Research lectures, seminars and events
The events listed in this area are research seminars, workshops and lectures hosted by Durham University departments and research institutes. If you are not a member of the University, but wish to enquire about attending one of the events please contact the organiser or host department.
|April 2018||June 2018|
Events for 15 May 2018
Structure and Representation: Thinking Ecologically About Policy and Structure; Learning How to Affect Change in Stable Structures - Workshop 2
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Demi Allen: A mass transference principle for systems of linear forms with applications to Diophantine approximation
In Diophantine approximation we are often interested in the Lebesgue and Hausdorff measures of certain
lim sup sets. In 2006, Beresnevich and Velani proved a remarkable result — the Mass Transference Principle —
which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic state-
ments for lim sup sets arising from sequences of balls in R
. Subsequently, they extended this Mass Transference
Principle to the more general situation in which the lim sup sets arise from sequences of neighbourhoods of “ap-
proximating” planes. In this talk I will discuss a recent strengthening (joint with V. Beresnevich) of this latter
result in which some potentially restrictive conditions have been removed from the original statement. This
improvement gives rise to some very general statements which allow for the immediate transference of Lebesgue
measure Khintchine–Groshev type statements to their Hausdorff measure analogues and, consequently, has some
interesting applications in Diophantine approximation.
A cubic polynomial equation in four or more variables tends to have many integer solutions, while one in two variables has a limited number of such solutions. There is a body of work establishing results along these lines. On the other hand very little is known in the critical case of three variables. For special such cubics, which we call Markoff surfaces, a theory can be developed. We will review some of the tools used to deal with these and related problems.
Joint works with Bourgain/Gamburd and with Ghosh
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