Friday, 17 January 2014

Peter Keevash proves the Existence Conjecture for combinatorial designs

Oxford Mathematics Professor Peter Keevash has proved the Existence Conjecture for combinatorial designs, answering a question of Steiner from 1853. 

A Steiner Triple System on a set X is a collection T of 3-element subsets of X such that every pair of elements of X is contained in exactly one of the triples in T. An example considered by Plücker in 1835 is the affine plane of order three, which consists of 12 triples on a set of 9 points. Plücker observed that a necessary condition for the existence of a Steiner Triple System on a set with n elements is that n be congruent to 1 or 3 mod 6. In 1846, Kirkman showed that this necessary condition is also sufficient.

 In 1853, Steiner posed the natural generalisation of the question: given integers q and r, for which n is it possible to choose a collection Q of q-element subsets of an n-element set X such that any r elements of X are contained in exactly one of the sets in Q? There are some natural necessary divisibility conditions generalising the necessary conditions for Steiner Triple Systems. The Existence Conjecture states that for all but finitely many n these divisibility conditions are also sufficient for the existence of general Steiner systems (and more generally designs).

In his paper Peter proves the Existence Conjecture, and more generally, shows that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition.

Friday, 17 January 2014

Roger Penrose at the Royal Institution

In October 2013, as the new Mathematical Institute opened, Roger Penrose gave a lecture at the Royal Institution talking about forbidden crystal symmetry in mathematics and architecture and its part in the Penrose tiling which adorns the entrance to the new building. Roger also took questions and answers about his work from the audience.

Tuesday, 14 January 2014

The Millennium Problems - films of the Clay Research Conference 2013

In October 2013 eminent mathematicians presented the latest thinking about some of the Millennium Problems as part of the Clay Mathematics Research Institute’s Annual Conference held here in the Andrew Wiles Building in Oxford. 

Peter Constantin (Princeton) addressed the Navier-Stokes Equations, Lance Fortnow (Georgia Institute of Technology) the P versus NP Problem, and Fernando Rodriguez Villegas (Austin) the Birch—Swinnerton–Dyer Conjecture.

Other contributions were talks by Edward Witten (Institute for Advanced Study, Princeton) on the Jones Polynomial of a Knot, Richard Thomas (Imperial College) on the work of Clay Research Awardee Rahul Pandharipande and Ingrid Daubechies (Duke) on Animation, Teeth and Skeletons. This last talk formed the link between the Clay Research Conference and the Opening Conference for the Andrew Wiles Building.

Wednesday, 8 January 2014

Architecture That Shook Oxford - the Andrew Wiles Building on film

As part of its 'Architecture That Shook Oxford' series Oxford Today, profiles the new Mathematical Institute, the Andrew Wiles Building. Dr William Whyte, Fellow of St John's College discusses both the building itself and what it says about the Oxford in the 21st Century.

Tuesday, 31 December 2013

Professor Frances Kirwan made a Dame Commander of the Order of the British Empire

Congratulations to Frances Kirwan, FRS, who has been honoured in the 2014 New Year Honours for services to mathematics. Frances, who specialises in algebraic and symplectic geometry, has been a Professor in Oxford since 1996, is a former President of the London Mathematical Society and is Chair of the United Kingdom Mathematics Trust.

Wednesday, 18 December 2013

Prime numbers - beauty and security

Watch Dr Richard Earl from Oxford Mathematics talk about prime numbers as part of the Christmas Science Lectures. Richard not only explains the intrinsic importance of prime numbers, but expands on their role in our everyday lives, notably their critical part in internet security.


Tuesday, 10 December 2013

Roger Penrose talks about his relationship with the Art of M C Escher

Sir Roger Penrose has a long-standing interest in and connection with M C Escher, the Dutch graphic artist best known for his mathematically inspired woodcuts and lithographs. Roger talks about a relationship that dates back nearly sixty years and also explains why art has been a consistent part of both his family life and his mathematics.


Monday, 9 December 2013

Roger Penrose explains the mathematics of the Penrose Paving

As you enter the new Mathematical Institute here in Oxford you are confronted with a pattern of beauty and intrigue. Designed by Sir Roger Penrose, Emeritus Rouse Ball Professor of Mathematics, and expanding on his discovery of almost thirty years ago, the Penrose Paving is constructed from just two different diamond-shaped granite tiles, each adorned identically with stainless steel circular arcs. There are various ways of covering the infinite plane with them, matching the arcs. But every such pattern is non-repetitive and contains infinitely many exact copies of what you see before you. 


A longer version, expanding on the mathematics is also available



Thursday, 5 December 2013
Tuesday, 3 December 2013