3-4 June 2015 -'Logic meets topology': model theory, o-minimality, definable groups, by Alessandro Berarducci (Leverhulme VP at QMUL/University of Pisa)

Venues and times:

• 13.00 - 17.00 on 3 June in Foster Court 219, UCL
• 9.00 - 13.00 on 4 June in Roberts 309, UCL

• UCL map

Course outline:

The course will provide an introduction to model theory and o-minimality. O-minimality can be seen as a realization of the idea of 'tame topology' inasmuch as it provides a category of topological spaces and maps where one does not encounter any of the annoying pathologies which infect general topology textbooks, and yet it is flexible enough to permit the development of significant parts of mathematics. Piecewise linear topology and semialgebraic or subanalytic geometry fit well into this context, which is far more general and allows for a uniform treatment.

The focus will mainly be on definable groups, since it is here that the interplay between logic and topology manifests itself more clearly. In general the study of definable groups plays an important role in model theory. In the stable case there is a long-standing algebraicity conjecture of Cherlin and Zilber concerning simple groups of finite Morley rank. In the o-minimal case, thanks to a fundamental result of Pillay, the connection is with Lie groups, rather than algebraic groups.

The course will present some of the main results highlighting the analogies with Lie groups, together with a list of open questions. Selected proofs will be given aiming to convey some working knowledge of the basic tools of o-minimality, such as cell decomposition, dimension and Euler characteristic.

Recommended reading:

• L. van den Dries. Tame topology and o-minimal structures. Cambridge UP, 1998.
• A. Strzebonski. Euler characteristic in semialgebraic and other o-minimal groups. Journal of Pure
  and Applied Algebra 96(2) (1994): 173201.
• A. Pillay. On groups and elds de nable in o-minimal structures. Journal of Pure and Applied
  Algebra 53(3) (1988):239255.
• A. Pillay. Type-de nability, compact lie groups, and o-Minimality. Journal of Mathematical Logic
  4(2) (2004): 147-162.
• A. Berarducci, M. Otero, Yaacov Peterzil, and Anand Pillay. A descending chain condition for
  groups de nable in o-minimal structures. Annals of Pure and Applied Logic 134(2-3) (2005): 303-313.

22-23 June - Polyhedral Combinatorics, by Jack Edmonds

Venue and times:

• 13.00 - 17.00 on 22 June (Room 505, Dept. of Mathematics, 25 Gordon Street, UCL)
• 9.00 - 13.00 on 23 June (Venue same as above)

• UCL map

Course outline:

For thousands of years, and even now in group theoretic geometry, the beautiful symmetries of a handful of polyhedra with a handful of facets have been at the center of refined mathematics. Since the advent of Turing's computers and operations research, beauty has been found in polyhedra regardless of symmetry, with facets as numerous as the stars. Linear-algebra theory is being nudged by great systems of linear inequalities as inputs.

• Day 1. Submodular Set Functions and Polymatroids
  The greedy algorithm. Combinatorics of (linear) dependence. Matroid partitioning and Lehman’s game. Duality and contractions. Exact matrix implementations. The ellipsoid method for exponentially large linear programs. Machine learning.
• Day 2. Some Pure Math of Classical Game Theory
  Bimatrix games and beautiful-mind Nash equilibria; Room partitioning and PPA.
  N-person cooperative games; set functions and cores. Farkas Lemma.
  Algorithms and algorithmic difficulties.

Resources

• Grotschel (ed.), Optimization Stories (on line)
• Schrijver, Combinatorial Optimization: Polyhedra and Efficiency (available digitally)

There will also be an independent and complementary course also by Jack Edmonds 'Existential Polytime and Polyhedral Combinatorics', 16-19th June at Queen Mary.

Now closed

19-20 Feburary - Truncated Hankel and Toeplitz Operators By Roman Bessonov (KCL and St Petersburg University)

Venues and times:

• 9am - 1pm on 19 Feburary in the Hardy Room, De Morgan House, 57-58 Russell Square, WC1B 4HS
• 9am - 1pm on 20 February, Hardy Room

• De Morgan House

Course outline:

This course will give an account of recent results on truncated Toeplitz and Hankel operators. While the subject has classical roots (Toeplitz matrices, nite section Wiener-Hopf operators, Commutant Lifting Theorem) its systematic study in more general settings has been started only recently. A characterization of these operators in terms of an operator identity (D. Sarason, 2007) discovers their hidden structure and serves as a base for our considerations.

Truncated Toeplitz/Hankel operators are compressions of usual Toeplitz/Hankel operators to shift-coinvariant subspaces of the Hardy space. This course therefore can be considered as a short introduction to the shift-coinvariant subspaces as well. Many specific features of these subspaces will be discussed: structure of reproducing kernels, Carleson embeddings, Clark measures, unit-component inner functions. The audience is expected to be familiar with a basic theory of the Hardy space.

• Lecture 1. De nitions, examples, and results. An operator identity for truncated Toeplitz and Hankel operators. Finite-rank operators. Carleson embeddings and truncated Toeplitz operators.
• Lecture 2. Unit-component inner functions. Weak factorizations of pseudo-continuable functions. Nehari and Hartman theorems for truncated Hankel operators.
• Lecture 3. Clark measures of inner functions. Duality theorems for shift-coinvariant subspaces of H1 . BMO-type criterium for boundedness and compactness of truncated Toeplitz/Hankel operators.
• Lecture 4. Relation to discrete Hilbert transform commutators. Open questions and conjectures.

22 - 23 May 2014 - Enumerative Combinatorics and Models of Polymers by Thomas Prellberg (QMUL)

Venues and times:

• 1pm - 5pm on 22 May in Gordon Square (23) 101, University College London
• 9am - 1pm on 23 May in Gordon Square (23) 102

http://www.ucl.ac.uk/maps

Many problems in mathematics and physics - including many in the modeling of polymers - can be rephrased as ”How many...”. Enumerative combinatorics seeks to answer these questions.
The course will give a general introduction to the world of enumeration and the techniques of generating functions - including asymptotic methods championed by the late Philippe Flajolet. In the second half the course I will show how we can apply these generating function methods to study polymers - especially adsorption, collapse and localisation.

The course will place particular emphasis on applications of the kernel method, a well-known method in algebraic combinatorics for solving functional equations, which has in recent years been extended significantly, and enables the derivation of generating functions for a variety of combinatorial and statistical mechanical problems.

Recommended reading:

• H Wilf, Generatingfunctionology, Academic Press 1994. (www.math.upenn.edu/wilf/gfology2.pdf)

Additional Optional reading:

• P Flajolet and R Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009. (http://algo.inria.fr/flajolet/Publications/book.pdf)
• EJJ van Rensburg, The Statistical Mechanics of Interacting Walks, Polygons, Animals, and Vesicles, Oxford Lecture Series in Mathematics and its Applications 18, Oxford University Press, 2002.
  

19 - 20 June - Introduction to the Theory of Jacobi Matrices and Orthogonal Polynomials by Sergei Naboko and Ian Wood (Kent University)

Venue and times:

• 1pm - 5pm on 19 June in Gordon Square (23) 101, UCL
• 9am - 1pm on 20 June in Gordon Square (23) 101

http://www.ucl.ac.uk/maps

This course will look at Jacobi matrices (finite or infinite tri-diagonal matrices), orthogonal polynomials and the connection between the two.

Orthogonal polynomials arise by applying to monic polynomials the Gram-Schmidt orthogonalisation procedure in the space of functions which are square integrable on an interval with respect to a non-negative weight.

The study of Jacobi matrices is the study of three term recurrence relations and the asymptotic behaviour of their solutions. This is a discrete analogue of the theory of differential equations from which many ideas can be borrowed. Various asymptotic methods are useful in the analysis.

The course will consider both classical and modern applications from the theory which has connections to the moment problem, complex analysis (via the Weyl-Titchmarsh m-function), and numerous applications to problems in physics and biology (e.g. Markov birth and death processes).

The aim of the course is to give a brief introduction to this active area of research.

Recommended reading:

• http://en.wikipedia.org/wiki/Orthogonal_polynomials
• http://en.wikipedia.org/wiki/Jacobi_operator

Additional optional reading:

Akhiezer, N. I. The classical moment problem and some related questions in analysis. Translated by N. Kemmer. Hafner Publishing Co., New York 1965.

Teschl, Gerald (2000), Jacobi Operators and Completely Integrable Nonlinear Lattices, Providence: Amer. Math. Soc. (freely available from Professor Teschl's homepage at the University of Vienna)