Geometry and arithmetics through the theory of algebraic cycles

Lead Research Organisation: University of Cambridge
Department Name: Pure Maths and Mathematical Statistics

Abstract

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.

Planned Impact

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.

Publications

10 25 50
publication icon
Charles Vial (Author) (2013) Algebraic cycles and fibrations in Documenta Mathematica

publication icon
Charles Vial (Author) (2013) Projectors on the intermediate algebraic Jacobians in New York Journal of Mathematics

publication icon
SHEN M (2016) THE MOTIVE OF THE HILBERT CUBE in Forum of Mathematics, Sigma

publication icon
Shen M (2016) The Fourier Transform for Certain HyperKähler Fourfolds in Memoirs of the American Mathematical Society

publication icon
Vial C (2017) On the motive of some hyperKähler varieties in Journal für die reine und angewandte Mathematik (Crelles Journal)

publication icon
Vial, C. (2017) Remarks on motives of abelian type in Tohoku mathematical journal

publication icon
Achter J (2017) On descending cohomology geometrically in Compositio Mathematica

publication icon
Achter J (2019) Parameter Spaces for Algebraic Equivalence in International Mathematics Research Notices

publication icon
Vial, C. On the motive of certain hyperKaehler varieties in journal fur die reine und angewandte mathematik

 
Description So far, I have discovered new and surprising geometric properties of hyperKaehler varieties. HyperKaehler varieties are important objects in geometry that come equipped with a very rich structure (which explains their significance). They are somehow closely related to so-called Calabi-Yau varieties, whose importance in theoretical physics and in our understanding of the universe cannot be underestimated.
Exploitation Route My findings have contributed to the academic knowledge in the field of algebraic geometry.
Sectors Other

 
Description My findings in the theory of algebraic cycles on hyperKaehler varieties have sparked new directions of research in pure mathematics. I have initiated a fruitful collaboration with Lie Fu (Lyon, France); we have now three articles accepted for publication. My collaboration with Jeff Achter and Sebastian Casalaina-Martin from the University of Colorado and the Colorado State University has grown deeper: we have proved further results of arithmetic significance concerning algebraic cycles.
First Year Of Impact 2016
Sector Other
Impact Types Cultural

 
Description Motives of hyperKahler varieties 
Organisation University of Amsterdam
Department Mathematics Department
Country Netherlands 
Sector Academic/University 
PI Contribution We conjectured the existence motivic structures for hyperKaehler varieties that are similar to that of abelian varieties. We established such existence in important cases.
Collaborator Contribution We conjectured the existence motivic structures for hyperKaehler varieties that are similar to that of abelian varieties. We established such existence in important cases.
Impact 2 published articles.
Start Year 2012
 
Description Motivic hyperKaehler resolution conjecture for generalized Kummer varieties 
Organisation Joseph Fourier University
Country France 
Sector Academic/University 
PI Contribution We promoted a conjecture of Ruan from homological equivalence to rational equivalence, and established it in the important case of generalised Kummer varieties.
Collaborator Contribution We promoted a conjecture of Ruan from homological equivalence to rational equivalence, and established it in the important case of generalised Kummer varieties.
Impact A preprint on the arXiv that has been submitted for publication.
Start Year 2016
 
Description Motivic hyperKaehler resolution conjecture for generalized Kummer varieties 
Organisation University of Lyon
Department Department of Mathematics
Country France 
Sector Academic/University 
PI Contribution We promoted a conjecture of Ruan from homological equivalence to rational equivalence, and established it in the important case of generalised Kummer varieties.
Collaborator Contribution We promoted a conjecture of Ruan from homological equivalence to rational equivalence, and established it in the important case of generalised Kummer varieties.
Impact A preprint on the arXiv that has been submitted for publication.
Start Year 2016
 
Description On descending cohomology geometrically 
Organisation Colorado State University
Department Department of Mathematics
PI Contribution We wrote 3 articles, 1 of which has been accepted for publication at Compositio Math, and 2 of which have been posted on the arXiv and have been submitted for publication.
Collaborator Contribution Joint articles
Impact Series of articles. Multi-disciplinary in that the projects deals with arithmetic questions related to geometry.
Start Year 2015
 
Description On descending cohomology geometrically 
Organisation University of Colorado Boulder
Department Department of Mathmatics
Country United States 
Sector Academic/University 
PI Contribution We wrote 3 articles, 1 of which has been accepted for publication at Compositio Math, and 2 of which have been posted on the arXiv and have been submitted for publication.
Collaborator Contribution Joint articles
Impact Series of articles. Multi-disciplinary in that the projects deals with arithmetic questions related to geometry.
Start Year 2015