Starts 19 Jun 2017

Ends 7 Jul 2017

Central European Time

ICTP-EAUMP School on Modern Functional Analysis | (smr 3133)

Starts 19 Jun 2017

Ends 7 Jul 2017

Central European Time

Nairobi - Kenya

Strada Costiera, 11
I - 34151 Trieste (Italy)

The summer school is intended to provide a thorough introduction to various topics in functional analysis, from introductory to more advanced, and with a particular focus on functional analytic methods used in current research.

The two introductory courses in the first week cover some of the fundamentals of functional analysis and of operator theory, before two more advanced courses in the second week will introduce ideas which establish a link between the methods of classical functional analysis and the modern theory of partial differential equations.

Finally, the two courses in week three will expose participants to some more advanced topics which tie together the areas of functional analysis, differential equations and geometry. Each of the courses will consist of a series of lectures and accompanying classes, which are intended to give participants an opportunity to deepen their understanding of the material introduced in the lectures.

All of the lecture courses will be delivered by internationally recognised researchers in analysis.

In the final week of the summer school, participants will have a chance to work independently on a small topic of their choice.

**COURSES**

**Week 1, 19-23 June - Introductory courses**

**Course 1:** An introduction to function spaces

**Lecturer:** **Dr Lassi Paunonen** (Tampere, Finland)

**Description:** An overview of the basic theory of Banach and Hilbert spaces, with a focus on function spaces which play a part in modern applications of functional analysis

**Syllabus:** Overview of basic properties of Banach and Hilbert spaces (completeness, separability, reflexivity, etc), some fundamental theorems in functional analysis (e.g. the Riesz Representation theorem and the Hahn-Banach theorem), examples with relevance to modern applications of functional analysis (Sobolev spaces, Bergman spaces, Hardy spaces etc)

**Course 2: **An introduction to operator theory

**Lecturer: Prof. Charles Batty** (Oxford, UK)

**Description:** An overview of some of the central theorems on linear operators between normed vector spaces, touching on the theory of unbounded linear operators

**Syllabus:** Overview of the basic theory of bounded linear operators between Banach spaces, spectral theory, some fundamental results in classical operator theory (uniform boundedness theorem, closed graph theorem, open mapping theorem, inverse mapping theorem), introductory material on the theory of closed unbounded operators (as time permits)

**Week 2, 26-30 June - Advanced courses**

**Course 3: **Advanced operator theory

**Lecturer: Dr David Seifert **(Oxford, UK)

**Description:** An overview of some more advanced topics in operator theory, making the connection with modern PDE theory

**Syllabus:** Compactness in normed vector spaces, compact operators (characterisation, properties, spectral theory, examples), operators with closed range, the closed range theorem, Fredholm theory (Fredholm index, Fredholm alternative, perturbation results), existence and uniqueness for simple elliptic PDEs via the Fredholm alternative

**Course 4:** The theory of operator semigroups and their applications

**Lecturer: Dr Sachi Srivastava** (University of Delhi)

**Description:** An introduction to the theory of strongly continuous operator semigroups and their applications

**Syllabus:** C0-semigroups, infinitesimal generators, spectral theory of generators, Lumer-Phillips theorem, Hille-Yosida theorem, examples, abstract Cauchy problems, mild solutions, applications to simple PDEs (heat equation, wave equation) and quantum systems

*Week 3, 3-7 July - Modern applications of functional analysis*

**Course 5:** The Laplacian on a Riemannian manifold

**Lecturers: Prof Claudio Arezzo** and **Dr Luca Di Cerbo** (ICTP)

**Syllabus:** Basic theory of differential and Riemannian manifolds. Differential forms and integration. L2 functions and forms. Gradient, divergence and Laplacian on a Riemannian manifold. Towards Hodge Theorem, as much as time permits.

**Course 6:** Distribution solutions to ordinary differential equations

**Lecturer:** Dr Ismail Mirumbe (Makerere, Uganda)

**Local organizing committee**

• James Katende (University of Nairobi)

• Damian Maingi (University of Nairobi)

• Bernard Nzimbi (University of Nairobi)

• Jared Ongaro (University of Nairobi, Chair)

• Patrick Weke (University of Nairobi)

**Scientific advisory committee**

• Leif Abrahamson (University of Uppsala, Sweden)

• Rikard Bogvad (Stockholm University, Sweden)

• Damiani Maingi (University of Nairobi, Kenya)

• Fernando Rodrigues Villegas (ICTP)

• David Ssevviiri (Makerere University, Uganda)

• Balazs Szendroi (University of Oxford, UK, main International Organizer)

• Patrick Weke (University of Nairobi, Kenya)

**EAUMP Country Coordinators**

• Juma Kasozi (Makerere University, Uganda)

• Sylvester E. Rugeihyamu (Dar es Salaam University, Tanzania)

• Jared Ongaro (University of Nairobi, Kenya)

• Michael Gahirima (University of Rwanda)

• Mubanga Lombe (University of Zambia)

The two introductory courses in the first week cover some of the fundamentals of functional analysis and of operator theory, before two more advanced courses in the second week will introduce ideas which establish a link between the methods of classical functional analysis and the modern theory of partial differential equations.

Finally, the two courses in week three will expose participants to some more advanced topics which tie together the areas of functional analysis, differential equations and geometry. Each of the courses will consist of a series of lectures and accompanying classes, which are intended to give participants an opportunity to deepen their understanding of the material introduced in the lectures.

All of the lecture courses will be delivered by internationally recognised researchers in analysis.

In the final week of the summer school, participants will have a chance to work independently on a small topic of their choice.

• James Katende (University of Nairobi)

• Damian Maingi (University of Nairobi)

• Bernard Nzimbi (University of Nairobi)

• Jared Ongaro (University of Nairobi, Chair)

• Patrick Weke (University of Nairobi)

• Leif Abrahamson (University of Uppsala, Sweden)

• Rikard Bogvad (Stockholm University, Sweden)

• Damiani Maingi (University of Nairobi, Kenya)

• Fernando Rodrigues Villegas (ICTP)

• David Ssevviiri (Makerere University, Uganda)

• Balazs Szendroi (University of Oxford, UK, main International Organizer)

• Patrick Weke (University of Nairobi, Kenya)

• Juma Kasozi (Makerere University, Uganda)

• Sylvester E. Rugeihyamu (Dar es Salaam University, Tanzania)

• Jared Ongaro (University of Nairobi, Kenya)

• Michael Gahirima (University of Rwanda)

• Mubanga Lombe (University of Zambia)