All seminars take place on Tuesdays at 4.00 pm in Room 505 on the 5th floor of the Mathematics Dept. See how to find us for further details. There will be tea afterwards in room 606.

If you require any more information on the Pure seminars please contact Dr Nadia Sidorova e-mail: n.sidorova AT ucl.ac.uk or tel: 020-7679-7864. If you would like to receive weekly announcements including titles and abstracts, you are welcome to join the seminar mailing list.

## 18 January 2011

Jozsef Solymosi - University of British Columbia

##### On geometric incidences

Abstract:

In this talk we review some old and new problems on geometric incidences. We present various bounds on point-line incidences. For a given arrangement of lines L and a set of points P the number of incidences is the number of point line pairs, {p.l}, (p from P and l from L) so that p is a point of l. We are interested about upper bounds on incidences over the Euclidean plane, 3-space, projective complex plane/space, and finite fields. The tools we are using for bounding the number of incidences are from algebra, discrete geometry, and algebraic geometry.

## 25 January 2011

Apala Majumdar - Oxford University

##### Passage from mean-field to continuum to liquid crystal theories

Abstract:

In this talk, we make quantitative comparisons between two widely-used liquid crystal modelling approaches - the continuum Landau-de Gennes theory and mesoscopic mean-field theories, such as the Maier-Saupe and Onsager theories. We use maximum principle arguments for elliptic partial differential equations to compute explicit bounds for the norm of static equilibria within the Landau-de Gennes framework. These bounds yield an explicit prescription of the temperature regime within which the LdG and the mean-field predictions are consistent, for both spatially homogeneous and inhomogeneous systems. We find that the Landau-de Gennes theory can make physically unrealistic predictions in the low-temperature regime. In my joint work with John Ball, we formulate a new theory that interpolates between mean-field and continuum approaches and remedies the deficiencies of the Landau-de Gennes theory in the low-temperature regime. In particular, we define a new thermotropic potential that blows up whenever the mean-field constraints are violated. The main novelty of this work is the incorporation of spatial inhomogeneities (outside the scope of mean-field theory) along with retention of mean-field level information.

## 01 February 2011

Sergei Konyagin - Moscow State University

##### Multiplicative translates of subgroups in residue rings

Abstract:

Please click here for Sergei Konyagin's abstract

## 08 February 2011

NO SEMINAR

## 15 February 2011

READING WEEK - NO SEMINAR

## 22 February 2011

Gui-Qiang Chen - Oxford University

##### Nonlinear Partial Differential Equations of Mixed Type in Mechanics and Geometry

Abstract:

Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some fundamental issues in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include transonic flow equations in fluid mechanics and the Gauss-Codazzi system for isometric embedding in differential geometry. In this talk we will discuss some recent developments in the analysis of nonlinear partial differential equations of mixed type through these examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.

## 01 March 2011

Colin McDiarmid - Oxford University

##### Colouring random geometric graphs

Abstract:

Please click here for Colin McDiarmid's abstract

## 08 March 2011

Andrew Dancer - Oxford University

##### Symplectic versus hyperkahler geometry

Abstract:

Hyperkahler geometry is a branch of Riemannian geometry that has a strong symplectic flavour. We show how several constructions in symplectic geometry, in particular cutting and implosion, have hyperkahler analogues.

## 15 March 2011

Francisco Santos - University of Cantabria

##### Counter-examples to the Hirsch conjecture

Abstract:

The Hirsch conjecture, stated in 1957, said that if a polyhedron is defined by $n$ linear inequalities in $d$ variables then its combinatorial diameter should be at most $n-d$. That is, it should be possible to travel from any vertex to any other vertex in at most $n-d$ steps (traversing an edge at each step). The unbounded case was disproved by Klee and Walkup in 1967. In this talk I describe my construction of the first counter-examples to the bounded case (polytopes). The conjecture was posed and is relevant in connection to linear programming since the simplex method, one of the "mathematical algorithms with the greatest impact in science and engineering in the 20th century", solves linear programming problems by traversing the graph of the feasibility polyhedron.

## 22 March 2011

Gabriel Paternain - Cambridge University

##### Transparent connections

Abstract:

I will discuss the inverse problem of reconstructing a connection from knowledge about its parallel transport. A connection is said be transparent if its parallel transport along every closed geodesic is the identity. I will explain how to describe all transparent SU(2)-connections over a closed negatively curved surface.