Browsing by Author "Saper, Leslie D"

$L^2$-cohomology is a cohomology theory on Riemannian manifolds. It agrees with de Rham cohomology in the compact case, but is often different in the non-compact case. In some informative examples of stratified spaces, $L^2$-cohomology of the regular part often agrees with the middle-perversity intersection homology of the stratified space, giving a de Rham-style theorem. Unfortunately, this doesn't always happen; we give a family of examples of real algebraic varieties for which the $L^2$-cohomology and middle-perversity intersection homology are not equal.\\

In both this family and in the case of complex singularities, it often happens that we can decompose the space into regions where the metric looks like that of a multiply-warped product, or like interpolations between such regions. An illuminating class of examples is that of normal complex surface singularities. In this case, the decomposition was begun by Hsiang and Pati and completed by Nagase, and this decomposition played a heavy role in the computation of $L^2$-cohomology.\\

Cheeger, Goresky, and MacPherson conjectured that the intersection cohomology of complex projective varieties and the $L^2$-cohomology of their regular part are isomorphic. One hope at the time of the conjecture would be that the proof would shed light on the local structure of complex algebraic singularities. If one instead looks at the local $L^2$-cohomology and ask that it is isomorphic to the local intersection homology, then the conjecture does imply restrictions on the cohomology which is only apparent after a closer look at the local geometry around the singularity.\\

In this thesis, we calculate the local $L^2$-cohomology for several examples of affine real and complex algebraic varieties with isolated singularities with the metric induced with the Euclidean metric. We give examples of real algebraic varieties where the local $L^2$-cohomology is not isomorphic to the middle intersection homology. We give another example where the local $L^2$-cohomology is not even a subspace of the cohomology of the link. We also calculate the local $L^2$-cohomology for a class of weighted homogeneous hypersurfaces; this class of examples includes the $A_k$-singularities in arbitrary dimension. \\

My research centers on the cohomology of arithmetic varieties. More specifically, I am interested in applying analytical, as well as topological methods to gain better insight into the cohomology of certain locally symmetric spaces. An area of research where the intersection of these analytical and algebraic tools has historically been very effective, is the classical theory of modular symbols associated to cusp forms. In this context, my research can be seen as developing a framework in which to compute modular symbols in higher rank.

An important tool in my research is the well-rounded retract for GL_n . In particular, in order to study the cohomology of the locally symmetric space associated to GL₃ more effectively I designed an explicit, combinatorial contraction of the well-rounded retract. When combined with the suitable cell-generating procedure, this contraction yields new results pertinent to the notion of modular symbol I am researching in my thesis.

A well known action on the space of Modular forms is done by Hecke operators, which also play an important role in modularity. This action can be further break down into steps, and form what we call a Hecke correspondence. On the other hand, through Belyi and Grothendieck there is a one to one correspondence between the equivalence classes of algebraic curves defined over $\Q$ equipped with Belyi functions and equivalence classes of dessins d'enfants. This applies in particular to modular curves. In my dissertation work, I will study the action of Hecke operators on certain dessins, namely, those that correspond to $X_0(N)$. This is done by defining a cohomology with coefficients in a local system on dessins, and have Hecke operators act on it. We will also construct a Hecke-equivalent isomorphism of the cohomology group with the space of cusp forms. We hope that this work can present the first step in studying the Hecke action on more general dessins.