Counting homomorphisms from surface groups to finite groups

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in $G$, to a sum over values of irreducible characters of $G$ weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.Comment: Comments welcome

A relative version of Rochlin's theorem

Rochlin proved \cite{VR} that a closed 4-dimensional connected smooth oriented manifold $X^4$ with vanishing second Stiefel-Whitney class has signature $\sigma(X)$ divisible by 16. This was generalized by Kervaire and Milnor \cite{kervaire_milnor_spheres} to the statement that if $\xi \in H_2(X;\mathbb{Z})$ is an integral lift of an element in $H_2(X; \mathbb{Z}/2\mathbb{Z})$ that is dual to $w_2(X)$, and if $\xi$ can be represented by an embedded sphere in $X$, then the self-intersection number $\xi^2$ is divisible by 16. This was subsequently generalized further by Rochlin (see Theorem \ref{matsumoto} below) and various alternative proofs of this result where given by Freedman and Kirby \cite{freedman1978geometric}, Matsumoto \cite{matsumoto}, and Kirby \cite{kirbybook}. We give further generalizations of this result concerning 4-manifolds with boundary. Given a smooth compact orientable four manifold $X^4$ with integral homology sphere boundary and a connected orientable characteristic surface with connected boundary $F^2$ properly embedded in $X$, we prove a theorem relating the Arf invariant of $\partial F$, and the Arf invariant of $F$, and the Rochlin invariant of $\partial X$. We then proceed to generalize this result to the case where $X$ is a topological compact orientable 4-manifold (which brings in the Kirby-Siebenmann invariant), $\partial F$ is not connected (which brings in the condition of being proper as a link), $F$ is not orientable (which brings in Brown invariants), and finally where $\partial X$ is an arbitrary 3-manifold (which brings in pin structures). The final result gives a ``combinatorial'' description of the Kirby-Siebenmann invariant of a compact orientable 4-manifold with nonempty boundary.Comment: Added several generalizations of the main result. Comments welcome

Morphological Phase Diagram for Lipid Membrane Domains with Entropic Tension

Circular domains in phase-separated lipid vesicles with symmetric leaflet composition commonly exhibit three stable morphologies: flat, dimpled, and budded. However, stable dimples (i.e., partially budded domains) present a puzzle since simple elastic theories of domain shape predict that only flat and spherical budded domains are mechanically stable in the absence of spontaneous curvature. We argue that this inconsistency arises from the failure of the constant surface tension ensemble to properly account for the effect of entropic bending fluctuations. Formulating membrane elasticity within an entropic tension ensemble, wherein tension represents the free energy cost of extracting membrane area from thermal bending of the membrane, we calculate a morphological phase diagram that contains regions of mechanical stability for each of the flat, dimpled, and budded domain morphologies

Acoustic suppression of the coffee-ring effect

We study the influence of acoustic fields on the evaporative self-assembly of solute particles suspended inside sessile droplets of complex fluids. The self-assembly process often results in an undesirable ring-like heterogeneous residue, a phenomenon known as the coffee-ring effect. Here we show that this ring-like self-assembly can be controlled acoustically to form homogeneous disc-like or concentrated spot-like residues. The principle of our method lies in the formation of dynamic patterns of particles in acoustically excited droplets, which inhibits the evaporation-driven convective transport of particles towards the contact line. We elucidate the mechanisms of this pattern formation and also obtain conditions for the suppression of the coffee-ring effect. Our results provide a more general solution to suppress the coffee-ring effect without any physiochemical modification of the fluids, the particles or the surface, thus potentially useful in a broad range of industrial and analytical applications that require homogenous solute depositions

Sulfur reduction in sediments of marine and evaporite environments

Transformations of sulfur in sediments of ponds ranging in salinities from that of normal seawater to those of brines saturated with sodium chloride were examined. The chemistry of the sediment and pore waters were focused on with emphasis on the fate of sulfate reduction. The effects of increasing salinity on both forms of sulfur and microbial activity were determined. A unique set of chemical profiles and sulfate-reducing activity was found for the sediments of each of the sites examined. The quantity of organic matter in the salt pond sediments was significantly greater than that occurring in the adjacent intertidal site. The total quantitative and qualitative distribution of volatile fatty acids was also greater in the salt ponds. Volatile fatty acids increased with salinity