quantum mechanics in the phase space
  • Semiclassical methods

Field theory for mathematicians (A. Alekseev)

Quantum field theory is a source of inspiration for a number of important concepts in modern mathematics. This course is an introduction into the subject. We will cover Lagrangian field theory based on the calculus of variations, the axiomatic approach due to Atiyah-Segal and the basics of the Feynman graphs perturbative technique. 

Program of the course:

  • Lagrangian field theory: basics of the calculus of variations, examples: free and interacting scalar fields, gauge theories, Yang-Mills theory in 2 dimensions, Chern-Simons theory in 3 dimensions.
  • Atiyah-Segal axioms of Quantum Field Theory. Topological Quantum Field Theories (TQFT). Classification of 2-dimensional TQFTs.
  • Perturbation theory: finite dimensional Gaussian integrals, Feynman graphs, applications to QFT.

prerequisites: Lie theory