For a finite group G, a G-module M, and a field F, an element u in H^d(G,M) is negligible over F if for each field extension L/F and every continuous group homomorphism from Gal(L^{sep}/L) to G, u is in the kernel of the induced homomorphism H^d(G,M) to H^d(L,M). Negligible cohomology was first introduced by Serre […]

What are the possible angles between two integer vectors in R^n? If we fix one such possible angle and one integer vector x, is there always another integer vector y that makes this angle with x? Assuming that x makes a given angle with some vector, how can we find the shortest such vector y? […]

In order to understand a topological space X, it is often easier to understand X in terms of an action by a group G. When X is a compact complex manifold, we often let G be products of S^1 or \C^* acting in a nice way ("holomorphically") on X. This simplifies several calculations of an […]

Biquandle module enhancements are invariants of knots and links generalizing the classical Alexander module invariant. A quiver categorification of these invariants was introduced in 2020. In this work-in-progress (joint with Yewon Joung from Hanyang University in Seoul) we take the next step by defining invariant quiver representations. As an application we obtain new polynomial knot […]