Hermite reciprocity and Schwarzenberger bundles

Hermite reciprocity refers to a series of natural isomorphisms involving compositions of symmetric, exterior, and divided powers of the standard $SL_2$-representation. We survey several equivalent constructions of these isomorphisms, as well as their recent applications to Green's Conjecture on syzygies of canonical curves. The most geometric approach to Hermite reciprocity is based on an idea of Voisin to realize certain multilinear constructions cohomologically by working on a Hilbert scheme of points. We explain how in the case of ${\bf P}^1$ this can be reformulated in terms of cohomological properties of Schwarzenberger bundles. We then proceed to study these bundles from several perspectives: We show that their exterior powers have supernatural cohomology, arising as special cases of a construction of Eisenbud and Schreyer. We recover basic properties of secant varieties $\Sigma$ of rational normal curves (normality, Cohen-Macaulayness, rational singularities) by considering their desingularizations via Schwarzenberger bundles, and applying the Kempf-Weyman geometric technique. We show that Hermite reciprocity is equivalent to the self-duality of the unique rank one Ulrich module on the affine cone $\widehat{\Sigma}$ of some secant variety, and we explain how for a Schwarzenberger bundle of rank $k$ and degree $d\ge k$, Hermite reciprocity can be viewed as the unique (up to scaling) non-zero section of $(Sym^k\mathcal{E})(-d+k-1)$.Comment: 26 page

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

In-vivo Single Cell Protein Interaction Investigation using Microfluidic Platform

We have developed a microfluidic platform to immobilize single living cells at a position in a microfluidic channel for monitoring cell responses to different stimuli over time. While the cell is trapped, the device maintains control over changing the channel media and flow rate. The captured cell may be released in order to capture another cell and repeat the experiment without disturbing the setup. We have demonstrated the monitoring of an individual yeast cell, in which the captured cell is exposed to α-factor ligand

Male infertility: role of genetic background

Male infertility represents one of the clearest examples of a complex disease with a substantial genetic basis. Numerous male mouse models, mutation screening and association studies reported over the last few years reveal the high prevalence of genetic causes of spermatogenic impairment, accounting for 10-15% of severe male infertility, including chromosomal aberrations and single gene mutations. Natural selection prevents the transmission of mutations causing infertility, but this protective mechanism may be overcome by assisted reproduction techniques. Consequently, the identification of genetic factors is important for appropriate management of the infertile couple. However, a large proportion of infertile males are diagnosed as idiopathic, reflecting poor understanding of the basic mechanisms regulating spermatogenesis and sperm function. Furthermore, the molecular mechanisms underlying spermatogenic damage in cases of genetic infertility (for example Yq microdeletions) are not known. These problems can be addressed only by large scale association studies and testicular or spermatozoal expression studies in well-defined alterations of spermatogenesis. It is conceivable that these studies will have important diagnostic and therapeutic implications in the future. This review discusses the genetic causes of male infertility known to date, the genetic polymorphisms possibly associated with male infertility, and reports novel results of global gene expression profiling of normal human testis by microarray technology

The muscarinic M3acetylcholine receptor exists as two differently sized complexes at the plasma membrane

The literature on GPCR (G-protein-coupled receptor) homo-oligomerization encompasses conflicting views that range from interpretations that GPCRs must be monomeric, through comparatively newer proposals that they exist as dimers or higher-order oligomers, to suggestions that such quaternary structures are rather ephemeral or merely accidental and may serve no functional purpose. In the present study we use a novel method of FRET (Förster resonance energy transfer) spectrometry and controlled expression of energy donor-tagged species to show that M3Rs (muscarinic M3 acetylcholine receptors) at the plasma membrane exist as stable dimeric complexes, a large fraction of which interact dynamically to form tetramers without the presence of trimers, pentamers, hexamers etc. That M3R dimeric units interact dynamically was also supported by co-immunoprecipitation of receptors synthesized at distinct times. On the basis of all these findings, we propose a conceptual framework that may reconcile the conflicting views on the quaternary structure of GPCRs