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Abstract We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one, and Hurwitz covers of zero residue loci in strata with projective dimension two. 

Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel–Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemann surfaces. This space carries actions of the foliated Laplacian derived from the \mathrm{SL}_{2}(\mathbb{R})-action as well as various differential operators related to relative period translations.In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum, exhibiting all Siegel–Veech transforms requires novel configurations of saddle connections. We also show that the continuous spectrum of the foliated Laplacian is much larger than the space of Siegel–Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations. 

Cell surfaces are often decorated with glycoconjugates that contain linear and more complex symmetrically and asymmetrically branched carbohydrates essential for cellular recognition and communication processes. Mannose is one of the fundamental building blocks of glycans in many biological membranes. Moreover, oligomannoses are commonly found on the surface of pathogens such as bacteria and viruses as both glycolipids and glycoproteins. However, their mechanism of action is not well understood, even though this is of great potential interest for translational medicine. Sequence-defined amphiphilic Janus glycodendrimers containing simple mono- and disaccharides that mimic glycolipids are known to self-assemble into glycodendrimersomes, which in turn resemble the surface of a cell by encoding carbohydrate activity via supramolecular multivalency. The synthetic challenge of preparing Janus glycodendrimers containing more complex linear and branched glycans has so far prevented access to more realistic cell mimics. However, the present work reports the use of an isothiocyanate-amine “click”-like reaction between isothiocyanate-containing sequence-defined amphiphilic Janus dendrimers and either linear or branched oligosaccharides containing up to six monosaccharide units attached to a hydrophobic amino-pentyl linker, a construct not expected to assemble into glycodendrimersomes. Unexpectedly, these oligoMan-containing dendrimers, which have their hydrophobic linker connected via a thiourea group to the amphiphilic part of Janus glycodendrimers, self-organize into nanoscale glycodendrimersomes. Specifically, the mannose-binding lectins that best agglutinate glycodendrimersomes are those displaying hexamannose. Lamellar “raft-like” nanomorphologies on the surface of glycodendrimersomes, self-organized from these sequence-defined glycans, endow these membrane mimics with high biological activity. 

Reconstructing the functions of living cells using nonnatural components is one of the great challenges of natural sciences. Compartmentalization, encapsulation, and surface decoration of globular assemblies, known as vesicles, represent key early steps in the reconstitution of synthetic cells. Here we report that vesicles self-assembled from amphiphilic Janus dendrimers, called dendrimersomes, encapsulate high concentrations of hydrophobic components and do so more efficiently than commercially available stealth liposomes assembled from phospholipid components. Multilayer onion-like dendrimersomes demonstrate a particularly high capacity for loading low-molecular weight compounds and even folded proteins. Coassembly of amphiphilic Janus dendrimers with metal-chelating ligands conjugated to amphiphilic Janus dendrimers generates dendrimersomes that selectively display folded proteins on their periphery in an oriented manner. A modular strategy for tethering nucleic acids to the surface of dendrimersomes is also demonstrated. These findings augment the functional capabilities of dendrimersomes to serve as versatile biological membrane mimics.