The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.
This paper covers three main topics. The first is addressing the question of interpolating between disparate index theorems on noncommutative two-tori. The second is to compute Hochschild cohomology for quantum special linear and special unitary groups. The third is producing an orthonormal basis for the vector space of matrix corepresentations...
The homotopy groups of bo^tmf are shown to be isomorphic to the homotopy groups of a wedge of suspensions of spectra related to integral Brown-Gitler spectra. We will then restate Mahowald's proof of the topological splitting of bo^bo and subsequently apply similar techniques to construct a map that realizing the...
This dissertation addresses the structure of the group of interval exchange transformations. The two primary topics considered are:
a) the classification of interval exchange actions for certain groups; and b) properties of the interval exchange group which are reflected in the dynamics of interval exchange maps.
In Chapter 3 a...
Let X be a quasi-projective complex variety. It follows from the work of Voevodsky that the motivic cohomology of X, denoted as $H^{p,q}(X)$ where q and p are integers with q nonnegative, can be represented in the triangulated category of motives over the field of complex numbers, denoted as $DM^{eff,-}_{Nis}$....
The Satake category is the category of perverse sheaves on the affine Grassmannian of a complex reductive group G. The global cohomology functor induces a tensor equivalence between the Satake category and the category of finite-dimensional representations of the split form of the Langlands dual group of G. We give...
Homotopy Gerstenhaber structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a B-infinity algebra by an associative algebra. Actions of B-infinity algebras on associative and B-infinity algebras are analyzed, extensions of B-infinity algebras by associative...
We prove the L<SUP>2</SUP>-convergence of polynomial ergodic averages of multiple commuting transformations for totally ergodic systems. We show that for each set of polynomials, each average is controlled by a particular characteristic factor introduced by Host and Kra, which is an inverse limit of nilsystems. We then investigate for which...
This thesis is devoted to the study of A-branes on symplectic tori and to the Mirror Symmetry conjecture. Using a method called Seidel's mirror map, we are able to reconstruct the homogeneous coordinate ring of a complex abelian variety using Lagrangian intersection theory on the mirror symplectic torus. Moreover, we...
We compute Lawson homology groups and semi-topological K-theory for certain "degenerate" varieties. "Degenerate" varieties are those smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. Our main method of study makes use of a technique of Bloch and...