# 2013-2014 intensive courses

## Introduction to the Theory of Dissipative Operators and Applications by Ian Wood and Sergey Naboko of Kent University

Monday 20 May

Tuesday 21 May

The course will give an introduction to the theory of dissipative operators and consider some applications. It will start with a review of some basic facts about self-adjoint operators, including the Spectral Theorem. Then dissipative operators will be defined and their spectral and resolvent properties investigated. Wood and Naboko will discuss completely non-self-adjoint dissipative operators, maximal dissipative operators. A principal focus of the course will be on self-adjoint dilations of these operators. Connections to Complex Analysis, strongly continuous semigroups and the extension theory of operators will be pointed out. Various examples of dissipative operators from Mathematical Physics will be presented.

The intension is to make the course self-contained as far as possible, but will assume basic familiarity with linear operators on Hilbert space. If you wish to do some preliminary reading before the course, the following publications are recommended:

- Akhiezer, Glazman: Theory of Linear Operators in Hilbert Space, Dover, 1993.
- Riesz; Sz.-Nagy: Functional analysis. Dover Publications, 1990, especially the appendix on dilation theory written by Sz.-Nagy.
- Sz.-Nagy, Foias, Bercovici, Kerchy: Harmonic Analysis of Operators on Hilbert Space, 2nd

edition, Springer, 2010.

## Schubert calculus on Grassmannians by Clelia Pech of King's College London

Monday 10 June

Tuesday 11 June

Enumerative problems are an important part of Algebraic Geometry. Their aim is often to count the number of objects (lines, curves ...) satisfying certain incidence conditions. For example, what is the number of circles tangent to three given circles in the plane? What is the number of plane conics through 5 points?

These kinds of questions have been extensively studied during the XIX century and form the bases of Schubert Calculus.

This intensive course will focus on Schubert calculus on Grassmannians, which parametrize vector subspaces of a given dimension in an ambient complex vector space. It involves the beautiful combinatorics of Young tableaux, which are also related to interesting problems in representation theory and the theory of symmetric functions.

Outline of the course:

- the Grassmannians and their projective embedding
- Plücker coordinates and Plücker relations
- Schubert classes, Poincaré duality
- Pieri and Giambelli rules
- the Littlewood-Richardson rule
- overview of recent generalizations : Schubert calculus on homogeneous spaces and quantum Schubert calculus.

The LTCC will be running the following intensives courses on the dates stated below:

- 2pm - 6pm on Wednesday 20 November - Room B05, Chadwick Building
- 9am - 1pm on Thursday 21 November - Room G08, Chadwick Building

This course will take place in the Chadwick Building, which is on Gower Street, at the UCL Campus:

www.ucl.ac.uk/find-us

Turbulence remains as the unsolved problem in classical (fluid) mechanics. Standard approaches comprise dimensional analysis, semi-empirical modelling of the unclosed terms in the Reynolds-averaged Navier–Stokes equations (RANS), and methods aiming at the resolution of the abundance of scales involved, as Large-Eddy or Direct Numerical Simulation (LES, DNS). Notwithstanding their undeniable ongoing progress, many aspects of the important class of wall-bounded and separated shear flows at large Reynolds numbers, Re, yet lack a sound theoretical understanding.

However, here advanced asymptotic techniques have proven promising more recently. The intention of this course is to focus on their impact on classical statistical/modelling methods an also novel findings. Outline of the course:

- Summary of governing equations, validity of Newtonian constitutive law for high-
*Re*flows, specification for isoviscous incompressible flow. - Fundamentals of the statistical theory (concept of local isotropy, Kolmogorov theory, 5/3-power law).
- Principles of averaging.
- Classical closures (based upon one-point correlations).
- Central topic: advanced asymptotic description of turbulent shear flows.

The course is prepared for being largely self-contained, but in case of preliminary reading the following classical textbooks are recommended:

- Schlichting, H., Gersten, K.: Boundary-Layer Theory (8th revised and enlarged ed.), Parts III–V, Springer, 2003.
- Tennekes, H., Lumley, J. L.: A First Course in Turbulence, MIT Press, 1972.
- Townsend, A. A.: The Structure of Turbulent Shear Flows (2nd ed.), Cambridge University

Press, 1976.