I am a mathematics PhD student at Princeton University (expected 2018) with a focus on Voevodsky's theory of motives and its connections and applications to stable homotopy theory, birational geometry, and arithmetic. Recently, I have been thinking about

• Integration of Voevodsky motives, motives of K-equivalent varieties, and connections to the existence of motivic t-structures;
• Triangulated categories of mixed motives and l-independence;
• Motivic zeta functions;
• Derived motivic measures; and
• (Stable) motivic homotopy theory and its connections to classical and etale homotopy theory.

Papers and Preprints.

1. Comparison of Stable Homotopy Categories and a Generalized Suslin-Voevodsky Theorem. Submitted. PDF
-You can find in this paper a proof that the constant sheaf functor induces, after inverting p, a fully faithful functor from the classical stable homotopy category to the motivic stable homotopy category over an algebriacally closed field of characteristic exponent p. Additionally, there is a proof of a homotopy-theoretic generalization of the etale version of the Suslin-Voevodsky comparison theorem.
2. Integration of Voevodsky motives. Preprint. PDF
-I develop a theory of integration for Voevodsky motives, and show that it circumvents some of the complications of motivic integration. Using this, I deduce new arithmetic and geometric information about K-equivalent varieties over fields. This generalizes the theorem of Kontsevich on the equality of Hodge numbers of K-equivalent complex varieties. Finally, I show that if there is a motivic t-structure with the expected properties on the category of rational geometric Voevodsky motives, then K-equivalent varieties have the same rational (Chow) motives, conditionally answering the rational version of a conjecture of Chin-Lung Wang. This basically implies that familiar cohomology groups cannot distinguish K-equivalent varieties. Another consequence is that if there is such a motivic t-structure, D-equivalent smooth projective complex varieties of general type have equivalent rational (Chow) motives, giving more evidence for a conjecture of Orlov.
3. A Universal Derived Motivic Measure. In preparation. Joint with Adeel Khan.