Abstract:

Differential and difference algebra were founded by Ritt and his students Kolchin and Cohn in the 1950s as the study of rings and modules endowed with an additional operator: a differential operator/derivation, or a difference operator/endomorphism. 

Using the methods of category theory, we elevate those areas to the level where they easily interact with fundamental areas of mainstream mathematics such as algebraic geometry and logic, and the progress is made by following established templates and higher principles, as opposed to ad-hoc reasoning.

Using the methods of Grothendieck’s topos theory and Lawvere-Tierney’s categorical logic, we make significant advances on homological algebra, algebraic geometry and cohomology theory in the difference context. 

Using the methods of Janelidze\s categorical Galois theory and our newly-developed theory of precategorical descent, we vastly generalise the 140-year old Picard-Vessiot style Galois theory for differential and difference fields to the context of differential and difference schemes, suitable for the treatment of parametic families of differential and difference equations. 

Bio-Sketch: 
Ivan Tomasic received his undergraduate degree in Mathematics from University of Zagreb, and his PhD from University of Edinburgh. He is currently a Reader in Pure Mathematics at Queen Mary University of London, where he joined in 2007 following positions in Leeds, Lyon and Zagreb.

He is interested in interactions of Logic (including Model Theory and Categorical Logic), and Algebraic Geometry through a long-term research programme of developing Relative Algebraic Geometry over a base topos, with a particular emphasis on applications of Topos Theory and Category Theory in Difference and Differential Algebra.