Energy in Yang-Mills on a Riemann Surface

Sengupta's lower bound for the Yang-Mills action on smooth connections on a bundle over a Riemann surface generalizes to the space of connections whose action is finite. In this larger space the inequality can always be saturated. The Yang-Mills critical sets correspond to critical sets of the energy action on a space of paths. This may shed light on Atiyah and Bott's conjecture concerning Morse theory for the space of connections modulo gauge transformations.Comment: 7 pages, 2 figures, Latex2e with epsfig, submitted to Journal of Mathematical Physic

Quantisation and the Hessian of Mabuchi energy

Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a K\"ahler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a function appearing in the study of balanced embeddings. E(k) is defined on the space of Hermitian endomorphisms of H^0(X, L^k), endowed with the L^2-innerproduct. We first prove that the leading order term in the asymptotic expansion of E(k) is D. We next show that if Aut(X,L) is discrete modulo scalars, then the eigenvalues and eigenspaces of E(k) converge to those of D. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature K\"ahler metric. As consequences of our results we prove that a certain estimate of Phong-Sturm is sharp and give a negative answer to a question of Donaldson. We also discuss some possible applications to the study of Calabi flow.Comment: 42 pages. Latest version is substantial revision. Main results now hold with no assumptions on spectral gaps. Applications and potential applications now included. Introduction rewritten to provide more context. To appear in Duke Mathematical Journa

Application of Thermoresponsive Polymer and Microfluidics to the Development of a Velocity-Dependent Cell-Sorting Microdevice

3rd Place, Denman Undergraduate Research ForumLow-cost velocity dependent cell sorting in 2D is a currently nonexistent technology for cancer research. The development of such a device would enable further research on the treatment of various deleterious cancers, such as Glioblastoma Multiforme (GBM), which metastasize based off the high motility of a single cell. Here we present a low-cost device capable of sorting these cells. Separation would enable development of highly specific therapeutic agents to limit cancer metastasis in patients. This device consists of microfluidics channels situated under microtextured Polydimethylsiloxane (PDMS) coated with the thermoresponsive polymer Poly(N-isopropylacrylamide) (PNIPAM). Cells are seeded on one end of the device and orient themselves parallel to the striations patterned into the PDMS; traveling further across the device over time. At a specific location (determined by velocity of target cells and time passed), low-temperature fluid can be passed through the microfluidic channel below which triggers a selective conformational change in the PNIPAM. This change shifts PNIPAM from nonpolar to polar, causing the polymer to release previously-adhered cells into solution in favor of binding to media. Establishing the PNIPAM layer capable of releasing cells while allowing them to adhere to microtextures on the PDMS involved a multi-step process. First, PDMS stamps are made of varying thickness, then they were placed in a plasma cleaner and exposed to Argon for 1,3, and 5 minutes at 30 Watts, 8-10 MHz, and ~1000microTorr. Then, samples were exposed to N-isopropylacrylamide (NIPAM) via immersion into a polymer solution and via dropping that solution onto samples and baked at 3 hours or 5 hours. Cell detachment analysis, goniometer experimentation, and SEM images showed that a 1 minute Argon gas exposure, with 1 minute of NIPAM immersion and a 3 hour bake yielded the most successful layer that lifted cells without inhibiting the PDMS microtexture. Future work involves optimizing the device to lift all cells exposed to the channel, as well as further corroborating its efficacy.A one-year embargo was granted for this item.Academic Major: Biomedical Engineerin