$N_\infty$-operads over a group $G$ encode homotopy commutative operations together with a class of equivariant transfer (or norm) maps. Their homotopy theory is given by transfer systems, which are certain discrete objects that have a rich combinatorial structure defined in terms of the subgroup lattice of $G$. In this talk, we will show that when $G$ is finite Abelian, transfer systems are in bijection with weak factorization systems on the poset category of subgroups of $G$. This leads to an involution on the lattice of transfer systems, generalizing the work of Balchin-Bearup-Pech-Roitzheim for cyclic groups of squarefree order. We will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg-Hill saturation conjecture. This is joint work with Evan Franchere, Usman Hafeez, Peter Marcus, Kyle Ormsby, Weihang Qin, and Riley Waugh.

The talk explores an interplay between three concepts from different areas of algebra and geometry: vector bundles from geometry and topology, valuations from commutative algebra and piecewise linear functions from convex geometry. A "vector bundle" over a geometric space X (such as a manifold) is, roughly speaking, an assignment of vector spaces to each point in X. Vector bundles are a central object of study in geometry and topology. We introduce the notion of a valuation with values in piecewise linear functions and see that these are the right gadgets to classify (equivariant) vector bundles on so-called "toric varieties". Examples include classification of all (equivariant) vector bundles on a projective space. This can be regarded as a reformulation of Klyachko's famous classification of toric vector bundles. This point of view leads to far reaching extensions which I will touch on if there is time. This is joint work with Chris Manon.

Nori proved in 2002 that given a complex algebraic variety $X$, the bounded derived category of the abelian category of constructible sheaves on $X$ is equivalent to the usual triangulated category $D(X)$ of bounded constructible complexes on $X$. He moreover showed that given any constructible sheaf ${\mathcal F}$ on ${\mathbb A}^n$, there is an injection ${\mathcal F}\hookrightarrow {\mathcal G}$ with ${\mathcal G}$ constructible and ${\rm H}^i({\mathbb A}^n, {\mathcal G})=0$ for $i>0$.

In this talk, I'll discuss how to extend Nori's theorem to the case of a variety over an algebraically closed field of positive characteristic, with Betti constructible sheaves replaced by $\ell$-adic sheaves. This is the case $p=0$ of the general problem which asks whether the bounded derived category of $p$-perverse sheaves is equivalent to $D(X)$,resolved affirmatively for the middle perversity by Beilinson.