Algebraic bordism spectra: Computations, filtrations, applications

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The projects belong to the field of motivic or A1-homotopy theory, an area of mathematics at the interface of algebraic geometry and homotopy theory. It is a relatively new and intensively developing discipline with the foundations established about twenty years ago. Despite its youth motivic homotopy theory has already led to striking results and innovations both in algebraic geometry and classical homotopy theory, including Voevodsky's proof of the Milnor and Bloch-Kato conjectures, new insights into the structure of stable homotopy groups of spheres by Isaksen, and techniques of slice filtrations that were successfully used by Hill, Hopkins and Ravenel in their work on the Kervaire invariant. Morel’s computation of the zeroth stable homotopy groups of motivic spheres implies that the Grothendieck-Witt ring of quadratic forms over a field constitutes an invariant of central importance in motivic homotopy theory.In homotopy theory an overwhelmingly rich variety of methods have been developed to gain a better understanding of cohomology theories. A particular example of such a technique is given by chromatic homotopy theory, which is based on the structure of the complex bordism spectrum MU and related cohomology theories. The algebraic analogue of MU is given by Voevodsky’s algebraic bordism spectrum MGL. This object has already been studied for some time. In particular, Levine and Morel obtained a geometric presentation for (a part of) the corresponding cohomology theory, and various authors applied algebraic cobordism to the study of the motivic stable homotopy category and to the study of algebraic varieties (via characteristic classes and oriented cohomology theories). Contrary to the classical homotopy theory picture, it turned out that in the motivic setting the approaches based on MGL miss some important piece of information, namely the quadratic orientation. This suggests one to look at other algebraic bordism spectra that take into account quadratic orientations.In the course of the current project we plan to study algebraic bordism spectra different from MGL, such as oriented algebraic bordism MSL, symplectic algebraic bordism MSp, framed algebraic bordism, and to investigate the interactions between these spectra and MGL. The supposed techniques involve spectral sequences based on connected and effective covers of Witt theory and hermitian K-theory as well as explicit geometric constructions. We also plan to apply cohomology theories related to algebraic bordism spectra (algebraic Brown-Peterson cohomology, algebraic Morava K-theories, etc.) to the study of algebraic varieties, in particular, to the study of homogeneous algebraic varieties and cohomological invariants.