Tropically constructed Lagrangians in mirror quintic threefolds

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve. As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

Stability of Vector Bundles on Curves and Degenerations

We introduce a weaker notion of (semi)stability for vector bundles on reducible curves that does not depend on a choice of polarization and suffices for many applications of degeneration techniques. We explore the basic properties of this alternate notion of (semi)stability. In a complementary direction, we record a proof of the existence of semistable extensions of vector bundles in suitable degenerations.

Designing LC VCOs Using Capacitive Degeneration Techniques

In this paper, we present a detailed analysis of VCOs using a capacitively degenerated negative resistance cell. The negative resistance cell using capacitive degeneration has a higher maximum attainable oscillation frequency and a smaller equivalent shunt capacitance when compared to the widely used cross-coupled negative-gm cell. These properties are of particular interest for the design of high-frequency and/or wide tuning range VCOs. The negative resistance provided by a traditional capacitively degenerated negative resistance cell is lower than that provided by a cross-coupled negative-gm cell. We present an active capacitive degeneration topology that overcomes this limitation. To validate this circuit topology we use two test vehicles. The first test vehicle is a 5.3 GHz VCO designed in a 0.25 μm CMOS technology and the second test vehicle is a 20 GHz VCO designed in a 0.25 μm BiCMOS technology. Measurement and simulation results from both test vehicles effectively demonstrate the efficacy of the capacitive degeneration technique.