arXiv research

Generalizes integration map to coinvariants of bounded functions.

The paper constructs Morse complexes for orbifolds and shows their homologies are orbifold invariants.

Let $M$ be a smooth manifold and $Γ$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $ρ:Γ\rightarrow \mathrm{Diff}(M)$ from $Γ$ to the group of diffeomorphisms of $M$. For any such action we associate a cohomology $\mathrm{H}(Ω(M)_Γ)$ which we call the cohomology of $Γ$-coinvariant …

The study finds infinite geodesics on manifolds with specific homotopy group properties.

Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representat…

We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construc…

Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H_2(IA_n) as a GL_n(Z)-module. Corollaries include a version of surjective representation stability for H_2(IA_n), the vanishing of the GL_n(Z)-coinvariants of H_2(IA_n), and the vanishing of the second rational homology grou…

Researchers establish a connection between knot homology and Lie algebra actions.

Study of modular representations in homology of congruence subgroups.

FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian pro…

Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group of positive scalar curvature metrics is…

This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic analysis on symmetric spaces, and special functions are discussed. The rank one case …

The category $\mathrm{FI}_G$ was first defined and explored by Sam-Snowden. Here, we develop more of the machinery of $\mathrm{FI}_G$-modules and find numerous examples to apply it to, extending the work of Church-Ellenberg-Farb and Wilson. In particular we develop a notion of character polynomials for $\mathrm{FI}_G$-…

The fundamental group $π$ of a Kodaira fibration is, by definition, the extension of a surface group $Π_b$ by another surface group $Π_g$, i.e. \[ 1 \rightarrow Π_g \rightarrow π\rightarrow Π_b \rightarrow 1. \] Conversely, we can inquire about what conditions need to be satisfied by a group of that sort in order to be…

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tau…

New computations show symplectic groups and mapping class groups have different properties regarding torsion.

New complex structures found in quantum SU(3) manifold.

This paper confirms Singer's conjecture for rank 4 in specific generic degrees.

Study of surface defects in gauge theories leads to duality and separation of variables.