On the involutions fixing the class of a lattice

With any integral lattice \Lambda in n-dimensional euclidean space we associate an elementary abelian 2-group I(\lambda) whose elements represent parts of the dual lattice that are similar to \Lambda. There are corresponding involutions on modular forms for which the theta series of \Lambda is an eigenform; previous work has focused on this connection. In the present paper I(\Lambda) is considered as a quotient of some finite 2-subgroup of O_n(\R). We establish upper bounds, depending only on n, for the order of I(\Lambda), and we study the occurrence of similarities of specific types.Comment: 11 pages LaTeX. To appear in Journal of Number Theor

Lattices with theta functions for G(√2) and linear codes

AbstractModular hermitian lattices over Z[i]and, in particular, unimodular lattices over Z[eπi4] give rise to modular forms for Hecke's group G(2)=<(1201), (01−10)>.Two general constructions of such lattices are performed, using codes over F2 and F9. Lattices with an extremal theta-function (i.e., with the largest minimum that Hecke's theory allows) are obtained in C2n for all n < 12, including the densest known sphere-packings of R4n for n = 1, 4, and 8

Complete Weight Enumerators of Generalized Doubly-Even Self-Dual Codes

For any q which is a power of 2 we describe a finite subgroup of the group of invertible complex q by q matrices under which the complete weight enumerators of generalized doubly-even self-dual codes over the field with q elements are invariant. An explicit description of the invariant ring and some applications to extremality of such codes are obtained in the case q=4

A representation theorem for algebras with involution

AbstractAlgebras with involution are represented as commutants of two adjoint vector- space endomorphisms

Vector bundles of rank four and A_3 = D_3

Over a scheme with 2 invertible, we show that a vector bundle of rank four has a sub or quotient line bundle if and only if the canonical symmetric bilinear form on its exterior square has a lagrangian subspace. For this, we exploit a version of "Pascal's rule" for vector bundles that provides an explicit isomorphism between the moduli functors represented by projective homogeneous bundles for reductive group schemes of type A_3 and D_3. Under additional hypotheses on the scheme (e.g. proper over a field), we show that the existence of sub or quotient line bundles of a rank four vector bundle is equivalent to the vanishing of its Witt-theoretic Euler class.Comment: 16 pages, final version; IMRN 2012 rns14

CATCH ME IF YOU CAN: DETECTING IMMUNE-CANCER CELL INTERACTIONS FOR BETTER IMMUNOTHERAPY

The tumor and tumor microenvironment (TME) consists of multiple cells in communication, including immune, fibroblasts, vascular cells., and cancer cells. Cellular communication between immune cells and cancer cells significantly influences patient response to immunotherapy and accounts for treatment resistance and interpatient response variability. Thus, it is important to study the complex interactions occurring in the TME, through proximity dependent cell labeling methods. In this dissertation, we used the enzyme-mediated intercellular proximity labeling strategy (EXCELL) to 1) detect and visualize immune cell labeling in a time- and concentration-dependent manner in vitro and 2) detect immune cell infiltrate in vivo. Using flow cytometry and confocal microscopy, we showed EXCELLs ability to label and detect immune cells with biotin. We showed that this process is both time- and concentration- dependent with an increase in immune cell labeling over time. Furthermore, we translated EXCELL from in vitro to in vivo showcasing EXCELL’s ability to label primary murine B and T cells with biotin post interacting with cancer cells. Overall, our findings suggest that the EXCELL method has great potential for improving our understanding of immune cell dynamics within the TME, ultimately leading to more potent pharmacological effects and cancer immunotherapy strategies.Doctor of Philosoph