Geometry and arithmetics through the theory of algebraic cycles

The basic objects of algebraic geometry are algebraic varieties. These are defined locally as the zero locus of polynomial equations. The main goal of algebraic geometry is to classify varieties. An approach consists in attaching invariants to varieties. Some invariants are of an arithmetic nature, e.g. the gcd of the degrees of closed points on X. Some are of a topological nature, e.g. the singular cohomology of the underlying topological space of X. Some are of a geometric nature, e.g. the Chow groups of X. A codimension-n algebraic cycle on X is a formal sum of irreducible subvarieties of codimension n and the Chow group CH^n(X) is the abelian group with basis the irreducible subvarieties of codimension n in X modulo a certain equivalence relation called rational equivalence. Roughly, rational equivalence is the finest equivalence relation on algebraic cycles that makes it possible to define unambiguously an intersection product on cycles. Moreover, the aforementioned invariants for X are encoded (or at least expected to be) in the Chow groups X. Therefore, in some sense, algebraic cycles are the finest invariants for algebraic varieties, and the theory of algebraic cycles lies at the very heart of geometry, topology and number theory.

I will integrate methods from K-theory, Galois cohomology and number theory to derive new results in the theory of algebraic cycles on varieties defined over finitely generated fields or other fields of arithmetic interest. Conversely, I will use the theory of algebraic cycles to derive new results of arithmetic
interest. In addition, the outcome of such results will shed new light on the geometry of such varieties. Thus, by its very nature, my research proposal on the theory of algebraic cycles is intradisciplinary within the mathematical sciences.

Algebraic geometry and number theory occupy a central place in pure mathematics and have interactions with many other fields ranging from representation theory, combinatorics, Lie theory, symplectic geometry, as well as more applied fields including cryptography (including elliptic curve cryptography) and error correction codes. Therefore, advances in the fields of algebraic geometry and number theory have potential impact on everyday life technology such as CD players and the internet through error correction codes, and security encryption for credit cards, etc.

The theory of algebraic cycles brings together algebraic geometry and number theory and is rooted at the very heart of those two central and most ancient fields of mathematics. As such, the significance of algebraic cycles within mathematics cannot be underestimated.

Problems around algebraic cycles are considered to be amongst the most difficult and most challenging problems in pure mathematics. Any significant result obtained as part of this proposal will attract interest from around the world, including many parts of Europe, Russia, the United States, India, China and Japan. Any progress in this field will enhance the international reputation of the Host Institution and, consequently, help Institutions across the UK recruit top-class academics and students.

The results of this work will be disseminated through papers that will first be uploaded on both the arXiv and my homepage at the Mathematical Department's website to ensure a fast and efficient dissemination of the research in order to maximize their usefulness and impact to academia. The results of this work will then be published in leading international journals to attest the quality of the research and increase the prestige of both the EPSRC and the Host Institution.

The knowledge will also be shared through the PI's travels to conferences and seminars, as well as by inviting visitors to the Host Institution. I will emphasize the cross-disciplinary perspective of my research and aim both at a general audience and at experts in my field. I am already familiar with this exercise as I have already presented the technical details of my work at several conferences, workshops and specialists' seminar series through Europe and Japan. I am also experienced at aiming at a general audience of mathematicians and I have recently been invited to give a Colloquium talk on my
research in Liverpool. There I have especially focused on the interactions of geometry and arithmetics in the theory of algebraic cycles.

Finally, the cross-disciplinary research undertaken will be disseminated at a graduate level through the master's program in mathematics (Part III Tripos) at Cambridge by designing graduate courses and setting up essays emphasizing the ubiquity of the theory of algebraic cycles. For instance, I designed
a graduate course at Cambridge for the Part III Tripos entitled `Galois cohomology'. This was a demanding task but I thoroughly enjoyed showing the students how techniques from algebra, algebraic geometry, number theory and homological algebra can be brought together and illuminate each other.
This way I am directly responsible for developing the students' ability to identify and analyze patterns, logic and critical thinking skills, ability to see relationships, and problem solving skills. Statistically, students with a postgraduate degree in the mathematical sciences (this includes master's degrees and PhDs) have the highest average starting salary among all UK holders of postgraduate degrees, according to Adrian Smith's report One Step Beyond (March 2010), p. 94: http://www.bis.gov.uk/one-step-beyond.

Thus teaching advanced topics such as my research to master's students makes those students very desirable to employers and feeds in the critical
pathway towards economic and societal impact.