Integral p_adic Hodge-theory and applications to p-adic deformation theory

We associate linear data to geometric objects like curves, surfaces or threefolds defined as zero-sets of algebraic equations over the integers or over a number domain that is annihilated by a power of a prime number. One can ask how much information about the original object can be captured in this way. The linear data are given by various cohomology theories which is a standard technique in many areas in Pure Mathematics including topology, complex and arithmetic geometry. This problem leads to what is called p-adic deformation theory; the project addresses the question in the case of crystalline cohomology or equivalently de Rham-Witt cohomology classes associated to algebraic cycles and how they deform p-adically. A related problem of particular interest is to identify an integral structure defined over the (p-adic) integers inside these linear data which leads to finer invariants of the geometric objects and encode more information. This method is very common in arithmetic geometry.

Academic impact will be achieved within the area of the proposal: p-adic Hodge-theory, relative Dieudonne-theory, deformation theory through publication of the results in leading academic journals and through presentations at relevant international conferences, workshops and seminar talks.

The proposal will provide training and development in research skills in p-adic arithmetic geometry for a Post-doctoral research assistant and will also help to build up capacity in the area of p-adic Hodge-theory which, compared to France, Germany and Japan, is slightly underrepresented in the UK. Capacity will be built in this proposal by interaction with other researchers and attendance at international conference throughout the project. The PI will also host a conference in Exeter in 2021 with high profile speakers.

It is common that in Pure Mathematics non-academic impact is not achieved immediately, but possibly in the longer term. Projects 2, 3 and 4 deal with " integral structures in p-adic cohomology theories " , especially with so called "Monsky-Washnitzer cohomology of affine varieties". These "objects" are not only of interest to p-adic Hodge-theoretists but also occur when using sophisticated cryptosystems, like elliptic/hyperelliptic curve cryptography where analogies of classical RSA- and Diffie-Hellman problems can be formulated by working with rational points of affine curves. The number of these rational points is closed related to the action of Frobenius on the above "objects". The proposed research makes these objects more accessible and may provide new techniques to promote further research in modern cryptography. Potential beneficiaries are banks, the GCHQ and IT security companies.

Public awareness of the contribution of mathematics to progress in science in the society as a whole remains an important challenge for all mathematicians including those working in Pure Maths. As an example the PI recently joined a debate in the Exeter Maths School on the genuine differences between Pure and Applied Maths research and will continue to promote such public relation measures in the future.