Motives of algebraic varieties

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Algebraic geometry, the study of shapes (algebraic varieties) defined by polynomial equations, is one of the oldest areas of modern mathematics with roots reaching back to the ancient Greek study of conic sections.  It is, however, also one of the most active areas of research in mathematics producing hundreds of scientific papers a year on a diverse array of topics with connections to many other areas such as number theory and high energy physics. Motives were introduced by the French mathematician Alexander Grothendieck, one of the founders of modern algebraic geometry, in the late 1960s as a "grand unified cohomology theory" bringing together several different theories used to study the shapes of objects in algebraic geometry. Since then, motives have become an indispensable tool in algebraic geometry.  They are intimately related to the study of algebraic cycles, one of the most difficult but fascinating subjects in the theory. I plan to work on four main projects in the area of motives and algebraic cycles: (1) motivic decompositions of projective homogeneous varieties, (2) periods and Feynman amplitudes, (3) variations of mixed Hodge structure and (4) Steenrod operations in algebraic geometry. The first project has direct applications to algebra.  The goal of the second is to study a deep relationship discovered by the physicist Dirk Kreimer between motives and the Feynman amplitudes which arise in calculations of interactions between elementary particles in high energy physics.  The third and fourth projects are loosely motivated by the circle of ideas surrounding the most important unsolved question in algebraic geometry, the Hodge conjecture, which asks when a shape can be defined by polynomial equations.