My research lies at the intersection of algebraic topology and algebraic geometry. It primarily focuses on the theory of mixed motives following Vladimir Voevodsky, intersection theory, and motivic cohomology. This theory fully integrates into the theory of topos and Grothendieck's six operations formalism.
Secondly, and more broadly, the theory of motivic homotopy extends the framework of homotopy of topological spaces to that of schemes. In its stable variant, it provides a framework for the study of cohomologies, of mixed (Weil) cohomologies, of absolute cohomologies in Alexander Beilinson's terminology like syntomic cohomology, but also for the theory of orientation, characteristic classes, and Gysin morphisms.
All these theories find their natural expression and full realization in the theory of higher homotopy developed by numerous mathematicians: Grothendieck, Quillen, Joyal, and Lurie. My companions and guides in this field have been Denis-Charles Cisinski and Bertrand Toën.
