I study applications of homological algebra to commutative rings. My dissertation was about numerical homological invariants of graded algebras called deviations. My primary current research project is about Koszul algebras and the homotopy Lie algebra. Lately I’ve been learning about Koszul duality from the perspective of operads.
Recent Publications
- A comparison of dg algebra resolutions with prime residual characteristic (with Josh Pollitz)
- Two types of dg algebra resolutions (the acyclic closure and minimal model) are vital computational tools in homological algebra. In characteristic zero they coincide, but they are radically different otherwise. Under some additional assumptions, we establish a comparison between these two resolutions, which allows us determine results about the structure and growth of the homotopy Lie algebra of a ring homomorphism.
- Axial constants and sectional regularity of homogeneous ideals (with Polymath)
- We introduce sectional regularity of a homogeneous ideal, which is a measurement of the regularity of the ideal’s generic sections with respect to certain linear spaces of various dimension. We relate this invariant to other studied invariants, such as axial constants, and show that axial constants and sectional regularity of the family of powers of an ideal grow linearly.
- Fractal Behavior of the Fibonomial Triangle Modulo Prime p, Where the Rank of Apparition of p is p+1 (with E. Kryuchkova)
- We establish that the fibonomial triangle, a generalization of Pascal’s triangle using Fibonacci numbers in place of consecutive integers in factorials, forms a fractal when taken modulo members of an infinite set of primes.
Papers in Progress
- Off Diagonal Deviations, Rigidity, and the Koszul Property
- We show that vanishing of off-diagonal deviations in homological degrees three and higher imply that a ring is a quotient of a Koszul algebra by a regular sequence. We show that eventual vanishing of off-diagonal deviations imply that all odd off-diagonal deviations must vanish, and explore implications for other numerical invariants, such as the slope of an algebra.
- Growth of Shifts of Deviations and the Radical of the Homotopy Lie Algebra